%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Progress in a Many-Minds Interpretation of Quantum Theory. %%% quant-ph/9904001, April 1999 %%% %%% %%% Plain TeX, 63 pages %%% %%% Matthew J. Donald %%% %%% web site: http://people.bss.phy.cam.ac.uk/~mjd1014 %%% %%% e-mail : mjd1014@cam.ac.uk %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\count17=0 %%to use pdfTeX comment out this line and uncomment the next \count17=1 \pdfoutput=\count17 %%to use plain TeX comment out this line and %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uncomment the previous %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ifnum\count17=1 \def\cmykBlue{1 1 0 0} \def\cmykBlack{0 0 0 1} \def\Blue{\pdfsetcolor{\cmykBlue}} \def\Black{\pdfsetcolor{\cmykBlack}} \def\pdfsetcolor#1{\pdfliteral{#1 k}} \def\setcolor#1{\mark{#1}\pdfsetcolor{#1}} \def\maincolor{\cmykBlack} \pdfsetcolor{\maincolor} \pdfinfo { /Title (Progress in a Many-Minds Interpretation of Quantum Theory) /Author (Matthew J. Donald) /CreationDate (April 1999) /ModDate (\number\year/\number\month/\number\day) /Subject (A Many-Minds Interpretation of Quantum Theory) /Keywords (quantum theory,many minds,philosophy of physics)} \pdfcompresslevel=9 \fi \magnification=1200 \hsize=13cm \headline={\hfil} \footline={\ifnum\count0 = 1 \hfil \else\hss\tenrm\folio\hss \fi} \topskip10pt plus30pt \def\<{{<}} \def\>{{>}} \def\dsize{\displaystyle} \def\text{\hbox} \def\newline{\hfil\break} \def\tr{\mathop{\rm tr}} \def\ent#1#2#3{{\rm ent}_{#1}(#2\,|\,#3)} \def\app#1#2#3{{\rm app}_{#1}(#2\,|\,#3)} \abovedisplayskip=3pt plus 1pt minus 1pt \belowdisplayskip=3pt plus 1pt minus 1pt \def\proclaim#1#2{\medskip\noindent{\bf #1}\quad \begingroup #2} \def\endproclaim{\endgroup\medskip} \def\A{{\cal A}} \def\B{{\cal B}} \def\H{{\cal H}} \def\sect{\S} \def\F{{\cal F}} \def\SM{{\cal SM}} \def\V{{\cal V}} \def\implies{\Rightarrow} \font\Bbb =msbm10 \def\Minkowski{{\hbox{\Bbb M}}} \def\blacksquare{\vrule height4pt width3pt depth2pt} \def\hcrh{\hfill \cr \hfill} \def\crh{\cr \hfill} \def\hcr{\hfill \cr} \font\smc =cmcsc10 \font\brm=cmbx12 \def\til{\lower 1.1 ex\hbox{\brm \char'176}} \ifnum\count17=1 \def\link#1#2{\leavevmode\pdfstartlink attr{/Border [0 0 0]} goto name{#1}\setcolor\cmykBlue #2\pdfendlink\setcolor\cmykBlack} \def\outlink#1#2{\leavevmode \pdfstartlink attr{/Border [0 0 0]} user{/Subtype /Link /A << /S /URI /URI (#1) >>} \setcolor\cmykBlue #2\pdfendlink\setcolor\cmykBlack} \def\name#1{\pdfdest name{#1} xyz} \def\pdfproclaim#1#2#3{\medskip\name{#3}\noindent{\bf #1}\quad \begingroup #2} \def\pdfeject{\eject} \else \def\link#1#2{{#2}} \def\outlink#1#2{{#2}} \def\name#1{} \def\pdfproclaim#1#2#3{\proclaim{#1}{#2}} \def\pdfeject{} \fi \vbox to 2cm{} \centerline{\bf Progress in a Many-Minds Interpretation of Quantum Theory.} \vbox to 1.5cm{} \centerline{\bf Matthew J. Donald} \vbox to 1.5cm{} {\bf \hfill The Cavendish Laboratory, JJ Thomson Avenue, \hfill Cambridge CB3 0HE, Great Britain.} \vbox to 0.75cm{} {\bf \hfill e-mail:\quad mjd1014@cam.ac.uk} \bigskip {\bf \hfill web site:\quad \catcode`\~=12 \outlink{http://people.bss.phy.cam.ac.uk/~mjd1014} {http://people.bss.phy.cam.ac.uk/\til mjd1014}} \vbox to 2cm{} \proclaim{abstract}{} In a series of papers, a many-minds interpretation of quantum theory has been developed. The aim in these papers is to present an explicit mathematical formalism which constitutes a complete theory compatible with relativistic quantum field theory. In this paper, which could also serve as an introduction to the earlier papers, three issues are discussed. First, a significant, but fairly straightforward, revision in some of the technical details is proposed. This is used as an opportunity to introduce the formalism. Then the probabilistic structure of the theory is revised, and it is proposed that the experience of an individual observer can be modelled as the experience of observing a particular, identified, discrete stochastic process. Finally, it is argued that the formalism can be modified to give a physics in which no constants are required. Instead, ``constants'' have to be determined by observation, and are fixed only to the extent to which they have been observed. \endproclaim \vbox to 1cm{} {\bf \hfill April 1999 \hfill \hfill quant-ph/9904001} \vfill \eject \centerline{\smc CONTENTS} \smallskip {\bf \parindent=54pt \item{1.\quad} \link{sec 1}{Introduction.} \item{2.\quad} \link{sec 2}{An abstract model of a finite information-processing structure.} \item{3.\quad} \link{sec 3}{Constraints and temporal development for switching structures.} \item{4.\quad} \link{sec 4}{Manifestations of switching structures in spacetime.} \item{5.\quad} \link{sec 5}{Life is like a game of chance.} \item{6.\quad} \link{sec 6}{Observing quantum probabilities.} \item{7.\quad} \link{sec 7}{Elementary models.} \item{8.\quad} \link{sec 8}{A model for private probabilities.} \item{9.\quad} \link{sec 9}{Physics without any physical constants.} \item{10.\quad} \link{sec 10}{Conclusion.} \item{Appendix} \link{hypothesis}{A Hypothesis.} \item{} \link{Ref}{References.}} \bigskip \pdfproclaim{1. Introduction.}{}{sec 1} \endproclaim Following von Neumann, the conventional formalism for quantum mechanics postulates that two quite distinct processes are involved in the time evolution of the wave-function of a quantum system. Process I -- ``collapse of the wave-packet'' -- is an abrupt change which supposedly occurs as a result of measurements. With appropriate probabilities, it replaces the wave-function by one of the eigenfunctions of some operator which is being measured. Process II, on the other hand, which is supposed to govern the system at all other times, is deterministic evolution under some version of the Schr\"odinger equation. However, no satisfactory account of process I has ever been given. In particular, it has never made been clear how a ``measurement'' should be distinguished from any other physical process, nor, given a measurement, how the ``operator being measured'' is to be defined, and nor how the ``abrupt'' change is to be made to conform with special relativity. Everett, therefore, made the brilliant suggestion that process I simply did not occur. Instead, he argued that process II was sufficient, and that the apparent occurrence of process I was entirely due to the way in which process II inevitably affected the physical structure of observers. Once again however, a detailed account was required. In particular, what is the physical structure of an observer? For many years now, I have been attempting to answer this question with the aim of providing a firm technical foundation for Everett's ideas. There should be no doubt that such a foundation is required, even in the context of decoherence theory -- the latest and most sophisticated version of the lesson that, as far as correlations to quasi-classical states of a given macroscopic object are concerned, an entirely quantum universe will, under normal circumstances, appear, for all practical purposes, to behave classically at the macroscopic level (\link{GJKKSZ}{Giulini et al.~(1996)}). When that lesson is absorbed, the problems remain: firstly to characterize unambiguously structures within such a universe which could provide a foundation for the correlations which we individually actually experience; and secondly to describe how those structures themselves can develop with time. The details of my work have so far appeared in three papers (\link{qtab}{Donald (1990, 1992, 1995)}). In \link{psoo}{Donald (1995)} in particular, I proposed a ``mathematical characterization of the physical structure of an observer''. This characterization forms part of a hypothesis, the current version of which is presented in the appendix to this paper. The hypothesis consists of a completely explicit set of definitions for the structure of an observer and for the set of possible future extensions of such a structure, together with definitions for the a priori probability of existence of an observer and for the probability of moving from a given structure to a given extension. In \link{qtab}{Donald (1990)}, with a detailed neurophysiological analysis, I began the work of arguing that humans do have the proposed type of structure and that the only entities which can with significant a priori probability possess such structure at the human level of complexity are those which we would be prepared to believe might be physical manifestations of consciousness. In \link{appalo}{Donald (1992)}, I gave an axiomatic definition of a priori probabilities for localized observers in quantum theory, in such a way as to be compatible with special relativity and quantum field theory. As a whole, the hypothesis constitutes a complete interpretation of quantum theory. According to this interpretation, an observer has a particular type of physical structure and the probabilities of changes in that structure (corresponding to new experiences) are as defined. Our observations of the physical world are explained if we have the type of physical structure suggested and if the suggested probabilities are empirically justified. If arguments show that these goals are not achieved, then it may be possible to save the interpretation by improvements in the hypothesis. In this paper, I shall present several such improvements. Starting in section \link{sec 2}{2}, I shall review the hypothesis, step by step, and in sections \link{sec 3}{3} and \link{sec 4}{4}, I shall explain the changes which I feel need to be made in the non-probabilistic parts of the hypothesis. Some elementary models are presented in section \link{sec 7}{7} in order to elucidate certain aspects of the interpretation. Although this paper could be read independently, much of the hypothesis is unchanged from \link{psoo}{Donald (1995)}, and explanation of some of the details is to be found there. Wherever possible, the parts of the hypothesis in this paper carry the same labels as the corresponding parts of the hypothesis in \link{psoo}{Donald (1995)}. The second purpose of this paper is to revise the probabilistic structure of the theory. This is a more interesting and fundamental revision as the nature of probability in many-minds interpretations has been a matter of some controversy (\link{LPS}{Lockwood et al.~(1996)}, \link{ommi}{Donald (1997)}). Given the framework already constructed, the solution proposed here is conceptually quite simple. In \link{appalo}{Donald (1992, 1995)}, a number was associated with each observer. Those numbers were referred to as ``a priori probabilities'', but I do not now believe that, by themselves, those numbers can form an adequate basis for a theory of observation. In section \link{sec 5}{5} of this paper therefore, I shall modify the theory in such a way that it becomes possible to model the experience of an individual observer as being equivalent to the experience of observing a particular, identified, stochastic process. In other words, life is like a game of chance. Once the stochastic process has been defined, it is necessary to show that its probabilities can be expected to be compatible with observed statistics. In section \link{sec 6}{6}, the nature of probability in a many-minds interpretation is discussed. It is then argued that, with probabilities defined by the stochastic process, a typical modern human observer will be aware of a world in which quantum theory is accepted and in which its detailed theoretical predictions are confirmed. In sections \link{sec 7}{7} and \link{sec 8}{8}, attention is turned to the observation of individual events with uncertain outcomes. After a preliminary discussion in section \link{sec 7}{7}, the full complexity of human neural processing is addressed in section \link{sec 8}{8}. It will be clear from a glance at the appendix that the proposed hypothesis is technically sophisticated. It is also speculative. Some elements are more speculative than others. In my opinion, the details are important, not merely because of the possibility that they might be correct, but because it is only by accepting the discipline of trying to construct a complete mathematical framework that one can come to understand the conceptual ramifications of the many-minds idea and test whether that idea is coherent. The overall aim is just to provide one theory which constitutes a complete mathematical framework for a many-minds interpretation and which is compatible with modern theoretical physics and, of course, with observation. The complexity of the hypothesis and its speculative nature may in themselves cast light on the nature of many-minds interpretations. Even the most speculative of the elements of the hypothesis is fiercely constrained by the other elements; by the goal of compatibility with quantum field theory, with special relativity, and with observation; and by the goal of being able to describe the existence of human observers in the framework of universal quantum theory. In my view, the difficulty of working out the details has been widely underestimated in the stream of preliminary sketches for interpretations which have gained popularity over the years (cf.~\link{BDV}{Bacciagaluppi, Donald, and Vermaas (1995)}, \link{psoo}{Dowker and Kent (1996)}). It has been my experience in the development of the present theory that apparently minor problems when pursued have usually turned into major problems. However, such problems become opportunities, when they force the refinement of the interpretation. Although it seems unlikely this process will be sufficient to lead to a theory which is unique in all its details; it can perhaps lead us further than might be expected. Except where grounds are given for modifying the details of \link{psoo}{Donald (1995)}, the many versions of the hypothesis discarded during its development to date are absent from this work. These failures and the difficulty in moving beyond them give me no doubt, not only that the present hypothesis could also be shown to be unsatisfactory, but even that unless there is some truth in the many-minds idea, then any version of it will ultimately be shown to be flawed in its detail. If the version of the many-minds idea presented here is true, then there is a law of nature which states that to any conscious entity there corresponds a physical structure characterized by the hypothesis. In other words, the hypothesis is proposed as a physical law in the same way that the standard model in particle physics might be so proposed. The only way to find a physical law is to try to describe the physical world in some particularly elegant fashion. In the case of the hypothesis, the idea is to provide a high-level abstract description of the functioning of a human brain; a description that is sufficiently abstract to be definable entirely within the mathematics of axiomatic quantum field theory. The elegance comes in trying to find the most simple such description sufficiently complete to express everything about the physical structure of the observer which is relevant to his or her mental life. Nevertheless, it should be realized that even if a physical law holds, there is no guarantee that it is sufficiently elegant to be distinguished from all alternative possibilities -- either minor variations or radical replacements. The search for an elegant description, refined by specific problems, is demonstrated in sections \link{sec 3}{3} and \link{sec 4}{4}. Here, for example, we meet a problem in individuating observers. This is solved by a mathematical translation of the fact that individual human brains are (usually) spatially separated from each other. The solution is elegant in that it fits naturally into the overall mathematical structure, but variations in the details of the solution are certainly possible. At present, the formalism is compatible with a fixed Lorentz invariant quantum field theory on Minkowski space which satisfies the Haag-Kastler-Schroer axioms. In section \link{sec 9}{9}, I shall speculate about further changes to the formalism and argue that, while remaining consistent with observation, one could allow for observation-dependent variations in the underlying quantum field theory. This suggests the possibility of a physics without physical constants and may be a preliminary step towards compatibility with more sophisticated physical theories. Eventually, I hope that some version of the present formalism might be useful in the development of axiomatic structures for theories like noncommutative geometry or quantum gravity or string theory. There are two general principles at the heart of my work which may make it useful in this context. These principles are that the formalism should define a {\sl minimal }structure for an observer and that an observer should be specified in entirely {\sl abstract }terms as a developing pattern of information. The result is that the formalism places minimal constraints upon the governing physical theory and is potentially adaptable to be compatible with the most exotic mathematical framework; requiring ultimately only the possibility of defining the structures used in minimal descriptions of individual observers. As our physical theories have become more sophisticated, it has constantly been found that in order to accomodate apparently esoteric empirical results, or even just theoretical advances, technical concepts which had once been considered fundamental have come to be seen as derived. If our idea of the world was based on the idea of a collection of particles with definite properties, then our entire philosophy may have been put at risk by the realization that particles do not have definite properties. In mathematical quantum field theory, not only the definite properties, but even the particles themselves have become derived concepts useful only in certain observational situations; including of course, scattering experiments. Wave-functions and eigenfunctions also cannot be considered as fundamental, unless at the level of the entire universe. It is thus to the advantage of the present formalism that it depends on generalizable concepts like patterns of spacetime relations, continuous paths, local geometry, and quantum states in the mathematical sense (rather than wave-functions). In terms of such high-level concepts it may be possible, (and will perhaps be required by quantum gravity), to allow even for time itself to appear only as a derived concept at the level of the observer. Allowing observation-dependent quantum field theory, as is proposed in section \link{sec 9}{9}, may be a hint of the kind of formal flexibility which may be needed for the implementation of this idea. In \link{ommi}{Donald (1997)}, the goals of this paper were announced and an introduction to ``many-minds'' interpretations was provided, together with discussions of and references to some alternative formulations. \pdfproclaim{2. An abstract model of a finite information-processing structure.}{}{sec 2} \endproclaim The first step in the interpretation is the modelling of an observer as a finite information-processing structure. This step is based on the preparatory intuition that human cognition works through patterns of neural firing developing over time. As a first approximation, a neuron can be thought of as a two-status entity: either a signal is passing along the neuron and it is ``firing'', or no signal is passing and the neuron is ``quiescent''. A human brain contains a finite number of neurons (around $10^{11}$) and the preliminary idea would be that a {\sl complete} description of everything relevant to the workings of the brain as the embodiment of a mind at a given moment, could be given by describing the pattern of neural firings in that brain over the lifetime of the person involved, up until the moment in question. There are many problems with this idea. From a physical point of view, neurons are macroscopic objects. Whether or not they only have two statuses as far as information processing is concerned, the family of quantum mechanical states which they visit over time is much too complex to express a simple binary opposition. To deal with this problem, it is necessary to focus attention not on neurons, but on substructures within neurons which do have a pair of quantum-mechanically simple statuses, directly tied to the firing or quiescent status of the neuron in they are contained. I call such substructures ``switches''. Much attention is devoted in \link{qtab}{Donald (1990, 1995)} to identifying and characterizing possible switches. An example might be a small piece of neural wall, whose electrical polarization will express the status of the encompassing neuron. It is the switch statuses which are the ``classical facts'' or the ``measured observables'' of the present interpretation. In the hypotheses of the previous papers of this series, the status of each switch was required to be ``observed'', or specified, or determined, exactly once per change of status and so, with a rather sloppy use of language, it was possible to blur the distinction between determinations of switch status and changes of switch status. The word ``switching'' was used to refer to both. However, as will be explained in section \link{sec 6}{6} of the present paper, it now appears necessary, in order to achieve a satisfactory probabilistic structure, to allow the possibility of switch status being determined more than once per change. For this reason, the distinction can no longer be blurred, and the fundamental events will now be referred to as ``determinations of switch status'' or ``determinations'' rather than as ``switchings''. It should be stressed that this is merely a linguistic amendment; no change is actually being proposed in the nature of the fundamental events. Patterns of switch status determinations will continue to be referred to as ``switching patterns'' or ``switching structures''. Many-minds interpretations are conceptually radical, so it should not be surprising that words must be used with some care. For example, a determination that a switch has status $C$ in some region is not something chosen by the observer. ``A determination'' is being used in the sense of ``a finding'' rather than ``a decision''. In fact it is very difficult entirely to avoid using language which gives an impression of an observer's future possibilities as being either, on the one hand, chosen or shaped by the observer; or, on the other hand, merely an expression of the observer's ignorance about a pre-determined unique future. Many-worlds theory rejects the latter idea because of the difficulty of constructing a theory of physical ``wave-packet collapse'' compatible, for example, with the experimental evidence for special relativity and for the violation of Bell's inequalities. The former idea should also be rejected if possible, because it makes the causal effects of an observer unphysical. The present theory is an attempt to show that both of these ideas can consistently be rejected; leaving us instead with the idea of the possible immediate futures of an observer at any moment, as being given by a well-defined set of possible extensions of his current history, each of which will occur with its own well-defined probability. Determinations are defined precisely by the hypothesis, but in essence they are the elementary facts from which the observer's world (or ``history'') is constructed. The facts are of the form that it appears to the observer as if a certain switch had a definite given status in a region of spacetime which is related in a given way to the other regions in which switch status is determined. At the neural level, a sequence of determinations of the polarization of a patch of neural membrane might be described in the form $LLHLLL$, where $L$ indicates that the electric potential inside the membrane is lower than that outside, and $H$ the opposite. This sequence would be indicative of one period of neural firing (the $H$), preceded and followed by periods of quiescence. For an abstract switch, a corresponding set of six switch determinations might be described by $CCOCCC$ with $C$ for ``closed'' and $O$ for ``open''. For the complete description of the mental processes of an observer, we shall require more geometrical information about a set of switch determinations than mere time-ordering. This brings us back to the idea of ``a pattern'' of neural firing (or more precisely ``a pattern'' of switch determinations). As will be explained below, it is essential for the understanding of probability that only a finite amount of information be required to describe an observer. The definition of a pattern which I introduce and discuss in \link{psoo}{Donald (1995)} amounts to a listing, called a ``docket'', of the spacetime relations between the spacetime sets in which switch statuses are determined. Thus a docket is a geometrical structure in spacetime defined as an equivalence class of ordered sequences $(A_i)_{i=1}^M$ of suitable spacetime sets. Two such sequences $(A_i)_{i=1}^M$ and $(B_i)_{i=1}^M$ will have the same docket if they have the same spacetime, or causal, arrangement -- in other words, if, for every pair $i$, $j$, $B_i$ is in the past of/spacelike to/in the future of $B_j$ exactly when $A_i$ is in the past of/spacelike to/in the future of $A_j$ -- and if one sequence can be continuously deformed into the other while the arrangement is essentially unaltered. With $M$ finite, only finitely many dockets are possible for $M$ sets. These ideas are sufficient to define an abstract ``pattern of switch determinations''. This provides us with part \link{A}{A} of the hypothesis in the appendix, according to which the structure of a mind is defined by an ordered sequence of $N$ switches and $M$ determinations of status, by a docket $d$ which defines the spacetime relations between the determinations, and by a function $\varphi$ which defines the switch $\varphi(m)$ which has its status determined in the $m^{th}$ determination. The status of switch $n$ is determined $K_n$ times and its $k^{th}$ determination is determination number $j_n(k)$. Taken together, $M$, $N$, $d$, and $\varphi$ define a ``minimal ordered switching structure $SO(M, N, d, \varphi)$''. In part \link{B3}{B3} of the hypothesis, irrelevant labels are removed from $SO(M, N, d, \varphi)$ to define a ``minimal switching structure $S(M, N, [d, \varphi])$''. All sorts of philosophical questions arise with the claim that such a definition is adequate as a description of everything which is relevant about the behaviour of a brain in its functioning as the embodiment of a mind. These questions are discussed at length in \link{psoo}{Donald (1995, 1997)}. The only shortcut I can offer to that philosophical discussion is to emphasize the following points: \smallskip \item{a.} Such an abstract pattern is an abstract analog of a pattern of neural firing and human cognition does seem to function through such a pattern. \smallskip \item{b.} A satisfactory theory of probabilities depends on only a {\sl finite} number of distinct future possibilities being available to any observer within a finite period. \smallskip \item{c.} In the context of an interpretation of quantum mechanics without a physical process of wave-packet collapse, there is no direct way to identify the sequence of quantum states occupied by a warm wet human brain. It is, for example, extremely difficult to see how a natural ``preferred basis'' of brain wave-functions could be identified. Because of this, we cannot make the assumption, inevitable in classical physics, that a brain is simply something given; ``out there''; existing in itself; a ready-made vessel for the mind. Thus quantum mechanics requires us to characterize the essential structure of a brain before we can identify the physical manifestations of such structures. \pdfproclaim{3. Constraints and temporal development for switching str\-uctures.}{}{sec 3} \endproclaim Even if it is accepted that for suitable $M$, $N$, $d$, and $\varphi$, the definition of \newline $S(M, N, [d, \varphi])$ gives an adequate ``description of everything which is relevant about the behaviour of a brain in its functioning as the embodiment of a mind'', it is not the case that every possible switching structure $S(M, N, [d, \varphi])$ is a description of something which we could interpret as such an embodiment. Indeed most switching structures are random and presumably meaningless. A satisfactory many-minds theory should define a priori probabilities so that the futures of human minds are not dominated by randomness. In order to do this, various constraints are imposed in the hypothesis on the physical manifestations of switching patterns. This disallows some manifestations which are not like individual functioning human brains; for example, because they are spatially disconnected. This section introduces some of these constraints. The temporal development of structures is also discussed. Pruning the apparently vast set of ``observer-like'' structures may be the most difficult part of developing a many-minds theory. There is always a temptation to abandon this task; to suggest that quantum theory should be viewed simply a theory of correlations (\link{Peacock}{Wheeler (1957)}, \link{Peacock}{Saunders (1996)}, \link{LPS}{Mermin (1998)}), and that every interaction between different physical systems is a species of observation. One drawback of this suggestion is that it provides no hope of understanding our own personal physical structures as observers. After all, the information-processing structure of the human brain appears not to depend on anything like the full complexity of the physical interactions of the atoms of which the brain is formed; we usually think of most of that complexity as mere thermal noise. Moreover, at least in relativistic quantum field theory, it is not all clear how ``physical systems'' are to be identified; even the idea of ``an electron'' is not without ambiguity. The most serious problem, however, in a proposal which at each moment, provides an infinite variety of ``observer-like'' entities, is that it is not clear that anything remains of the concept of temporal succession except for a vague idea about similarity between past and future (cf.~\link{Bell}{Bell (1976, 1981)}, \link{Butterfield}{Butterfield (1996)}). Given the problems with the idea that there are no rules constraining the definition of the physical structure of observers, we must either abandon the many-minds interpretation or propose some rules. Of course, such a proposal will only be correct, if the rules are facts about reality or physical laws, rather than mere invention. The main test of whether or not they could be factual will consist of asking whether or not they are consistent and allow us to explain belief in quantum theory. If they pass this test, then they can annex all the empirical evidence usually taken to support the (hopelessly inconsistent) conventional theory. However, as mentioned in the introduction, physical laws are also frequently required to be, in some sense, ``elegant'', or ``simple'', or even ``beautiful''. In the present context, this might mean something like ``unfussy'' or ``not subject to special conditions'', and, in these terms, it could be suggested that the constraints introduced in this section are ugly and that the hypothesis might be, at best, accurate phenomenological description rather than fundamental law. That is for the reader to decide, and to improve on if s/he can. It should be noted, however, that in compensation for these constraints, not only does one gain a possibly complete and consistent interpretation of quantum theory, but also, as we shall see in section \link{sec 9}{9}, one may be able to reduce, perhaps to zero, the vast amount of information which would be conventionally required to specify exact initial conditions. Although initial conditions and physical laws are neatly separated in conventional physics, this does not mean that they should be ignored in the total count of the complexity or fussiness of a theory. In \link{psoo}{Donald (1995)}, I mentioned the possibility of using the details of the theory with its allowance for fluctuations in switch states to construct arbitrary switching structures with a priori probability of around $(1.14)^{-M}$ and I suggested that these structures were unimportant because their a priori probability was so low. These structures were referred to as ``artificial'' because they are not reflections of patterns of causal correlations implicit in the universal quantum state. Unfortunately, I now think, that with the version of the hypothesis presented in that paper, the future of a structure $S(M, N, [d, \varphi])$ which did describe a human brain would be dominated by the large number of possible ways in which small numbers of new short-term artificial switches could arise. This brings us to the first of the new constraints to be imposed in this paper: a requirement of homogeneity between switches, which will make negligible the total probability of such occurrences. The details of this requirement will be discussed in the \link{sec 4}{next} section. Another type of possible switching structure which would satisfy the definition of $S(M, N, [d, \varphi])$ could be formed by the amalgamation of the switching structures describing several individuals and would describe the brains of all those individuals simultaneously. This would mean, for example, Alice and Bob existing not only as separate individuals: as ``Alice'' and as ``Bob'', but also as the joint individual ``Alice-n'-Bob''. Now the answer to the question ``What would it be like to be ``Alice-n'-Bob''?'' is simply that it would be like being Alice and Bob separately but in the same ``world''. At first sight, one might then think that ``Alice-n'-Bob'' would be a better description of reality than the individual-minds theory in which Alice has her world and Bob separately has his. However, inevitably one is led by this idea to try describe the entire human race by one vast joint structure and I do not believe that this can be done satisfactorily. There would be correspondingly vast ambiguities in such a structure; there would be an uncertainty of at least a twenty-fifth of a second (the light travel time to the antipodes) in the specification of an instant in time for such a structure; and the problem of whether humans also exist as individuals would arise. In order to rule out such joint structures (at least with significant probability), I therefore propose in this paper a modification of the hypothesis which will force a switching structure to be manifested by a connected set of switches. This also will be discussed in detail in the \link{sec 4}{next} section. I shall refer to the problem of the extension of a switching structure from one individual to more than one as the ``breeding problem''. Another problem relating to the existence of superfluous structures was introduced in \link{psoo}{Donald (1995)}. This was the ``trimming problem'' which arises from the possibility of taking a switching structure describing a single individual and removing a few of the switches to leave a structure which may have higher a priori probability but very little less information. The problem is to decide which of these structures should be associated with a given person. The trimming problem will be resolved in this paper by modifying the order in which structures are allowed to develop and the definition of a priori probability so that trim structures will tend to become fuller. The development of structures is governed by part \link{BB}{B} of the hypothesis, which defines what it means for one structure to be an immediate successor to another. Two different ways in which a structure can be extended are allowed by part \link{BB}{B}. Either there is a single new determination of the status of an existing switch (\link{BB}{B1}), or a new switch is introduced (\link{BB}{B2}). However, it is required by \link{A5}{A5} that, for a switch to be part of the structure, it needs to have opened and closed at least twice. Thus any new switch must have at least four determinations of status. If new switches could be added with fewer determinations, then all the constraints imposed by parts \link{C}{C} to \link{F}{F} of the hypothesis would not be expressed and an unrestricted increase in switch numbers would be possible. Because four determinations are needed in \link{BB}{B2}, the introduction of a new switch cannot be instantaneous. This means that, although in general part of the meaning of a human switching structure will include a ``psychological present moment'', successors of such a structure may involve new determinations of status localized before that moment. Otherwise, the entire structure would have to be reduced for a certain interval to a single switch. In the case of \link{BB}{B1}, such extensions back in time are also permitted. The single new determination can be made at any time during the history of the switch; subject to the later constraint (\link{C13}{C13}) which prevents a pile-up of re-determinations of status. In general, the possibility of extension back in time resolves the trimming problem by allowing growth to occur wherever it is possible. This means that status determinations of high a priori probability can become part of the switching structure before earlier determinations of low a priori probability, but that those earlier determinations can nevertheless still eventually be included. However, a fundamental arrow of time remains built into the theory both by the forward direction of switch paths (part \link{C}{C}) and by the inductive definition of a priori probability (part \link{GG}{G}). In section \link{sec 8}{8}, the existence of the ``psychological present'' will be explained as an empirical consequence of the theory. \pdfproclaim{4. Manifestations of switching structures in spacetime.}{} {sec 4} \endproclaim Once switching structures and their successors have been defined, the remaining parts of the hypothesis are concerned with the definition of probabilities. This requires the introduction of the set of ``physical manifestations'' of a structure. In parts \link{C}{C} to \link{F}{F}, individual manifestations are considered. In part \link{GG}{G}, the probability of existence of an abstract structure is defined by maximizing the probabilities of its individual manifestations. An individual manifestation, defined in \link{GG}{G1} as a pair $((\sigma_m)^M_{m=1}, W)$, can be thought of as a quantum world in which an appropriate pattern of switch behaviour exists. It is a world defined by a sequence of quantum states ($(\sigma_m)^M_{m=1}$ in part \link{F}{F}) and by a sequence of geometrical information ($W$ in part \link{C}{C}). Taken together, these describe the switches moving through spacetime and changing in status in such a way as to model the abstract structure defined in parts \link{A}{A} and \link{BB}{B}. In developing a many-minds interpretation of quantum theory, it is natural to construct such possible models of an observer's world. They correspond closely to conventional ideas about reality in quantum physics and use both of von Neumann's processes -- deterministic evolution in the definition of the Heisenberg picture states, and ``collapse'' in the sequential change from one state to another. Nevertheless, ultimately, these conventional ideas break down. Each switching structure has continuously many possible individual manifestations, each of which describes how the world might be for the observer whose experiences are constituted by the given structure. But there is no way to choose between these possibilities. Only the structure is experienced. All of the possible worlds enter into the universal quantum state in similar ways. They are all indistinguishable as far as the observer is concerned. All will be used here in the calculation of a priori probability. They are not ``real'' worlds, neither are they ``worlds'' in the Everett or DeWitt sense (\link{Butterfield}{DeWitt and Graham (1973)}); they are just mathematical constructs which mirror the unique experience of a single observer characterized by one completely-defined abstract switching structure. It is, of course, no more than a hypothesis that the total set of all such possibilities -- the equivalence classes of individual manifestations -- should be relevant to a complete interpretation of quantum theory. Whether that hypothesis is successful is for the reader to judge. Part \link{C}{C} of the hypothesis is concerned with the idea of a switch as a two-status object moving through spacetime and having determined status in specified regions. Thus, in part \link{C}{C}, for each individual manifestation, a specific sequence of sets $(A_i)_{i=1}^M$ is identified which has the docket $d$ defined in part \link{A}{A}. For $n = 1, \dots, N$, a spacetime path $x^n(t)$ is defined on a parameter interval $[0, T_n]$. This is the path along which switch $n$ moves, although it is only ``in operation'' in a subinterval $[S_n, T_n]$. The switch status is determined at parameter times $t_{nk}$ for $k = 1, \dots, K_n$. At parameter time $t$, switch $n$ occupies the spacetime region $\Lambda_n(t)$. $\Lambda_n(t)$ is a Poincar\'e transform, defined by $x^n(t)$ and by a path $L^n(t)$ in the restricted Lorentz group, of a set $\Lambda$. $\Lambda$ is the set, common to all the switches, at which comparisons are made. \link{C4}{C6} demands that the path $x^n(t)$ followed by switch $n$ and the boosts imposed on that switch by $L^n(t)$ be consistent. The requirement of connectedness which is the fundamental step in making joint switching structures like ``Alice-n'-Bob'' of negligible probability is imposed by part \link{C8}{C10} of the hypothesis. This requires that between any two spacelike separated points within regions where switch status is determined, there is a spacelike path lying entirely within the spacetime set traced out by switch paths. By itself, however, \link{C8}{C10} will not solve the breeding problem. No constraint is imposed on the shape of the set $\Lambda$. By giving $\Lambda$ very long fine hairs, it would be possible to connect arbitrarily distant switches without any significant loss of a priori probability. This possibility is ruled out by \link{C8}{C11} which requires that, at any moment, only a maximum number ($C$) of other switches can contact a given individual switch. $C$ -- the ``contact number'' -- may be the most arbitrary part of the entire hypothesis. Some value has to be given to $C$, not only for the breeding problem, but also to prevent switches piling up on top of each other. In \link{psoo}{Donald (1995)}, only the latter problem was considered, and $C$ was defined to be zero. Now, however, I propose that $C$ should be defined to be thirteen. Thirteen is one more than the maximum number of identical spheres which can make simultaneous contact with another sphere of the same size in three dimensions. Thus thirteen allows for a switch to have a maximal number of nearest neighbours, and a replacement. The fact that observers are localized is fundamental in the proposed interpretation. An observer only interacts with and has information about limited aspects of the universe. At the level of an individual geometrical manifestation $W$, these aspects are defined by a limited set $\B(W)$ of observables (bounded, but not necessarily self-adjoint operators) on which the quantum states of the switches are defined. The sets $\B(W)$ are defined by parts \link{D}{D1} and \link{D}{D2} of the hypothesis, while in part \link{D}{D3}, quantum states on general sets of bounded operators are defined. As required for a theory of localized entities, the states in \link{D}{D3} are not necessarily pure. The quantum state of a switch when it is in the spacetime region $\Lambda$ will be on the local algebra $\A(\Lambda)$ -- the algebra of all observables measurable within $\Lambda$ (\link{GJKKSZ}{Haag (1992)}). The Poincar\'e transformation $(x^n(t), L^n(t))$ is used to transform the state of a switch in $\Lambda_n(t)$ back to $\Lambda$. The two possible statuses (``open'' and ``closed'') of switch $n$ are defined by two orthogonal projections $P_n$ and $Q_n$ in $\A(\Lambda)$. $\B(W)$ is generated from transforms of the $P_n$ and $Q_n$ and from the algebras $\A(\Lambda_n(t))$ for $t \in [S_n, T_n]$. In parts \link{E}{E} and \link{F}{F} the quantum states of the switches are characterized. The first four items in part \link{F}{F} are a formal expression of the idea, introduced in \link{qtab}{Donald (1990)}, that ``A switch is something spatially localized, the quantum state of which moves between a set of open states and a set of closed states, such that every open state differs from every closed state by more than the maximum difference within any pair of open states or any pair of closed states.'' These open and closed states are the states in which the switch status is determined. They will be referred to as ``determination states''. The final item of part \link{F}{F} requires that the determination states cannot change arbitrarily between different switches. This is the homogeneity requirement referred to in the \link{sec 3}{previous} section. New switches are only allowed if they have determination states which are close to the corresponding states of some other contemporary switch in the structure. The requirement is sufficiently loose to permit some variation in determination states -- as is neccessitated, for example, by changes in neural temperature -- but it is also tight enough to disallow artificial switches. The loss of a priori probability caused by a single arbitrary collapse sufficient to mimic switching can be as low as $(1.14)^{-1}$, but only if the switch mimicked is of a very simple type. However, the most likely ways in which structures of human complexity can arise is through the causal correlations within a functioning brain in which the switches are something like patches of neural membrane. With the homogeneity requirement, the determination states of the switches are no longer independent. Artifical switches, imposed in a functioning brain, must also involve collapses to patches of neural membrane. These will be of extremely low a priori probability unless the membrane required is already in existence and has, in conventional terms, some possibility of behaving in the required fashion. In part \link{E}{E}, the identity of the switches over time is considered. As has already been mentioned, identity over time has frequently been seen as a problem in many-worlds and many-minds interpretations of quantum theory (\link{Bell}{Bell (1976, 1981)}, \link{Butterfield}{Butterfield (1996)}). Part \link{E}{E} defines the path followed by a switch through spacetime as being that path along which the local quantum state changes least. In parts \link{E}{E} and \link{F}{F}, finite sequences of quantum states are considered. This is a reflection of the fundamental problem of the interpretation of quantum mechanics that although quantum states can be found at any instant which provide apparently accurate descriptions of any physical system, compatible with all that is observed about the system at that instant, that compatibility property is not preserved over time. This is the Schr\"odinger cat problem -- a quantum state can be found to describe the cat as it is observed at the beginning of the experiment, but, by the end of the experiment, the time propagation of that state is sure to be some sort of superposition or mixture, which is neither compatible with a live cat, nor with a dead cat. Similarly, in the human brain, a switch can have a well-defined quantum state at one instant, but thermal and other fluctuations will soon result in that quantum state describing a mixture of switch statuses. The quantum state must be then replaced (or ``collapsed'') in order (loosely speaking) to purify the mixture. In order to avoid an analogue of the quantum Zeno paradox (\link{appalo}{Donald (1992)}), the number of such replacements must be limited, and it is natural to allow one replacement per determination of switch status -- ``one collapse per measurement''. Each such change of state should allow the path $x^n(t)$ of switch $n$ to change direction. However, in \link{psoo}{Donald (1995)} the path of switch $n$ was only allowed to alter in response to a determination of the status of switch $n$ itself. This was a mistake and resulted in an inadequate hypothesis, because it is possible for the observer to gain information from only a part of his brain which would affect how he sees his entire brain moving. The mistake is corrected in the hypothesis given in the appendix to this paper, by introducing in part \link{C}{C} a second sequence of parameter times on path $x^n(t)$: $(t'_{nm})_{m=m_n^i}^{m_n^f}$ such that for $m = m_n^i, \dots,m_n^f$, $t'_{nm}$ is the time at which the $m^{th}$ change of state affects switch $n$ and the local state changes from $\sigma_{m-1}$ to $\sigma_{m}$. By \link{C8}{C8}, the spacetime point $x^n(t'_{n m})$ is required to be in the closure of the causal future of at least $m$ of the points $x^{n'}(t_{n'k'})$ so that the $m^{th}$ change of state cannot occur except as an effect of at least $m$ prior status determinations. \pdfproclaim{5. Life is like a game of chance.}{}{sec 5} \endproclaim The indeterministic nature of quantum theory indicates that we are sometimes confronted with a world in which any one of several possible different futures may occur. According to the present theory, this happens practically all the time, because of the stochastic nature of neural processing and the level of detail in which it is analysed. The different futures referred to here involve different switching structures and therefore different experiences, rather than the many alternative manifestations of a single switching structure discussed in the \link{sec 4}{previous} section. In a many-minds interpretation, all such different futures have similar claim to reality. Some possibilities may be more likely than others, but we know from experience that the most likely does not always happen. We do, however, seem to see a fairly typical world. Indeed, we can only believe that we have evidence about probability if we believe the assumption that we do see such a world. Such beliefs do appear to be consistent. Why they should be true is the mystery at the heart of probability, and the more one thinks about it, the more mysterious it seems to become (\link{LPS}{Papineau (1995)}). Nevertheless, this mystery can be left unexplained. The goal for physics is to tell us what the world is like. In this paper, I propose that, for us, the world like a discrete stochastic process. However mysterious probability may ultimately be, we have no problem in understanding what it is like to observe such a process. Games of chance have surrounded us from infancy. Board game players, for example, know what it is like to wait until two dice show doubles in order to be released from some trap. Poker players know the rarity of a really good hand. We all know what it means to ``know the odds''. At the core of this proposal are three essential elements, which indeed are central to the entire hypothesis. The first is the idea, discussed further in \link{ommi}{Donald (1997)}, that the stochastic process should be discrete, which requires that at any moment the future possibilities can be enumerated. It is only because of this discreteness that the process could be simulated; that it is like a game of chance; that it is the sort of process with which we are familiar. The second essential element is that the stochastic process should be well-defined. The probabilities, in other words, are objective. They are facts which have exact numerical values, and it is the purpose of physical theory to find the physical laws by which they are defined. In this section, a possible definition will be put forward. The third essential element is that the stochastic process should be able to explain our observations. Study of this element will occupy sections \link{sec 6}{6}, \link{sec 7}{7}, and \link{sec 8}{8}. According to the present theory, everything which is relevant about the functioning of a brain in its embodiment of a mind can be encoded in a finite switching structure. In part \link{BB}{B} of the hypothesis, the immediate successors of a given structure are defined. In order to define a discrete Markov process, it is only necessary to define the probabilities for moving from a given structure to its immediate successors. This will be done in three steps, corresponding to \link{G2}{G2}, \link{G2}{G3} -- \link{G5}{G7}, and \link{G7}{G8} of the hypothesis. In \link{G2}{G3} -- \link{G5}{G7}, a definition is given for the a priori probability of a minimal switching structure $S(M, N, [d, \varphi])$, and in \link{G7}{G8} the jumping probability for moving from a given structure to a given successor is defined to be proportional to the successor's a priori probability. However, before considering these definitions, it is necessary to recall the mathematical function defined in \link{G2}{G2}, which lies at the heart of the definition of a priori probability. Here the relation between that function and the idea of decoherence will be emphasized as this will be fundamental to the analysis of observations in subsequent sections. The definition of a priori probability is developed in \link{otre}{Donald (1986, 1992, 1995)}, where it is based on the introduction of a function $\app{\B}{\sigma}{\rho}$ of a set of operators $\B$ and two quantum states (in the sense of definition \link{D}{D3}) $\sigma$ and $\rho$ on $\B$. This function is interpreted in \link{otre}{Donald (1986, 1992)}, consistent both with an axiomatic definition and with a broad range of appropriate properties, as giving ``the probability, per unit trial of the information in $\B$, of being able to mistake the state of the world on $\B$ for $\sigma$, despite the fact that it is actually $\rho$''. In other words, $\app{\B}{\sigma}{\rho}$ provides a probability for the observation on $\B$ of the appearance of a generalized ``collapse'' from local state $\rho$ to local state $\sigma$. The collapse is generalized because we are working at the level of a macroscopic but localized observer whose state will inevitably be mixed rather than pure. For such an observer, different possibilities will arise as different approximately-decoherent components of the local mixed state rather than as eigen-components of a wave-function. Perhaps the most important property of $\app{\B}{\sigma}{\rho}$ is that it generalizes the idea that the a priori probability of seeing a state $\sigma$ in a mixture of the form \name{5.1} $$\rho = p\sigma + (1-p)\sigma_d \eqno{(5.1)}$$ where $\sigma_d$ is disjoint from $\sigma$, is the coefficient $p$ of $\sigma$. Thus \name{5.2} \item{5.2)} Suppose that, on a set $\B$, $\rho = p\sigma + (1-p)\sigma_d$ for some $0 \leq p \leq 1$, in other words, that $\rho(B) = p\sigma(B) + (1-p)\sigma_d(B)$ for all $B \in \B$, and suppose that there exists a projection $Q \in \B$ such that $\sigma(Q) = 1$, $\sigma_d(Q) = 0$. Then, in \link{appalo}{Donald (1992)}, it is proved, from the definition, that \name{5.3} $$\app{\B}{\sigma}{\rho} = \rho(Q) = p. \eqno{(5.3)}$$ An approximate form of this result also holds, in the sense that if $\sigma(Q) \sim 1$ and $\sigma_d(Q) \sim 0$, then $$\app{\B}{\sigma}{\rho} \sim \rho(Q) \sim p, \eqno{(5.4)}$$ where the $\sim$ are given an explicit definition in (5.1) of \link{appalo}{Donald (1992)}. (\link{5.3}{5.3}) equates three different expressions for quantum probability: $\app{\B}{\sigma}{\rho}$; the ``expected value'' $\rho(Q)$ of the projection $Q$ in the state $\rho$; and the coefficient $p$ in (\link{5.2}{5.2}). The ``expected value'' of an appropriate projection in an appropriate state is the expression in terms of which empirical results and textbook quantum mechanics are most easily and most often calculated. However, to use this idea it is necessary to choose ``appropriate'' projections and states. It is here that one looks to ``the classical world'' or ``the macroscopic level'' or ``the measured operator''; and that, because none of these concepts can be precisely defined, conventional quantum mechanics becomes an art rather than a science. The paradox of ``Wigner's friend'' exemplifies the importance of not using ``inappropriate'' projections. The goal in this paper is to explain how ``appropriate expectations'' $\rho(Q)$ can be underpinned by precisely-defined probabilities. (\link{5.2}{5.2}) expresses an idea of local ``decoherence''. This is one of the most important themes in the foundations of quantum theory, but it is a theme which is prey to a well-known confusion. On one hand, there is the idea of adding a classical probabilistic structure to a quantum theory, so that one can speak of the (proper) mixture $p\sigma + (1-p)\sigma_d$ as being the state $\sigma$ with probability $p$ and the state $\sigma_d$ with probability $1-p$. On the other hand, there is similar mathematics which can be used to describe the restrictions of states to localized systems. It is this mathematics which is often applied through the idea that a pure state is an expression of complete information about a local system, but that when such a system has interacted with an environment, complete information is usually not locally available and the local state can be expressed as an (improper) mixture of decohering possibilities. We deal here with these improper mixtures. The aim is to express the fundamental probabilistic structure of quantum theory in terms of the mathematics of (\link{5.2}{5.2}) -- (\link{5.3}{5.4}). By relating improper decoherent mixtures to genuine probabilities of precisely specified events, the traditional confusion will be explained and the problems of the ambiguity and inexactness of such mixtures will be solved. On a technical level, there are several ways in which the idea of decoherence of quantum states might be expressed. For example, \name{5.5} \item{5.5)} With (\link{5.2}{5.2}), we might say that $\sigma$ is decoherent in $\rho$ on $\B$ and that $\sigma$ and $\sigma_d$ are mutually decoherent in $\rho$ on $\B$. In the circumstances of (\link{5.3}{5.4}), we might say that $\sigma$ and $\sigma_d$ are approximately mutually decoherent in $\rho$ on $\B$. \name{5.6} \item{5.6)} (\link{5.2}{5.2}) implies that, for all $B \in \B$ such that $QB, BQ \in \B$, $\rho(QB) = \rho(BQ) = \rho(Q) \sigma(B)$ and we might say that $Q$ is a decohering projection for $\rho$ on $\B$ which, with probability $\rho(Q) = p$, reduces $\rho$ to $\sigma$ on $\B$. As long as $p = \rho(Q) > 0$, (\link{5.2}{5.2}) also implies that there is an extension $\rho'$ of $\rho$ such that $\sigma = \dsize{Q \rho' Q \over \rho'(Q)}\Big|_\B$. \pdfeject \name{5.7} \item{5.7)} The opposite of decoherence is also important. This is the idea that a pure state is an expression of complete information about a physical system. \medskip The properties of $\mathop{\rm app}$ are such that much of this can be expressed by taking $\app{\B}{\sigma}{\rho}$ to measure (or to define) the probability of $\sigma$ as a decoherent part of $\rho$ on $\B$. As far as \link{5.7}{5.7} is concerned, the set $\B$ defines the localized observables about which information is available. The coherence of pure states is expressed by the following property, which holds for any projection $P \in \B$: \name{5.8} $$\rho(P) = 0 \text{ and } \app{\B}{\sigma}{\rho} > 0 \implies \sigma(P) = 0. \eqno{(5.8)}$$ This implies that if we have total information -- if $\B = \B(\H)$ (the set of all observables) -- and if $\rho = |\Psi\>\<\Psi|$ is pure, then $\app{\B(\H)}{\sigma}{\rho}$ is zero unless $\sigma = |\Psi\>\<\Psi|$. This has a generalization for any C$^*$-subalgebra ${\cal C}$ of $\B(\H)$: if $\rho$ and $\sigma$ are states on $\B$ with ${\cal C} \subset \B \subset \B(\H)$, and if $\rho|_{\cal C}$ (the restriction of $\rho$ to ${\cal C}$) is pure, then $\app{\B}{\sigma}{\rho}$ is zero unless $\sigma|_{\cal C} = \rho|_{\cal C}$. \medskip Now we turn to the definition for the a priori probability of a minimal switching structure $S(M, N, [d, \varphi])$, which was developed in \link{appalo}{Donald (1992, 1995)}. Corresponding to a structure $S(M, N, [d, \varphi])$, \link{B3}{B3} of the hypothesis gives us a set of ordered switching structures $SO(M, N, d', \varphi')$ and for each of these orderings, part \link{C}{C} gives us a set $GSO(M, N, d', \varphi')$ of possible geometrical structures $W$, where each $W$ is a sequence of the form $$W = (x, \Lambda, \theta, (T_n, (t_{nk})^{K_n}_{k=1}, (t'_{nm})_{m = m_n^i}^{m_n^f}, x^n(t), L^n(t), P_n, Q_n)^N_{n=1})$$ corresponding to the elements defined in part \link{C}{C}. Finally, for each geometrical structure, parts \link{E}{E} and \link{F}{F} of the hypothesis give us a set ${\cal N}(W)$ of sequences $(\sigma_m)^M_{m=1}$ of restrictions of quantum states to a set of observables $\B(W)$ defined in part \link{D}{D}. The aim now is to use the function $\app{\B}{\sigma}{\rho}$ to define the a priori probability of the sequences $(\sigma_m)^M_{m=1}$. These are sequences of quantum states, and the move from $\sigma_m$ to $\sigma_{m+1}$ can be interpreted as the appearance of a generalized collapse in precisely the sense discussed above. It is then natural to assume that successive collapses should be treated as independent events and to impose a product structure on the developing probabilities. There are, however, two problems. One is to know where to start the sequence, and the other is to know on which set of operators we have information at any given collapse. The initial quantum state, we shall denote by $\omega$. This may be thought of as the state corresponding to the ``universal wave-function'' of Everett. All a priori probabilities are defined relative to this state. As we shall see in section \link{sec 9}{9}, it is plausible to think of $\omega$ as being an essentially featureless background, which carries only the texture of the observed universe. Nevertheless, this texture has to be imposed on all the states in any sequence of collapses. For this to occur, the mathematics, explained in section 9 of \link{appalo}{Donald (1992)}, requires that the same set of operators be used at each collapse throughout any given sequence. This leads us to \link{G2}{G3} of the hypothesis in which the a priori probability of a sequence of states $(\sigma_m)^M_{m=1}$ on a set of operators $\B$ given an initial state $\omega$ is defined by the function $\app{\B}{(\sigma_m)^M_{m=1}}{\omega}$ which satisfies \name{5.9} $$\app{\B}{(\sigma_m)^M_{m=1}}{\omega} = {\textstyle\prod\limits^M_{m=1}} \app{\B}{\sigma_m}{\sigma_{m-1}} \eqno{(5.9)}$$ where $\sigma_0 = \omega$. $\app{\B}{(\sigma_m)^M_{m=1}}{\omega}$ gives us a definition for the a priori probability of individual sequences of local quantum states. In \link{G2}{G4}-- \link{G5}{G6}, the a priori probability of a geometrical structure $W$ is defined inductively by looking for the maximum a priori probability to which a sequence of elements of ${\cal N}(W)$ can approximate, given that the sequences of initial portions of those elements also approach maximal a priori probability. The inductive nature of this definition imposes a causal structure according to which the most likely states for a given determination of switch status are influenced by states earlier in sequence, but not by later states. In \link{G7}{G7}, the a priori probability of a switching structure $S(M, N, [d,\varphi])$ is defined by taking the supremum over the a priori probabilities of the elements $W \in GSO(M, N, d', \varphi')$ and over the possible labellings $SO(M, N, d', \varphi') \in S(M, N, [d, \varphi])$. \medskip Finally, in \link{G7}{G8}, the probability of moving from $S(M, N, [d, \varphi])$ to an immediate successor $S(M', N', [d', \varphi'])$ is defined to be proportional to the a priori probability of that successor. The constant of normalization in \link{G7}{G8} is significant. For any given structure $S(M, N, [d, \varphi])$, there is a finite set of possible immediate successors, denoted by $\Xi(M, N, d, \varphi)$, and each of the elements of this set has finite a priori probability. The number $$\xi = \sum \{\app{}{S(M', N', [d', \varphi'])}{\omega}: S(M', N', [d', \varphi']) \in \Xi(M, N, d, \varphi)\}$$ is thus finite. $\xi$ is the sum of the a priori probabilities of the possible immediate successors of $S(M, N, [d, \varphi])$. If $\xi$ was equal to $\app{}{S(M, N, [d, \varphi])}{\omega}$ then $S(M, N, [d, \varphi])$ could be thought of as splitting into its distinct successors and it would be natural to define the jumping probability to any one of these successors to be $$\app{}{S(M', N', [d', \varphi'])}{\omega}/\app{}{S(M, N, [d, \varphi])}{\omega}.$$ However, the analysis of a priori probability provided in \link{appalo}{Donald (1992)} gives no reason to believe that $\xi$ should equal $\app{}{S(M, N, [d, \varphi])}{\omega}$. If $\xi > 1$ then the successors more than exhaust the present and it is reasonable to choose the constant of normalization for the jumping probability to be $1/\xi$. On the other hand, if $\xi < 1$, I speculate that it is appropriate to normalize the jumping probabilities by $1/\app{}{S(M, N, [d, \varphi])}{\omega}$ and to introduce a probability of extinction equal to $1 - \xi/\app{}{S(M, N, [d, \varphi])}{\omega}$. This is speculation because there can be no direct empirical evidence. The proposed theory considers each individual observer separately, and no observer can know of, let alone report, his own extinction. Nevertheless, I do not think that a many-minds theory can be plausible without such an extinction probability. Indeed, without extinction, ``most'' switching structures would be very large and individually very improbable and we might wonder why we should observe ourselves to be comparatively small and probable. \eject \pdfproclaim{6. Observing quantum probabilities.}{}{sec 6} \endproclaim All the definitions have now been reviewed. The hypothesis defines a stochastic process on a set of entities. That process could, in principle, be simulated, using a suitable lattice quantum field theory. Given an initial structure, the probability of going to any other structure, or of hitting any set of structures, is well defined. What remains is to establish the connections between this process and our observations. The complexities of the hypothesis are needed because of the difficulty of giving a conceptually and mathematically complete formulation of the foundations of quantum mechanics. But it is also necessary to demonstrate the relationship between such a complete formulation and the empirical adequacy for all practical purposes of conventional quantum mechanics. This is a complicated issue. There should be no doubt that the hypothesis is conceptually radical. Nothing is ``real'' except the switching structures of individual observers (each considered separately), the initial condition $\omega$, the underlying quantum field theory, and the objective probabilities defined by the hypothesis. Out of these ``elements of reality'', each separate observer must construct his experiences and learn to guess at what his future may bring. This is done by the observer being aware of his structure as awareness of an ``observed world''. How this might be possible is considered in \link{psoo}{Donald (1995, 1997)}. It is a sophisticated form of the doctrine that one is aware of the external world entirely through being aware of the history of one's own brain. The ``observed world'', the world we see about us, is only a mental representation. In order to understand that representation, we make further representations of it. Much of our mental processing is involved with such representations of representations. These representations have their biological explanation in terms of their survival value; ``survival value'', of course, implying enhanced relative probability. Among the most sophisticated representations available to the modern human observer, are the theoretical representations of modern physics. These theoretical representations of his observed world are used by the observer to predict, at least statistically, his future observations. The purpose in this section is to begin to explain, within the framework of the hypothesis, why these predictions are usually successful. This may seem a rather limited goal. A more thorough analysis would involve a much more careful and philosophical discussion of the nature of awareness and of representations, as well as an investigation of more ordinary aspects of the observed world. However, although this will not be provided in this paper, it is my belief that what is provided is sufficient to indicate the essential outlines of such an analysis. Understanding of the quantum level gives a foundation for conventional ideas about the reduction of the everyday to that level. The relationship between the probabilistic predictions of conventional quantum theory and those of the hypothesis has already been discussed at some length in my previous papers; particularly in \link{appalo}{Donald (1992)}. Much of this discussion remains relevant and is compatible with the improvements presented here. Elementary quantum theory, quantum statistical mechanics, and decoherence theory produce a picture of the local quantum states of a macroscopic system as being, to a good approximation, decoherent mixtures of correlated states weighted by numbers which reflect the probabilities defined by elementary quantum theory. The central purpose of the hypothesis is to analyse the information in such an approximate decoherent mixture of correlations, and to decompose the mixture with probabilities determined by the weights. Thus, ultimately, the probabilities of elementary quantum mechanics and the probabilities defined by the hypothesis agree because they all reduce to weights in quantum mixtures as exemplified by (\link{5.3}{5.3}). Nevertheless, the introduction of an explicit stochastic process on the space of switching structures is a significant step which makes plain an incompleteness in \link{appalo}{Donald (1992, 1995)}. Those papers, I believe, provide a correct definition of ``a priori probability'' and correctly identify high probability switching structures and the states to which they are correlated. What was lacking, however, was an adequate theory for sets of observers, and the suggestion in this direction, made in section 9 of \link{psoo}{Donald (1995)}, now strikes me as insufficient and wrong. In classical probability theory, if $\Pr(a) = {2\over 3}$ and $\Pr(b) = {1\over 3}$, then in $N$ independent trials, a string of $N$ $a$'s is more likely than any other single string, while for $N$ large, most members of the set of all possible strings, by simple counting, contain about as many $b$'s as $a$'s. Probability theory applied to sets of strings tells us, however, that, what we are likely to see, at least for $N$ large, is a ``typical'' string, which will contain about $N/3$ $b$'s. More precisely, what the mathematics tells us is that for $N$ large, there is a set of strings, all of which have about $N/3$ $b$'s, which has probability close to one. Similarly in the present theory, what we are likely to see will be determined by what is ``typical'', and to make sense of this, it is not enough just to have a definition of the most likely observers, nor is it enough merely to be able to count observers. We also have to have a definition for probabilities of sets of observers. The stochastic process provides this. Nevertheless, these objective definitions are only a foundation for probabilistic reasoning which, in general, will involve both objective probabilities and probabilities as degrees of rational belief. When we ask how likely it is that we will observe some outcome for an observation, we need to take into account how much knowledge is available to us. According to classical physics, outcomes are determined, but we still say that a fair die will show three with probability ${1 \over 6}$, because we do not expect to know the exact initial conditions. The present theory is indeterministic but our ignorance often goes beyond that indeterminism. To discover exactly how likely it is that one should observe some outcome, one would have to know exactly what switching structure one started from, and define precisely the set of switching structures corresponding to the observation of the particular outcome. As pointed out in section 9 of \link{psoo}{Donald (1995)}, neither of these is achievable. Moreover, it is usually not possible to be precise about the extent of our knowledge. Objective probabilities have to be precisely definable in order to be objective; if life is like a game of chance, then the chances must be given in the rules. Rational belief, on the other hand, is by its nature a vague concept. As a result, there will always be some imprecision in the idea of ``typical'', even beyond the question of what is sufficient for a probability to be ``close to one''. But after all, we can only identify an occurrence as ``typical'' if it can occur repeatedly and in a way which is, at least to some extent, unaffected by circumstances. For example, we can claim that coins will typically land heads about as often as they land tails without needing to specify which coin is tossed, or when, or by whom, or even the currency. Similarly, when we discuss the outcomes of observations, we shall refer only quite vaguely to repeatable physical situations which humans like us might observe. This imprecision is not important. Although the definitions of the hypothesis must be exact, exactness is irrelevant when it comes to explaining how those definitions relate to the insubstantial mental construct which, according to the present theory, is the everyday world. Here, all that is needed, and all that is possible, are explanations at a practical level. In the next three sections, we shall consider the relationship between what is typical according to the hypothesis and empirical quantum probabilities. Partly because of the imprecision of the idea of ``typical'', and partly because of the complexity of the theory, it will clearly not be possible to give an explicit analysis from first principles of the properties of the manifestions of typical switching structures. Instead, we shall consider aspects of the consistency of the theory and the way in which it supports and completes the intuitions developed by decoherence theory. On the basis of arguments already given, it will be assumed that the development of the hypothesis has been successful and that a valid many-minds interpretation has been achieved. The relationship between the probabilities defined in the hypothesis and those of conventional quantum theory will then be examined. The central question in this section is why a typical modern observer should be aware of a world in which the detailed theoretical predictions of conventional quantum theory are confirmed. This will be explained using conventional quantum techniques, without a detailed analysis of single events at the neural level. Subsequently, we shall consider the probabilistic analysis of individual events; first, in section \link{sec 7}{7}, with the study of elementary models; and then, in section \link{sec 8}{8}, at a more realistic level. It will be useful to express in precise terms some aspects of the assumptions to be made: \pdfproclaim{Definition 6.1}{}{6.1} Given an initial state $\omega$ and $\varepsilon > 0$, an $\varepsilon$-manifestation of \newline $S(M, N, [d, \varphi])$ consists of a manifestation $((\sigma_m)^M_{m=1}, W)$ of $S(M, N, [d, \varphi])$, such that, for $m = 1, \dots, M$, $$\displaylines{ |\mathop{\rm app}({\cal N}(W), \B(W), m, \omega) - \app{\B(W)}{(\sigma_i)^{m}_{i=1}}{\omega}| < \varepsilon \cr \text{and } |\app{}{S(M, N, [d, \varphi])}{\omega} - \app{}{W}{\omega}| < \varepsilon. }$$ A property which, for all sufficiently small $\varepsilon > 0$, holds for every $\varepsilon$-manifestation of $S(M, N, [d, \varphi])$ will be referred to as a necessary property of the physical structure of $S(M, N, [d, \varphi])$. \endproclaim The definitions imply that there are $\varepsilon$-manifestations for any $\varepsilon > 0$. \eject \pdfproclaim{Assumption 6.2}{}{6.2} For $S(M, N, [d, \varphi])$ corresponding to a typical human observer, it is a necessary property that its manifestations $((\sigma_m)^M_{m=1}, W)$ involve quantum states $\sigma_m$ which, at least for large $m$, are quantum states describing an appropriate and persistent human brain. \endproclaim \proclaim{Assumption 6.3}{} In the circumstances of \link{6.2}{6.2}, it is also necessary that the $\sigma_m$ have high probability extensions which describe the world observed by $S(M, N, [d, \varphi])$. \endproclaim Although both assumptions use concepts which need some elucidation, essentially assumption \link{6.2}{6.2} says that the hypothesis is sufficient to characterize the physical structure of human observers, while assumption \link{6.2}{6.3} says that the observer's representation of the world of appearances is accurate. Both assumptions are fundamental for a many-minds interpretation, and arguments which can be seen as explaining and justifying them are to be found throughout this series of papers. The consistency arguments which use assumption \link{6.2}{6.2} will be based on the assumption that our own observations, of ourselves and of our society, give us an accurate picture of a ``typical human observer'' and of an ``appropriate'' brain for such an observer. That brain should be ``persistent'' in the sense that, for all sufficiently small $\varepsilon > 0$, if $S(M^s, N^s, [d^s, \varphi^s])$ is a significantly probable successor of $S(M, N, [d, \varphi])$, then the brain described by an $\varepsilon$-manifestation of $S(M, N, [d, \varphi])$ should be suitably similar to the relevant part of the history of the brain described by the $\varepsilon$-manifestations of $S(M^s, N^s, [d^s, \varphi^s])$. In referring to ``the world observed by $S(M, N, [d, \varphi])$'' in assumption \link{6.2}{6.3}, it is assumed that sufficient structure is contained within $S(M, N, [d, \varphi])$ for the corresponding observer to be able to construct a mental representation of a world. Assumption \link{6.2}{6.3} also refers to the idea of a high probability extension of a state $\sigma_m$. This was discussed at length in \link{appalo}{Donald (1992)}. For an example, suppose that the structure $S(M, N, [d, \varphi])$ corresponds to a particular person who is watching a football match. According to \link{6.2}{6.2}, manifestations $((\sigma_m)^M_{m=1}, W)$ exist for which the sequences of states $(\sigma_m)^M_{m=1}$ give an accurate description of the spectator's brain. Information about the present position of the ball will be contained both in $S(M, N, [d, \varphi])$ and in $(\sigma_m)^M_{m=1}$. This means that, if the theory is consistent, then there should be probable extensions $\rho'$ of $\sigma_M$ (the present moment state of the observer's brain) to sets of observables $\B$ which are contained in some local algebra $\A(\Lambda)$, where $\Lambda$ contains the observed ball position, such that $\rho'$ restricted to $\B$ is a possible quantum state for a football. $\rho'$ will depend on the initial state $\omega$ in the same way that, in conventional physics, our personal observations will not be sufficient to determine the exact current state of a football and for the remaining information we could look to the initial conditions for the universe. The following definition provides one way of defining suitable states $\rho'$: \pdfproclaim{Definition 6.4}{}{6.4} Given a switching structure $S(M, N, [d, \varphi])$, a set of operators $\B$, an initial state $\omega$, $\varepsilon > 0$, and an $\varepsilon$-manifestation $((\sigma_m)^M_{m=1}, W)$ of $S(M, N, [d, \varphi])$ define a state $\rho'$ on $\B$ to be an $\varepsilon$-prediction for $S(M, N, [d, \varphi])$ of the current state on $\B$, if $\rho' = \sigma_M'|_\B$ for some state $\sigma_M'$ such that there exists a sequence $(\sigma'_m)^M_{m=1}$ with $\sigma'_0 = \omega$, and $\sigma'_m|_{\B(W)} = \sigma_m|_{\B(W)}$ with $$\app{\B_1}{(\sigma'_m)^M_{m=1}}{\omega} \geq \sup\{ \app{\B_1}{(\sigma''_m)^M_{m=1}}{\omega} : \sigma''_0 = \omega, \sigma''_m|_{\B(W)} = \sigma_m|_{\B(W)}\} - \varepsilon $$ where $\B_1 = \B(W) \cup \{ B C : B \in \B, C \in {\cal C}(W)\}$. ($\B_1$ is the minimal set containing $\B$ and $\B(W)$ which also allows correlations between $\B$ and the switch projections experienced by the observer to be expressed (\sect 3 of \link{appalo}{Donald (1992)}.) A property which, for all sufficiently small $\varepsilon > 0$, holds for every $\varepsilon$-prediction will be referred to as a property predicted by $S(M, N, [d, \varphi])$ of the state on $\B$. \endproclaim Although this is a reasonable definition, it is not unmodifiable. Indeed, an alternative definition could be based on postulate eight of \link{appalo}{Donald (1992)}. However, the idea of a high probability extension is not a fundamental concept within the theory, and so does not need to be absolutely and unambiguously defined. $\B$ and $\Lambda$ in the example are clearly fuzzy, and, in general, there need not be a unique limiting state $\rho'$ for $\varepsilon \rightarrow 0$. Nevertheless, if the present theory has any validity, then, in appropriate circumstances, both suitable sequences $(\sigma_m)^M_{m=1}$ and suitable extensions $\rho'$ of $\sigma_M$ will exist ``for all practical purposes''. This will be sufficient as a basis for an explanation of ordinary observation in the framework of the full theory which will in turn make the theory plausible. Ultimately, extensions as defined by \link{6.4}{6.4}, or in any other way, are merely means of expressing the fundamental ``elements of reality'' referred to at the start of this section. Such extensions are the best possible theoretical representations of the world seen by the observer; that is, of the world of which, through his switching structure, the observer has created a mental representation. Our ordinary physical knowledge has been developed to make predictions about the world we observe. We can use that knowledge, both to choose appropriate extensions, given partial information about an observer's structure; and, given an extension, to make predictions about the observer's subsequent structure. Because we understand our own switching structures through the representations of external worlds that we construct from them, the easiest way for us to predict the likely temporal development of any switching structure is, first to construct a theoretical representation of the corresponding observed world -- in other words, an extension -- and then use that extension to predict future observations. Although this is the easiest way, it is not the fundamental way, which is, of course, that defined by the hypothesis. Agreement between the two ways will be a consequence of properties of the a priori probability function and of the fundamental quantum dynamics. In the most likely extended states, the information in a switching structure is correlated with observables external to the observer. The a priori probability function requires the quantum states of the observer to change as slowly as is possible, given the information available. No change in quantum state, in the Heisenberg picture used throughout this work, would correspond to evolution of observables under Hamiltonian dynamics. Our knowledge of conventional quantum theory allows us to model that evolution on either internal or external observables, and we can then use the correlations to relate one set to the other. Changes in information also can be correlated between internal and external observables. Statistical mechanics is one field in which it is particularly valuable to be able to claim that a definition like \link{6.4}{6.4} provides the best possible theoretical representation of the state of the observed world. For suitable sets $\B$, $\rho'$ will be the predicted state of an observed thermal system. Such states certainly need not be pure. A fairly exactly specified non-zero (von Neumann) entropy may well be a predicted property, in the sense of \link{6.4}{6.4}, for a thermal system. The observer may only have information about macroscopic parameters, and the states predicted by \link{6.4}{6.4} will then be something like the maximum entropy state given fixed values of those parameters. This means that in the present framework, it is not necessary to rely on ergodic theory in order to explain why, in suitable circumstances, statistical mechanical systems behave as if they occupied Gibbs equilibrium states. In a sense made precise by definition \link{6.4}{6.4}, the observed state {\sl is} an ``ensemble''. One of the major difficulties in the analysis of quantum probability is the variety of different notions of ``probability'' which are involved. A distinction between an ontological idea of probability as objective and an epistemological idea of probability as degree of rational belief has already been drawn. It is now useful to make two further distinctions, between public and private and theoretical and empirical probabilities. First is the theoretical concept of public (observer independent) quantum probability. Depending on the circumstances, this may be calculated using the expected value of a projection, or the square of a transition amplitude, or the coefficient of a component of the expansion of a density matrix into disjoint pure states, or an a priori probability defined by the function app of (\link{5.9}{5.9}). Sometimes, as in equation (\link{5.3}{5.3}), different calculations will give the same answer. The corresponding empirical notion can be expressed by, for example, the published long-term relative frequencies of particular types of observational results. A second theoretical notion is expressed by the probabilities defined by the stochastic process of the hypothesis. These are private in the sense that they apply to an individual observer. The corresponding relative frequencies will be experienced in the awareness of that observer. According to the present theory, public probabilities are only shadows of the fundamental private probabilities. Relations between the public notion of quantum probability and the private notion are subtle and give rise to two closely-related theses: \pdfproclaim{Thesis 6.5}{}{6.5} A typical modern human observer should be aware of a world in which quantum theory is accepted and in which its detailed theoretical predictions are confirmed. \endproclaim \proclaim{Thesis 6.6}{} Under apppropriate circumstances, there is a fairly direct agreement between the private and public probabilities. \endproclaim It is a consequence of thesis \link{6.5}{6.5} that the present theory is an interpretation of quantum theory. In section \link{sec 8}{8}, a model of the functioning of a human brain which is consistent with thesis \link{6.5}{6.6} will be expounded. In the remainder of the present section, the relation between the theses will be examined, and it will be argued that \link{6.5}{6.5} would hold even under conditions much weaker than \link{6.5}{6.6}. \link{6.5}{6.6} involves the study of precisely-specified individual experiences, but for \link{6.5}{6.5}, it is sufficient to consider only broadly-characterized events with nearly-certain ``typical'' outcomes. The model of brain functioning in section \link{sec 8}{8} separates quantum probabilities from the counting of neural events and geometries and shows that agreement between private and public probabilities follows from appropriate assumptions of indifference in the counting between possible alternative observations. The possibility of \link{6.5}{6.6} failing while \link{6.5}{6.5} succeeded might arise if these indifference assumptions failed to hold for some particular class of personal experience. For example, a significant difference between the hypothesis of this paper and the version of \link{psoo}{Donald (1995)} is that in the earlier version, the determinations of switch status were required, in the language of part \link{F}{F}, to alternate between ``open'' and ``closed''. Under this assumption, the number of structures corresponding to a high level of neural activity was raised relative to the number corresponding to a low level. This would breach the indifference assumption when an observer was presented with a pair of alternatives which would give rise to significantly different levels of activity. As an example, one might consider an observer listening to the sound of a Geiger counter and presume that more neural activity would follow clicking than silence. Because the fundamental probabilities are private, ultimately the only empirical tests of either \link{6.5}{6.5} or \link{6.5}{6.6} consist of asking yourself whether it is in accord with your own personal observations. As far as \link{6.5}{6.5} is concerned, we do (do we not?) experience a world in which it appears that satisfactory tests of the public notion of quantum probability abound, and, that so far, when ``appropriate'' calculations have been used, those tests have been passed. This means that we do have tests of \link{6.5}{6.5}. Indeed, an explanation of why the present theory predicts that typical observers will experience a world in which the details of quantum theory are publicly confirmed will allow the present theory to annex all the public empirical evidence for those details. As far as \link{6.5}{6.6} is concerned, suppose once again that it failed, and that the probability of some possible observation which involved a high level of neural activity was raised relative to an alternative involving a low level. This would lead to a situation in which a typical observer would tend to find himself seeing more of the high activity alternative than he would expect; either from quantum mechanical calculation, or from reports from other laboratories. Thus, with the Geiger counter example, he might find that his personal measurements, with enhanced probability of clicking, corresponded to radioactive lifetimes that were shorter than expected. If he performed his experiments in the company of a colleague, then, because of the most basic quantum correlations, he would see her seeing the same results, and thus he would find her agreeing with him about the puzzling discrepancy; but if he asked her to repeat the experiments in his absence, then he would be most likely to see her reporting to him that the discrepancy had disappeared. In such a case, each individual separately would typically believe that his own personal observations were somewhat unusual. The performance of quantum experiments in public would even allow a public consensus about these anomalies, at least among the audience for the experiments. Turning now to sufficient conditions for \link{6.5}{6.5}, recall that in classical theory, if we are given a source of strings and we have reason to believe that the letters are produced independently and with constant probabilities, then we can estimate those probabilities by assuming that the string we happen to have been given is typical -- or more precisely that the letters in it have typical relative frequencies. The laws of large numbers tell us that this is a consistent procedure and that the longer our test string, the better our estimates will be likely to be. Similarly, in quantum theory, we can use the assumption that suitable observations have been typical to estimate the probabilities of simple repeated individual events. The present theory will be consistent as an interpretation of quantum theory only if those estimates are likely to agree with the theoretical predictions of elementary quantum theory. Moreover, the classical laws of large numbers explain how the empirical basis for probabilistic reasoning can be reduced to the observation of properties with sharply peaked distributions, and the aim here is to show how the same reduction can be made in quantum theory. Experimental tests of probabilistic theories always depend on the laws of large numbers to produce significant and near certain answers from noisy and uninteresting random data. This is particularly important in the present theory, because, in situations in which the noisy and uninteresting random data is not directly observed, only the significant answers affect the switching structures and the set of manifestations of the observer. Then, because of the near certainty of those answers, the details of the definition of private probabilities are irrelevant. In order to be aware of a world in which quantum theory is accepted, it is not necessary to be aware of the detailed history by which the experimental evidence for quantum theory has been built up. It is only necessary to be aware of a limited range of almost inevitable facts about the world, involving the sort of summaries of experimental evidence which we could expect to learn at second-hand, either from colleagues, or from machines, or even from textbooks. For example, these facts might include the relation between the observed spectrum of hydrogen and the calculated eigenvalues of the corresponding Schr\"odinger equation. In conventional terms, any account of the observation of a hydrogen spectrum would ultimately depend on the existence of many individual atomic excitations, but in the present theory, even direct observation of the spectral lines no more provides evidence for specific excitations, than observation of a double slit interference pattern provides evidence for the passage of electrons through specific slits. When we look at an interference pattern formed by an appropriately prepared stream of electrons hitting a screen, we can almost certainly expect to see a pattern which will show the wave-like behaviour of the electron and which may be used to confirm the de Broglie relation between electron energy and wave-length. We will not however see any evidence of the order in which individual electrons have struck the screen. And because, according to the present theory, only the observations of an individual observer are authentic, with each individual considered separately; this means that, in this situation, there is no actual order in which individual electrons struck the screen. The present theory has built in to it a natural ``coarse graining'' which means that no definite existence is required beyond the direct personal experience of individual observers. An observation may have both typical and specific aspects. In many circumstances, it is permissible to focus on the typical aspects, because the classical probabilistic structure defined by the hypothesis allows us simply to aggregate the specific aspects. For example, when we look at a complicated bubble chamber photograph, we do observe specific quantum events of low a priori probability. The corpuscular behaviour that is revealed, however, will be typical of all such photographs. When we have looked at the photograph, we will have seen something which, in its precise details, was of low probability. Because those details will have affected our internal structure, this probability is ``private'', and its analysis will depend on the validity of thesis \link{6.5}{6.6}. Nevertheless, we will be part of a large set of observers, who in looking at the photograph, both see evidence for particulate structure and share a common history up to the moment of starting to examine the photograph. As a future of the common history, that set will have had high probability. Among all our predictable information about quantum theory, there is much information about the values of quantum probabilities. Indeed, as observers of quantum probabilities, we are, more often than not, in the position of the gambler's wife who sees only the nightly remorse, and not the card-by-card ups and downs. \pdfproclaim{example 6.7}{}{6.7} Let strings ${\bf s} = (s_n)_{n=1}^N$ of $a$'s and $b$'s be generated by $N$ independent and identically distributed classical trials, with $\Pr(s_n = a) = p$, $\Pr(s_n = b) = 1 - p$. The laws of large numbers tell us that the set of all strings with approximately $pN$ $a$'s has probability close to one. For example, for $\delta > 0$ and $\eta > {1\over 2}$, let $X^N_{\delta, \eta}$ be the set of strings with between $pN - \delta N^{\eta}$ and $pN + \delta N^{\eta}$ $a$'s. Then, applying Chebyshev's inequality to the random variable which gives the relative frequency of $a$ in ${\bf s}$, gives $$\Pr(X^N_{\delta, \eta}) \geq 1 - \delta^{-2} p(1-p) N^{1-2\eta}$$ and so $\Pr(X^N_{\delta, \eta}) \rightarrow 1$ as $N \rightarrow \infty$. Suppose that $\rho$ is a density matrix on a Hilbert space $\H$, that $P$ is a projection, and that $\rho(P) = p$. Let $(\H_n, \rho_n, P_n)_{n=1}^N$ be a sequence of $N$ isomorphic copies of $(\H, \rho, P)$. Set $Q_n = 1- P_n$. Let $\H^N = \otimes_{n=1}^N \H_n$ and $\rho^N = \otimes_{n=1}^N \rho_n$. $\rho^N$ provides a quantum model of the classical distribution with $s_n = a$ corresponding to $P_n$ and $s_n = b$ corresponding to $Q_n$, in the sense that the expected value in the state $\rho^N$ of any projection of the form $\otimes_{n=1}^N R_n$ where $R_n$ is either $P_n$ or $Q_n$ is the same as the probability of the corresponding string. Sums of these projections have expected values which are the same as the probabilities of the corresponding sets of strings. For example, there is a projection $P_{X^N_{\delta, \eta}}$ on $\H^N$ corresponding to the set $X^N_{\delta, \eta}$ and $\rho^N(P_{X^N_{\delta, \eta}}) \rightarrow 1$ as $N \rightarrow \infty$. The relative frequency operator $F^N$ is defined by $$F^N = \sum_{M=0}^N \textstyle{M \over N} P_{S^N_M}$$ where $P_{S^N_M}$ \vadjust{\kern1pt} is the projection corresponding to the set $S^N_M$ of strings of length $N$ with exactly $M$ $a$'s. Direct calculations, or standard results on the binomial distribution, yield $\rho^N(F^N) = p$ and $\rho^N((F^N - p)^2) = p(1-p)/N$ (cf. \link{GJKKSZ}{Hartle (1968)} and the papers by DeWitt and by Graham in \link{Butterfield}{DeWitt and Graham (1973)}). \endproclaim For an application of this example, suppose that an observer becomes aware that many independent repetitions of an appropriate experiment on identically prepared systems have been performed. If he has sufficient information about the systems used, then this awareness would mean that the existence of the experiments would be a prediction of the observer in the sense of \link{6.4}{6.4}. Thus, at the beginning of the experiment, it would be possible to construct sets $\B_n$ which could be modelled by $\B(\H_n)$, such that the state predicted by the switching structure of the observer on $\otimes_{n=1}^N \B_n$ would be a product state which could be modelled by $\rho^N$. The observer can now use example \link{6.7}{6.7} to predict that, as far as public quantum probabilities are concerned, the most likely observed proportion of experiments with result $a$ will be $p$. More precisely, at the time that the experiment is set up, the observer can predict, in the sense of \link{6.4}{6.4}, that expectations of the observables modelling $P_{X^N_{\delta, \eta}}$, or $F^N$, or $(F^N - p)^2$ will be as given by example \link{6.7}{6.7}, and, through the analysis of appropriate collapse-free quantum dynamics, he can extend such expectations to observables modelling public records of the proportion of experiments with result $a$. Suppose that subsequently the observer is told in roughly what proportion of the experiments result $a$ was observed, but that he does not observe the individual results. Then his own personal quantum states will become directly correlated to a limited range of values for $F^N$. In theory, to make a complete model of this situation, one would start from manifestations of a particular switching structure $S(M, N, [d, \varphi])$. One would then construct its successors and predict properties of the manifestations of those successors. One might do this, for example, by an explicit dynamical model of a von Neu\-mann-type measurement of $F^N$, or, perhaps, just by considering chains of correlation from sums of eigen-projections of $F^N$ (like $P_{X^N_{\delta, \eta}}$), to possible records of the relative frequency, and thence to possible neural observations of those records. A ``collapse'' process is necessarily involved in this situation, as it is not inevitable that the observed relative frequency will be close to $p$. However, the point of example \link{6.7}{6.7} is that far more weight in $\rho^N$ is attached to such relative frequencies. If the observer did observe all the individual results, then thesis \link{6.5}{6.6} would be relevant because there would be the possibility of a systematic bias in the individual observations. However, as long as he is only aware of, or influenced by, the overall proportion of different results and particularly if he is just making a simple choice between whether the relative frequency is close to $p$ or not, the large relative difference of weights in $\rho^N$ will continue to dominate in the probabilistic analysis. In fact, as will be discussed in section \link{sec 8}{8}, the nature of the human brain, and the way that it is modelled by the hypothesis, mean that $S(M, N, [d, \varphi])$ will have a huge number of relevant successors, differing in a myriad of fine details of neural processing. Nevertheless, because they are not correlated to individual results, those fine details will not systematically affect the observed proportion of $a$ results, and correlation to sums of eigen-projections of $F^N$ will ensure that, in all but a probabilistically negligible set of successors, the proportion will be close to $p$. This implies that, in this situation, the proportion observed by the ``typical'' observer will agree with the proportion predicted by elementary quantum theory. Example \link{6.7}{6.7} indicates how the hypothesis can provide a framework in which the large $N$ mathematics of the quantum laws of large numbers can be applied without either having to pass to the limit $N = \infty$ or having to use the multitude of individual systems as our conceptual basis. It is not necessary to pass to the limit $N = \infty$ to derive classical probabilities from example \link{6.7}{6.7}, because the hypothesis defines objective classical probabilities for individual finite observers. On the other hand, the elementary projections $\otimes_{n=1}^N R_n$ need not be considered as being more fundamental than composite projections like $P_{X^N_{\delta, \eta}}$. It is the individual observer who is fundamental and whose structure provides the natural coarse-graining. Given this, it becomes possible to generalize the analysis of example \link{6.7}{6.7} to a wide variety of circumstances in which records of predictable quantum probabilities are observed. It also becomes possible to apply many other schemes in which large $N$ mathematics has been used to reveal quasi-classical structures (e.g.~\link{GJKKSZ}{Hepp (1972)}, \link{LPS}{Namiki et al.~(1997)}), and to address the conceptual issues which have made those schemes problematic (\link{Bell}{Bell (1975)}, \link{GJKKSZ}{Giulini et al.~(1996},\sect 9.1 -- 9.3)). In this section, it has been argued that the hypothesis is compatible with empirical evidence for quantum theory. This means that that evidence can be used in turn as an important part of the justification for the hypothesis. This is a consistency argument which starts with the development of the hypothesis as a way of making sense of the empirical evidence. Then it is supposed that the hypothesis does describe quantum states which are states of a brain processing definite information, and that those states can be extended to describe systems external to the brain. Finally, by considering the properties that such systems could be expected to have, and how those properties will evolve and correlate with possible future states of the observer, it is argued that a typical observer will be aware of empirical evidence for quantum theory. The consequence of this is that the overall picture is consistent. Such consistency between observation and theory is what it means to have an interpretation. What remains is to consider whether the details of the hypothesis provide a plausible theory of individual observations of individual events with uncertain outcomes. \pdfproclaim{7. Elementary Models.}{}{sec 7} \endproclaim There are many interwoven aspects to the hypothesis. In this section, I shall attempt to explain some of these aspects by showing how the hypothesis develops from Everett's original picture. The consistent histories theory will also be briefly discussed, and analogies will be drawn to the present theory. Following Everett, most elementary versions of the many-minds interpretation assume that, if we wish to describe the observation of a quantum experiment, then we can split the Hilbert space $\H$ of the universe naturally into a tensor product $\H = \H_O \otimes \H_{ex}$ where $\H_O$ is a space of wave-functions for the observer and $\H_{ex}$ is a space of wave-functions for his experimental apparatus and for the world external to him. Then, it is assumed that the true state of the universe ($\omega$) is a pure state, $\omega = |\Psi\>\<\Psi|$ and that $\Psi$ has a decomposition of the form \name{7.1} $$\Psi = \sum_{r=0}^{\infty} \sqrt{p_r} \psi_r \otimes \varphi_r \eqno{(7.1)}$$ where $(\varphi_r)_{r=0}^{\infty}$ is an orthonormal set of wave-functions representing the distinct results of the experiment and the rest of the universe, while $(\psi_r)_{r=0}^{\infty}$ is a sequence of wave-functions for the observer, with $\psi_r$ representing the observer observing the result represented by $\varphi_r$. (\link{7.1}{7.1}) then models correlations between wave-functions of the observer and wave-functions of the rest of the universe by proposing that $\psi_r$ is correlated with $\varphi_r$, and models the probabilities of different observations by proposing that observation $r$ has probability $p_r$. In my earlier papers, I have argued that almost every aspect of this model is an over-simplification. Nevertheless, equation (\link{7.1}{7.1}) does introduce the intuition which powers the many-minds interpretation. The hypothesis in the appendix to this paper is no more than an attempt to solve the problems of (\link{7.1}{7.1}) while preserving the intuition. Even within the framework of (\link{7.1}{7.1}), one important forward step in the analysis of $\omega$ is possible. As observers we undoubtedly interact only with very limited aspects of the entire universe. Because of this, we need only consider restrictions of $\omega$ to sets of observables $\B(t)$ accessible to the observer at time $t$. Appeal to the existence of such sets allows us to assume that $\omega$ is a mixture of macroscopically different ``observer states''. Such an assumption will not be correct at the global level of the ``wave-function of the universe'', but, as we are assured by decoherence theory (\link{GJKKSZ}{Giulini et al.~(1996)}), it is almost surely true for appropriate local restrictions of such a wave-function. Let us therefore denote by $\B(t)$ some suitable choice of local observables. Working in the Heisenberg picture, the time dependence of the splitting of $\omega$ can then be expressed in the time dependence of $\omega|_{\B(t)}$. Part \link{D}{D} of the hypothesis defines a sophisticated version of $\B(t)$, but, even in the framework of (\link{7.1}{7.1}), we can take $\B(t) = \B(\H_O) \otimes 1$, where $\B(\H_O)$ is the set of all bounded operators on the Hilbert space $\H_O$. This choice allows us to replace (\link{7.1}{7.1}) by \name{eq7.2} $$|\Psi\>\<\Psi|\big|_{\B(\H_O) \otimes 1} = \sum_{r=0}^{\infty} p_r |\psi_r\>\<\psi_r|. \eqno{(7.2)}$$ In (\link{eq7.2}{7.2}), only the observer states are considered and correlations with the observed system are lost. This is also true in the hypothesis. With the full hypothesis applied at the human level, the idea, as discussed in the \link{sec 6}{previous} section, is that the external observed system can be ``reconstructed'' from the internal structure of the observer -- in other words, we ``observe'' (exist as) not the ``real world'', but (as) that apparent shadow of the apparently real world formed by the functioning of our brains. However it is also useful to note that, to a good approximation, correlations with the observed system can be restored in (\link{eq7.2}{7.2}) by appropriate enlargements of the set of operators considered. For example, according to various models of environmental decoherence, it would be not be unreasonable to write $\H_{ex} = \H_{sys} \otimes \H_{env}$, where $\H_{sys}$ represents wave-functions of the system under observation and $\H_{env}$ represents the rest of the universe including the environment of that system, and to replace (\link{7.1}{7.1}) by \name{7.3} $$\Psi = \sum_{r=0}^{\infty} \sqrt{p_r} \psi_r \otimes \varphi_r \otimes \chi_r \eqno{(7.3)}$$ where $(\varphi_r)_{r=0}^\infty$ (resp.~$(\chi_r)_{r=0}^\infty$) is an orthonormal set in $\H_{sys}$ (resp. $\H_{env}$) (cf.~(3.15) of \link{GJKKSZ}{Giulini et al.~(1996)}). We can now take $$\B(t) = \B(\H_O) \otimes \B(\H_{sys}) \otimes 1 = \B(\H_O \otimes \H_{sys}) \otimes 1,$$ where $\B(\H_{sys})$ is the set of all bounded operators on the Hilbert space $\H_s$ and this allows us to replace (\link{eq7.2}{7.2}) by \name{7.4} $$|\Psi\>\<\Psi|\big|_{\B(\H_O \otimes \H_{sys}) \otimes 1} = \sum_{r=0}^{\infty} p_r |\psi_r\>\<\psi_r| \otimes |\varphi_r\>\<\varphi_r|. \eqno{(7.4)}$$ In (\link{7.4}{7.4}), the correlations between wave-functions $\psi_r$ and $\varphi_r$ continue to be expressed. Nevertheless, (\link{7.4}{7.4}) is only an approximation, and there is considerable ambiguity in the choice of the space $\H_{sys}$. Similar ambiguities will arise in any analysis of correlation at the level of the full theory (e.g.~in definition \link{6.4}{6.4}). The language of (\link{5.6}{5.6}) applies in this scenario: $|\psi_r\>\<\psi_r| \otimes 1 \otimes 1$ (respectively $1\otimes |\varphi_r\>\<\varphi_r| \otimes 1$, $|\psi_r\>\<\psi_r| \otimes |\varphi_r\>\<\varphi_r| \otimes 1$) is a decohering projection for $|\Psi\>\<\Psi|$ on $\B(\H_O) \otimes 1 \otimes 1$ (resp. $1 \otimes \B(\H_{sys}) \otimes 1$, $\B(\H_O \otimes \H_{sys}) \otimes 1$, reducing $|\Psi\>\<\Psi|$ to $|\psi_r\>\<\psi_r|$ (resp. $|\varphi_r\>\<\varphi_r|$, $|\psi_r\>\<\psi_r| \otimes |\varphi_r\>\<\varphi_r|$) with, in all cases, probability $p_r$. The most fundamental problem with (\link{7.1}{7.1}) -- (\link{7.3}{7.4}) lies in the identification of ``wave-functions for the observer'' and in the indexing of such wave-functions. My proposed solution to this problem is part \link{A}{A} of the hypothesis and the idea of switching structures. Using this idea, suggests a generalization of (\link{eq7.2}{7.2}) of the form \name{7.5} $$\omega\big|_{\B(t)} = p_0 \rho\big|_{\B(t)} + \sum_{r = 1}^R p_r \sigma_{O[r]}\big|_{\B(t)} \eqno{(7.5)}$$ where $0 \leq p_r \leq 1$ for $r = 0, 1, \dots, R$, $\sum_{r = 0}^R p_r = 1$, $\rho$ is an arbitrary quantum state disjoint from the $\sigma_{O[r]}$, $\{O[1], \dots, O[R]\}$ is a set of possible observers, and $\sigma_{O[r]}$ is a quantum state of the universe in which the physical structure of the observer in question is a manifestation of $O[r]$. Parts \link{C}{C}, \link{E}{E}, and \link{F}{F} of the hypothesis are concerned with the mapping from switching structures $O[r]$ to possible states $\sigma_{O[r]}$. Assuming that the $\sigma_{O[r]}$ are disjoint in the sense that there exist projections $Q[r] \in \B(t)$ such that $\sigma_{O[r]}(Q[s]) = \delta_{r s}$ and $\rho(Q[s]) = 0$, (\link{7.5}{7.5}) and (\link{5.3}{5.3}) yield \name{7.6} $$\app{\B(t)}{\sigma_{O[r]}}{\omega} = \omega(Q[r]) = p_r. \eqno{(7.6)}$$ This ``a priori probability'' agrees with the standard probabilistic interpretation of quantum mixtures applied to equation (\link{7.5}{7.5}). The next stage in the elaboration of the hypothesis is to consider with more care the time development of an observer. This suggests a modification of (\link{7.5}{7.5}) to read \name{7.7} $$\sigma_{O[r^{m}]}\big|_{\B(m +1)} = p[r^{m}, 0] \rho_{r^{m}, 0}\big|_{\B(m+1)} + \sum_{r^{m+1} = 1}^{R^{m+1}} p[r^{m}, r^{m+1}] \sigma_{O[r^{m}, r^{m+1}]}\big|_{\B(m+1)}. \eqno{(7.7)}$$ In (\link{7.7}{7.7}), $t$ has been replaced by a discrete marker $m$ for the steps of a developing process and the state of a structure at step $m$ is split into states for its possible immediate successors at step $m +1$. (\link{7.6}{7.6}) becomes \name{7.8} $$\app{\B({m+1})}{\sigma_{O[r^{m}, r^{m+1}]}}{\sigma_{O[r^{m}]}} = \sigma_{O[r^{m}]}(Q[r^{m+1}]) = p[r^{m}, r^{m+1}]. \eqno{(7.8)}$$ For notational convenience, (\link{7.7}{7.7}) will be rewritten as \name{7.9} $$\sigma_{O[r^{m}]}\big|_{\B({m+1})} = \sum_{r^{m+1} = 0}^{R^{m+1}} p[r^{m}, r^{m+1}] \sigma_{O[r^{m}, r^{m+1}]}\big|_{\B(m+1)}. \eqno{(7.9)}$$ In (\link{7.9}{7.9}), the ``left-over'' state $\rho_{r^{m}, 0}$ has been absorbed into the main sum by setting $\sigma_{O[r^{m}, 0]} = \rho_{r^{m}, 0}$. Similar remainders will occur frequently below. When (\link{7.1}{7.1}) or (\link{eq7.2}{7.2}) is presented in elementary versions of many-minds or many-worlds interpretations, something closer to (\link{7.9}{7.9}) is often being invoked, because usually the assumption is made that there is a quantum measurement being considered and that the apparatus for that measurement already exists. These assumptions are false in a universal quantum theory without collapse. In such a theory, the universal uncollapsed quantum state $\omega$ superposes all possible situations that might occur on planets like ours by this stage in the development of the universe. (\link{7.7}{7.7}), (\link{7.8}{7.8}), and (\link{7.9}{7.9}) model only one time step but generalize immediately to a succession of steps. Thus, set $\sigma_{O[\,]} = \omega$ and, for $m = 0, 1, \dots, M-1$, consider a sequence of equations analogous to (\link{7.9}{7.9}): \name{7.10} $$\displaylines{ \sigma_{O[ r^1, \dots, r^{m-1}, r^{m}]}\big|_{\B(m+1)} \hcrh = \sum_{r^{m+1} = 0}^{R^{m+1}} p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]\, \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}\big|_{\B(m+1)}, \hfill \llap{(7.10)} }$$ where there exist projections $Q[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}] \in \B(m+1)$ such that $$\sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}(Q[r^1, \dots, r^{m-1}, r^{m}, s^{m+1}]) = \delta_{r^{m+1} s^{m+1}}. \eqno{(7.11)} $$ (\link{7.10}{7.10}) and (\link{7.10}{7.11}) model a single event in which an observer $O[ r^1, \dots, r^{m-1}, r^{m}]$, defined by his entire history, has the possibility of jumping to one of a set of possible futures $$\{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}] : r^{m+1} = 0, \dots, R^{m+1}\}$$ with corresponding probabilities $p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]$. The possibility $r^{m+1} = 0$ corresponds to extinction. (\link{7.10}{7.10}) is identical to (\link{7.9}{7.9}) except that the history of each observer has been made explicit in the notation. The next proposal however is not merely notational. Suppose that, for $m = 0, 1, \dots, M-1$, \name{7.12} $$\displaylines{ \sigma_{O[r^1, \dots, r^{m-1}, r^{m}]}\big|_{\B(M)} \hcrh = \sum_{r^{m+1} = 0}^{R^{m+1}} p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]\, \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}\big|_{\B(M)} \hfill \llap{(7.12)} }$$ where there exist projections $Q[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}] \in \B(M)$ such that $$ \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}(Q[ r^1, \dots, r^{m-1}, r^{m}, s^{m+1}]) = \delta_{r^{m+1} s^{m+1}}. \eqno{(7.13)}$$ In (\link{7.12}{7.12}) and (\link{7.12}{7.13}), the set of observables $\B(m+1)$ has been replaced by the set $\B(M)$. This is an important change, which, in a related form, is discussed at length in \link{appalo}{Donald (1992)}. All states in the sequence have to be localized to the same set of observables, if the full influence of earlier states on later states is to be expressed. Whether or not is it justifiable to assume that the states $\sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}$ are decoherent both on $\B(m+1)$ and on $\B(M)$ depends on the precise definition of these sets and is the reason why the set $\B(W)$ of \link{D}{D2} of the hypothesis is not taken to be a von Neumann algebra. In the elementary version of the many-minds interpretation, the extension from single to multiple observations is made by a replacement of (\link{7.3}{7.3}) by \name{7.14} $$\displaylines{ \Psi = \sum_{r^1=0}^{\infty}\sum_{r^2=0}^{\infty} \dots \sum_{r^M=0}^{\infty} \sqrt{p[r^1] p[r^2]\dots p[r^M]} \hcrh \psi_{[r^1, r^2, \dots , r^M]} \otimes \varphi_{r^1} \otimes \varphi_{r^2} \otimes \dots \otimes \varphi_{r^M} \otimes \chi_{[r^1, r^2, \dots , r^M]}, \hfill \llap{(7.14)} }$$ where now separate independent Hilbert spaces $\H_{sys^1}$, $\H_{sys^2}$, \dots, $\H_{sys^M}$ have been introduced for each successive observation. (\link{7.14}{7.14}) is essentially the decomposition of $\Psi$ proposed by Everett in his treatment of ``memory sequences'', together with an added result-dependent environmental decoherence term $\chi_{[r^1, r^2, \dots , r^M]}$. This additional term means that (\link{7.4}{7.4}) can be replaced by \name{7.15} $$\displaylines{ |\Psi\>\<\Psi|\big|_{\B(\H_{O} \otimes \H_{sys^1} \otimes \dots \otimes \H_{sys^M}) \otimes 1} = \sum_{r^1=0}^{\infty}\sum_{r^2=0}^{\infty} \dots \sum_{r^M=0}^{\infty} p[r^1] p[r^2]\dots p[r^M] \hcr |\psi_{[r^1, r^2, \dots , r^M]}\>\<\psi_{[r^1, r^2, \dots , r^M]}| \otimes |\varphi_{r^1}\>\<\varphi_{r^1}| \otimes |\varphi_{r^2}\>\<\varphi_{r^2}| \otimes \dots \otimes |\varphi_{r^M}\>\<\varphi_{r^M}|. \cr \hfill \llap{(7.15)} }$$ Conceptually, (\link{7.15}{7.15}) is very different from the sequence (\link{7.12}{7.12}). (\link{7.15}{7.15}) relies on a set of observables ($\B(\H_{O})$) localized to the present time with which the observer can describe his current memories of the past, while in the extension from (\link{7.10}{7.10}) to (\link{7.12}{7.12}) in the theory of this paper, it is assumed that $\B(1) \subset \B(2) \subset \dots \subset \B(M)$ and that $\B(m)$ is a set of observables which is localized in the past of the observer (cf.~part \link{D}{D} of the hypothesis). In many-minds quantum theory, nothing is definite unless it is part of the structure of an observer. Thus, in a theory based on (\link{7.15}{7.15}), an observer's past is not definite, except in as far as it can be reconstructed from his present time structure. This is one source of the problem, referred to in sections \link{sec 3}{3} and \link{sec 4}{4}, of identity over time. The culmination of the instant-to-instant approach suggested by (\link{7.15}{7.15}) is the idea (\link{BDV}{Barbour (1994)}) that existence in time is an illusion -- that an observer exists only momentarily and that ``the past'' is only a representation of instantaneous physical ``memory traces''. This idea is rejected in the hypothesis, according to which the past of an observer {\sl is }part of his structure. I have underpinned this proposal by arguing (in \link{ommi}{Donald (1997)}) that the analysis of the relationship between mind and brain can, at the very least, be considerably simplified if we take our awareness to be constructed from our past as well as from our present. As a model for the hypothesis, (\link{7.12}{7.12}) is only a model at the level of the individual manifestations described in section \link{sec 4}{4}. For this to be successful when applied to a switching structure $S(M, N, [d, \varphi])$ modelled by $O[r^1, \dots, r^{M-1}, r^M]$, we need to invoke assumptions \link{6.2}{6.2} and \link{6.2}{6.3}, and assume that, for some sufficiently small $\varepsilon > 0$, there is an $\varepsilon$-manifestation $((\sigma_m)^M_{m=1}, W)$ of $S(M, N, [d, \varphi])$ such that, for $m = 1, \dots, M$, $\sigma_m$ is given by $\sigma_{O[r^1, \dots, r^{m-1}, r^m]}$, and $\B(W)$ by $\B(M)$. If this is possible, then, from definition \link{6.1}{6.1}, \name{7.16} $$|\app{}{S(M, N, [d, \varphi])}{\omega} - \app{\B(W)}{(\sigma_m)^{M}_{m=1}}{\omega}| \leq 2\varepsilon. \eqno{(7.16)}$$ If we now assume that we can take $\varepsilon = 0$, then (\link{7.12}{7.12}), (\link{7.12}{7.13}), (\link{5.3}{5.3}), and (\link{5.9}{5.9}) yield \name{7.17} $$\eqalignno{ \app{}{S(M, N, [d, \varphi])}{\omega} &= \app{\B(W)}{(\sigma_m)^{M}_{m=1}}{\omega} =\app{\B(M)}{(\sigma_{O[r^1, \dots, r^{m-1}, r^m]})^{M}_{m=1}}{\omega} \cr &= p[r^1, \dots, r^{M-1}, r^M] p[r^1, \dots, r^{M-1}] \dots p[r^1]. &{(7.17)} \cr}$$ When we extend this model to the immediate successors of $S(M, N, [d, \varphi])$, we will model those successors by $$\{O[r^1, \dots, r^{M-1}, r^{M}, r^{M+1}] : r^{M+1} = 1, \dots, R^{M+1} \}$$ and we will replace (\link{7.12}{7.12}) and (\link{7.12}{7.13}) with \name{7.18} $$\sigma_{O[r^1, \dots, r^{m}]}\big|_{\B(M+1)} = \sum_{r^{m+1} = 0}^{R^{m+1}} p[r^1, \dots, r^{m}, r^{m+1}]\, \sigma_{O[r^1, \dots, r^{m}, r^{m+1}]}\big|_{\B(M+1)} \eqno{(7.18)}$$ where there exist projections $Q[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}] \in \B(M+1)$ such that $$ \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}(Q[ r^1, \dots, r^{m-1}, r^{m}, s^{m+1}]) = \delta_{r^{m+1} s^{m+1}}. \eqno{(7.19)}$$ for $m = 0, 1, \dots, M$. Using these equations, and assuming that we can continue to set $\varepsilon = 0$ in expressions analogous to (\link{7.16}{7.16}), the normalization constant $\xi$ of \link{G7}{G8} can, by the same steps that led to (\link{7.17}{7.17}), be calculated to be $$\eqalignno{ \xi &= \sum_{r^{M+1}=1}^{R^{M+1}} p[r^1, \dots, r^{M}, r^{M+1}] p[r^1, \dots, r^{M}] \dots p[r^1] \cr &= \sum_{r^{M+1}=1}^{R^{M+1}} p[r^1, \dots, r^{M}, r^{M+1}] \app{}{S(M, N, [d, \varphi])}{\omega} \cr &= (1 - p[r^1, \dots, r^{M}, 0]) \app{}{S(M, N, [d, \varphi])}{\omega}. \cr}$$ This implies that $\xi \leq \app{}{S(M, N, [d, \varphi])}{\omega}$ and it follows that the probability of moving to successor $O[r^1, \dots, r^{M-1}, r^{M}, r^{M+1}]$, as given by \link{G7}{G8}, will be \newline $p[r^1, \dots, r^{M}, r^{M+1}]$, and the probability of extinction will be $p[r^1, \dots, r^{M}, 0]$. These probabilities agree with those given in connection with (\link{7.10}{7.10}). This long chain of suppositions shows that the hypothesis is a development of elementary models like (\link{7.12}{7.12}) and (\link{7.18}{7.18}), and that the probabilities it produces stem from such elementary models. However, although it may be a useful starting point, revealing some of the points at issue and linking directly to other interpretations, in fact, (\link{7.12}{7.12}) is not a particularly accurate model of the way in which the hypothesis represents the changing quantum states of a human brain. There are two problems with the suggested type of map $O[r^1, \dots, r^{m-1}, r^m] \rightarrow \sigma_{O[r^1, \dots, r^{m-1}, r^m]}$ from the set of observers into a set of quantum states. One problem, which will be discussed in the \link{sec 8}{next} section, is that although such a map can be part of a good model of the hypothesis, the map concerned may be effectively many-to-one rather than one-to-one. The second, and even more fundamental important problem, is that no explicit definition of any such map has been suggested, and, in fact, there does not seem to be any precise choice of state corresponding to a given observer. This problem is solved in the hypothesis essentially by calculating probabilities through a supremum over the entire set of possible observer manifestations. (\link{7.12}{7.12}) and (\link{7.12}{7.13}) model a theory in which a finite number ($M$) of events occur, and at event $m$ there are $R^m + 1$ possible distinguishable outcomes with probabilities which depend on the previous outcomes. Combining the whole sequence gives \name{7.20} $$\displaylines{ \omega\big|_{\B(M)} = \sum_{r^M=0}^{R^M} \sum_{r^{M-1}=0}^{R^{M-1}} \dots \sum_{r^1=0}^{R^1} p[r^1, \dots, r^{M-1}, r^M] p[r^1, \dots, r^{M-1}] \hcrh \dots p[r^1] \sigma_{O[r^1, \dots, r^{M-1}, r^M]}\big|_{\B(M)} \qquad \qquad \llap{(7.20)} }$$ and, extending (\link{7.12}{7.13}), we may suppose that there exist projections \newline $Q[r^1, \dots, r^{M-1}, r^M] \in \B(M)$ such that $$\sigma_{O[r^1, \dots, r^{M-1}, r^M]}(Q[s^1, \dots, s^{M-1}, s^M]) = \delta_{r^{1} s^{1}} \dots \delta_{r^{M-1} s^{M-1}} \delta_{r^{M} s^{M}}. \hfill \eqno{(7.21)}$$ (\link{7.20}{7.20}) and (\link{7.20}{7.21}) express $\omega$ on $\B(M)$ as a decoherent decomposition of alternative observer states. Indeed, in the language of (\link{5.6}{5.6}), $Q[r^1, \dots, r^{M-1}, r^M]$ is a decohering projection for $\omega$ on $\B(M)$ which reduces $\omega$ to $\sigma_{O[r^1, \dots, r^{M-1}, r^M]}$ with probability $$\eqalignno{ \omega(Q[r^1, \dots, r^{M-1}, r^M]) &= \app{\B(M)}{\sigma_{O[r^1, \dots, r^{M-1}, r^M]}}{\omega} \cr &= p[r^1, \dots, r^{M-1}, r^M] p[r^1, \dots, r^{M-1}] \dots p[r^1]. &{(7.22)} \cr}$$ At the level of the full hypothesis, no such simple decomposition into different alternatives is assumed -- indeed, the set $\B(W)$ which corresponds to $\B(M)$ depends on the entire structure of the observer and on the precise manifestation considered, so that there is no appropriate single set of observables on which different observer states are defined. The full hypothesis also does not identify single projections like $Q[r^1, \dots, r^{M-1}, r^M]$ which at a stroke reduce $\omega$ to the current observed state. (\link{7.20}{7.20}) -- (\link{7.20}{7.22}) are oversimplifications; artifacts of the model of this section. Nevertheless, in view of the properties of the function $\mathop{\rm app}$ mentioned in section \link{sec 5}{5}, it is necessary, if the hypothesis is to be correct, that, for any individual observer, there should, for all sufficient small $\varepsilon > 0$, be $\varepsilon$-manifestations $((\sigma_m)^M_{m=1}, W)$ for which each state $\sigma_{m+1}$ is in some sense a decoherent part of the prior state $\sigma_{m}$ on $\B(W)$. For this, it is sufficient, in the framework of this section, to replace (\link{7.12}{7.12}) and (\link{7.12}{7.13}) by a set of equations for each $m = 0, \dots, M-1$ and each $r^1$, \dots, $r^{m-1}$, $r^{m}$ of the form \name{7.23} $$\displaylines{ \sigma_{O[r^1, \dots, r^{m-1}, r^{m}]}\big|_{\B(M)} = p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]\, \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}\big|_{\B(M)} \hcrh \hfill + (1 - p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]) \sigma^d_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}\big|_{\B(M)} \qquad \hfill \llap{(7.23)} }$$ where there exists a projection $Q[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}] \in \B(M)$ such that $$\eqalignno{ \sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}(Q[ r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]) &\sim 1 \cr \text{and}\quad \sigma^d_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}(Q[ r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]) &\sim 0. \qquad &{(7.24)} \cr}$$ (\link{7.23}{7.23}), (\link{7.23}{7.24}), and (\link{5.3}{5.4}) yield \name{7.25} $$\displaylines{ \qquad \app{\B(M)}{\sigma_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}}{\sigma_{O[r^1, \dots, r^{m-1}, r^{m}]}} \hcrh \hfill \qquad \quad \sim \sigma_{O[r^1, \dots, r^{m-1}, r^{m}]}(Q[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]) \hcrh \sim p[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]. \hfill \llap{(7.25)} }$$ (\link{7.23}{7.23}) and (\link{7.23}{7.24}) differ from (\link{7.12}{7.12}) and (\link{7.12}{7.13}) not only in that (\link{7.23}{7.24}) is approximate, but also in that the state $\sigma^d_{O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]}$ which expresses \newline $O[r^1, \dots, r^{m-1}, r^{m}]$ not becoming $O[r^1, \dots, r^{m-1}, r^{m}, r^{m+1}]$, is not required to be decomposable into distinguishable and meaningful alternatives. Only in estimating probabilities will it be necessary to consider more than one alternative simultaneously, but even for this, as we shall see in the \link{sec 8}{next} section, only a limited range of alternatives will be relevant at any moment. The step-by-step, approximate decoherence of (\link{7.23}{7.23}) and (\link{7.23}{7.24}) is much more plausible than (\link{7.20}{7.20}) -- (\link{7.20}{7.22}); indeed it is no more than the approximate decoherence ubiquitous in localized thermal macroscopic systems. Nevertheless, it would be absurd to claim that the mere invokation of decoherence, and the replacement of ``quantum probabilities'' by ``classical probabilities'' is, by itself, the solution to all the problems of quantum theory. The crucial difficulty is not to find structures to which appropriate ``classical probabilities'' can be assigned, but to decide which of many such structures are significant and to find a way of precisely characterizing those structures. Consider, for example, the theory of consistent histories. There are several similarities between that theory and the hypothesis of this paper. Both theories present abstract structures which are purported to be sufficient to characterize a ``classical'', ``quasi-classical'', or ``observed'' world. Both theories aim to understand quantum uncertainties in terms of classical probability theory. Both theories are developments from conventional quantum mechanics and are built on sets of projections, thought of as ``yes-no'' questions. In the hypothesis, these projections are defined in part \link{F}{F}. In consistent histories, consistent families of projections $(P^1_{r^1})_{r^1=0}^{R^1}$, \dots, $(P^M_{r^M})_{r^M=0}^{R^M}$ on $\omega$ are considered. Such families are required to satisfy some version of the consistency conditions \name{7.26} $$\displaylines{ \omega(P^1_{r^{1}} \dots P^{M-1}_{r^{M-1}} P^{M}_{r^{M}} P^M_{s^M} P^{M-1}_{s^{M-1}} \dots P^1_{s^{1}}) \hcrh = \delta_{r^1, s^1} \dots \delta_{r^{M-1}, s^{M-1}} \delta_{r^M, s^M} \ \omega(P^1_{r^{1}} \dots P^{M-1}_{r^{M-1}} P^{M}_{r^{M}} P^{M-1}_{r^{M-1}} \dots P^1_{r^{1}}). \hfill \llap{(7.26)} }$$ However, in my opinion, consistent history theorists have been far too easily satisfied with the consistency conditions. The core problem with the theory is to explain why any particular set of histories from the continuum which satisfy (\link{7.26}{7.26}) should be physically natural, or important, or should apply to us as observers. The consistency conditions make it straightforward to define sets of numbers which satisfy the axioms of classical probability theory, but without some fundamental set of histories, it is not clear what those numbers mean. Probabilities can be defined only after the fundamental entities have been identified. \link{GJKKSZ}{Griffiths (1998)} attempts to evade this problem by assuming that the observer is not part of the physical system considered. Under this assumption, consistent histories is merely a theory of experimental observations; telling us nothing about ontology. In particular, the problem of understanding the nature of observers is entirely unaddressed. In the hypothesis, the fundamental entities are defined as information processing structures. Then the continuous variations in the possible physical manifestations of such structures can be dealt with by dealing with all possibilities simultaneously, and basing the theory not on individual possibilities, but on equivalence classes of them. \pdfproclaim{8. A model for private probabilities.}{}{sec 8} \endproclaim In section \link{sec 6}{6}, two theses concerning the relationship between the probabilities of the hypothesis and those of conventional quantum theory were introduced and discussed. It was argued that a modern human observer, who is ``typical'' in the probability of the hypothesis, should be aware of a world in which quantum theory is accepted. In this section, we shall turn to thesis \link{6.5}{6.6}, and consider a model for the immediate observation of an individual event. In section \link{sec 7}{7}, it was shown that the hypothesis is a development of elementary models of state change in observation, like (\link{7.12}{7.12}) and (\link{7.18}{7.18}), and that it produces satisfactory probabilities for those models. This is a preliminary step towards thesis \link{6.5}{6.6}. However, it is only preliminary, because, in fact, the models of section \link{sec 7}{7} are not adequate as representations of observation at the neural level. In this section, a more sophisticated model will be presented. This will be a model for a brain acquiring possible information over a time period short in conventional terms but long enough to contain many determinations of switch status -- a period long enough for a single glance, for example. The goal, in this model, is to argue for a more sophisticated version of a relationship like (\link{5.3}{5.4}) or (\link{7.25}{7.25}) between a fundamental (hypothesis-defined) probability and the ``expected value'' of a suitable projection. This will provide short-term agreement between the probabilities of the hypothesis and those of conventional theory. By taking the long term as a developing succession of such ``glances'', this agreement can be extended to arbitrary time intervals. We shall suppose that at some moment an observer predicts, in the sense of definition \link{6.4}{6.4}, that the state on some set of observables $\B$ is $\rho$, and that $\B$ contains projections $P^{\B}_a$ and $P^{\B}_b$ with $P^{\B}_a + P^{\B}_b = 1$ such that, according to conventional theory, the outcome of some observation on $\B$ will be $a$ with probability $\rho(P^{\B}_a)$ and $b$ with probability $\rho(P^{\B}_b)$. The aim is to link these numbers with probabilities defined by the stochastic process of the hypothesis. We shall begin by analysing the build-up of information within the brain of the observer. This will uncover many complexities. One result of those complexities is that any outcome will be be observed in many different ways -- by many different futures of the observer. However, the analysis will indicate that quantum probabilities can be separated from geometrical and combinatorial complexities. This will make it reasonable to assume that the complexities will not bias the outcome. There are two reasons why single step probabilities in the sto\-chastic process defined by the hypothesis cannot, in general, be directly equated with the probabilities assigned to experimental results in elementary quantum mechanics. One has to do with the number of distinct successors at each step, and the other with the amount of information recorded in a single step. There is a radical difference in the counting of successors with the model of (\link{7.18}{7.18}) and with the application of the hypothesis to more realistic neural models. Underlying (\link{7.18}{7.18}) is the elementary assumption that each distinct successor corresponds to a distinct orthogonal projection. But, according to the full version of the hypothesis, successors can differ by only seemingly inconsequential alterations in the precise geometrical pattern of determinations of switch status and this may have essentially negligible effect on the corresponding quantum states. I shall refer to such differences as being ``minor''. They arise because of the abstract way in which information is defined in the hypothesis. For example, according to the hypothesis, two switching structures may differ by only the time-ordering of a single pair of determinations. It was proposed in \link{qtab}{Donald (1990, 1995)}, that a status determination corresponds a snapshot of some property of neural membrane which is linked to neural firing and in \link{psoo}{Donald (1995)}, it was estimated that a human brain might have a switching rate (roughly equivalent to a rate of status determination) as high as $10^{15}$ switchings per second. Then we might consider two situations such that in situation 1, one knows part of neuron A is firing at spacetime point $(t_1, {\bf x}_A)$ and part of neuron B is firing at space-time point $(t_2, {\bf x}_B)$, while in situation 2, part of neuron A is firing at $(t_2, {\bf x}_A)$ and part of neuron B is firing at $(t_2, {\bf x}_B)$. If $c^2(t_1 - t_2)^2 - ({\bf x}_A - {\bf x}_B)^2 > 0$, we shall have different casual information in these two situations, but that difference will be of negligible neurophysiological relevance if $t_1 - t_2 < 10^{-4}$s. It will be argued in this section that the existence of a multitude of successors which vary only by minor differences has two important positive effects. One is that the ``probability of extinction'' (defined in \link{G7}{G8}) can be expected to be zero during the normal functioning of a human brain because the inevitable loss of a priori probability due to ``collapse'' to any specific future switching structure will be outweighed by the existence of a large number of similar structures. The other positive effect is that the concentration of the possibility of minor differences to time intervals during which many switch status determinations have already been made allows the existence of a ``present moment'' for an observer to be explained; despite the fact that the hypothesis allows new determinations to be included at essentially arbitrary times. In \link{appalo}{Donald (1992)}, another model (``postulate nine'' of that paper) was presented. In that model, a priori probabilities were defined using observables beyond those local to the brain of the observer (cf. Definition \link{6.4}{6.4}), and it was supposed that when an observer observed, at a macroscopic level, the outcome of an experiment, that observation would take place within a single step from (in the notation of section \link{sec 7}{7}), say $O[r^1, \dots, r^{M-1}, r^{M}]$ to $O[r^1, \dots, r^{M-1}, r^{M}, r^{M+1}]$. For a more adequate model, in which attention is restricted to observables within the brain, we shall have to take account of the fact that only a tiny amount of information is expressed by each determined switch status in the brain. The brain is a very noisy system, in which information is accumulated in small pieces scattered over many parallel channels. Although, because of the number of different channels, the time required for sufficient accumulation to distinguish between different outcomes may be quite short, inevitably there will be a background of many essentially simultaneous determinations which are irrelevant to the particular observation being studied. We shall continue to use assumptions \link{6.2}{6.2} and \link{6.2}{6.3}. Indeed given a switching structure $S(M, N, [d, \varphi])$, we shall denote by $((\sigma^{\varepsilon}_m)^M_{m=1}, W^{\varepsilon})$ an $\varepsilon$-manifestation for which $\varepsilon$ is sufficiently small to have the properties required by those assumptions. We shall assume a strong form of persistency (assumption \link{6.2}{6.2}) by assuming that if $((\sigma^{\varepsilon}_m)^M_{m=1}, W^{\varepsilon})$ is an $\varepsilon$-manifestation of $S(M, N, [d, \varphi])$, then, for any successor $S(\hat M, \hat N, [\hat d, \hat \varphi])$ which is relevant to our calculations, we can (ignoring for simplicity the re-orderings allowed by \link{B3}{B3}) find an $\varepsilon$-manifestation of the form $((\hat \sigma^{\varepsilon}_m)^{\hat M}_{m=1}, \hat W^{\varepsilon})$ such that $\hat \sigma^{\varepsilon}_m = \sigma^{\varepsilon}_m$ for $m = 1, \dots, M$. The stochastic process is built on the elementary step from a structure \newline $S(M, N, [d, \varphi])$ to an immediate successor $S(M', N', [d', \varphi'])$. To calculate the probability of this step explicitly, one would \name{eq8.1} need to know the normalization constant $$\displaylines{ \xi(S(M, N, [d, \varphi])) = \sum \{\app{}{S(M', N', [d', \varphi'])}{\omega}: \hcrh S(M', N', [d', \varphi']) \in \Xi(M, N, d, \varphi)\} \qquad \qquad \llap{(8.1)} }$$ of \link{G7}{G8}, and so one would need to be able to list all the immediate successors -- the elements of $\Xi(M, N, d, \varphi)$. If the hypothesis is to be successful, then $\xi$ should be closely approximated by the sum of the a priori probabilities of successors which, for all sufficiently small $\varepsilon$, have $\varepsilon$-manifestation brain states $\sigma'{}^{\varepsilon}_{M'}$ which can be analysed in classical terms as processing information which is a meaningful continuation of the information in $\varepsilon$-manifestation brain states $\sigma^{\varepsilon}_M$ of $S(M, N, [d, \varphi])$. It is at this point that it is important to be able to rule out the possibility discussed in section \link{sec 3}{3} of the future of a human brain being ``dominated by the large number of possible ways in which small numbers of new short-term artificial switches could arise''. Then we have to consider what sort of classically meaningful continuations arise. Here we turn to conventional neurophysiology. Suppose, as discussed in \link{qtab}{Donald (1990, 1995)}, that switchings correspond to changes, linked to neural firing, in some property of neural membrane. A fully functioning brain state $\sigma^{\varepsilon}_M$ will have a very large number of possible meaningful continuations. By part \link{BB}{B} of the hypothesis, any continuation will correspond either to a new switch or to a single new determination of the status of an existing switch. For simplicity, and without essential loss of generality, I shall mainly discuss the latter case, involving the determination at some moment of the status of a patch of neural membrane. Where that patch has been, and also where it will be in the short term future, is largely determined, through part \link{E}{E} of the hypothesis, by the geometrical structure in $W^{\varepsilon}$ and the quasi-classical neural structure represented by $\sigma_M^{\varepsilon}$. Most of the possible variation, for a given switch, therefore lies in the moment at which the new status is specified, and, of course, in the actual status. The hypothesis defines a finite structure. This implies that there are only a finite number of possible distinguishable ``moments'' (or equivalence classes of moments) at which a new switch status can be determined. However, the docket $d$ of $S(M, N, [d, \varphi])$ carries information which is sufficient to determine a very fine division of times up to the present moment in the brain. A different docket is defined whenever the causal relations between any pair of switch determinations is changed. A single new determination during a period when there are $10^{15}$ switchings per second, defines a different docket $d'$ for every change of about $10^{-15}$ seconds in the instant of that new determination. On the other hand, there will only be one docket $d'$ for a new determination (on a given switch) in the strict causal future of all the determinations in $d$. As far as the classical meaning of the firing pattern of a brain is concerned, a change of say $10^{-5}$ seconds in the instant of a single determination would clearly be a ``minor'' difference. As neural firing changes only on a millisecond timescale, it would also be ``minor'' in relation to the local quantum state. Thus many terms in (\link{eq8.1}{8.1}) of equivalent meaning will correspond to a new determination during a period when many other determinations have already been made. Considered together, this large set of terms will have far higher probability than the comparatively far smaller set corresponding to a new determination during a period with few other determinations. However, \link{C13}{C13} of the hypothesis acts as a constraint on this process, by requiring that a switch cannot repeat status more rapidly than it has already changed status. This constraint means that we cannot continue indefinitely to add new determinations within a given time period. Taken together, these imply that the past of a switch will tend to ``fill'' with determinations until no more can be added because of \link{C13}{C13}. As a result, it is reasonable to assume that a developing structure will have a fairly precisely defined future edge or ``present moment'' and that there will be a strong tendency for ensuing determinations to be made close to that future edge. This effect will be enhanced by the fact that the more determinations are made within a given spatial and temporal locality within the brain, the more the status of any new determination within that locality will tend to be fixed by what has already been determined. A status which is compatible with many neighbouring determinations will have higher a priori probability than a determination which probes the undecided. We shall turn now to consider in detail the observation of some stochastic outcome; in particular, the observation on a set $\B$ of either $a$ or $b$. We are concerned with the initial acquisition by an observer of sufficient information to determine which outcome will be seen. That initial acquisition may require many determinations, but will often appear to take place very quickly -- in the first glance -- at the level of ``pre-conscious'' processing. Guessing at some numbers just for the sake of argument, it is, I think, not unreasonable, given a switching rate of $10^{15}$Hz, to suppose a ``present moment'' defined to within a millisecond, which is certainly shorter than the ``psychological moment'' -- and then we might consider information from determinations made within a millisecond, scattered over 10$^5$ neurons (0.1\% of retinal cells), during which perhaps 10$^{12}$ irrelevant determinations would also be made elsewhere in the brain. For a model in which the probability of observing of a given outcome can be estimated, consider first a single fixed sequence of switching structures \newline $(S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}$ which is sufficiently long that, by its conclusion, a definite outcome will be known. Suppose that that outcome is $a$, which happens with conventional probability $\rho(P^{\B}_a)$. When appropriate, $((\sigma^{\varepsilon}_m)^{M}_{m=1}, W_{M}^{\varepsilon})$, will denote an $\varepsilon$-manifestation for $S(M, N_M, [d_M, \varphi_M])$. (The persistency assumption is implicit here, in that we do not write $\sigma^{\varepsilon}_{m, M}$.) We shall suppose that in this sequence, relevant determinations have been made at instants $M_s$ for $s = 1, \dots, S$, and that the other determinations are irrrelevant. In order to model the build-up of information to the point at which the observed outcome is definite, we shall begin by modelling each separate relevant determination using (\link{5.2}{5.2}) and (\link{5.3}{5.3}). This requires the existence of projections which, in the sense of (\link{5.6}{5.6}), are ``decohering'' on to the determined statuses. We can expect to be able to find projections to characterize particular events in any sufficiently large quantum system, because, for example, such events will be characterized by the values of certain observables, and we can then specify those values using the projections provided by the spectral theorem. The present situation will certainly be sufficiently large in this sense. The switches discussed in \link{qtab}{Donald (1990, 1995)} have length scales of order $10^{-9}$m or $10^{-8}$m. Even a region as small as $(10^{-9}$m$)^3$ can have of order $100$ thermally-active degrees of freedom in the warm dense environment of the living brain. The switch size is chosen to allow the existence of the projections $P_n$ and $Q_n$ of \link{C13}{C14} which discriminate switch status, but it will be possible to find projections in $\B(W_{M_s}^{\varepsilon})$ which carry more information about the local state of the switch than do those projections, because $\B(W_{M_s}^{\varepsilon})$ is much larger than the algebra to which $P_n$ and $Q_n$ are required to belong, and because $P_n$ and $Q_n$ are required to define properties repeated over the whole lifetime of the switch. Suitable interactions with the environment of a system are sufficient to make a given projection decohering. In the present situation, the set $\B(W_{M_s}^\varepsilon)$ specifies states only in limited substructures of the brain, leaving plenty of local environment into which any coherence will rapidly disappear. Moreover, switch status represents local neural firing status and neural firing works as a method of communicating information because it is a macroscopic, thermally irreversible, ``all-or-nothing'' process which involves transition from one macro\-scopically-distin\-guishable metastable status of the local neural membrane to another. Because of the metastability, status determination can be expected to have the property assumed in conventional descriptions of quantum measurement that the determined status will be ``amplified'' across a wider region. Such amplification into the environment is more than enough for the required decoherence. This means that it is reasonable to suppose that, for each relevant determination, we can find a projection $P^s \in \B(W_{M_s}^{\varepsilon})$, which, in the language of (\link{5.6}{5.6}), is decohering for $\sigma^{\varepsilon}_{M_s - 1}$ on $\B(W_{M_s}^{\varepsilon})$, and reduces $\sigma^{\varepsilon}_{M_s - 1}$ to $\sigma^{\varepsilon}_{M_s}$. We shall suppose that $B P^s, P^s B \in \B(W_{M_s}^\varepsilon)$ for all $B \in \B(W_{M_s}^\varepsilon)$, and that the constraints imposed by part \link{F}{F} of the hypothesis can be modelled by supposing that in becoming aware of the new determination, the state of the observer changes from $\sigma^{\varepsilon}_{M_s - 1}$ to $\sigma^{\varepsilon}_{M_s}$ where \name{8.2} $$\sigma^{\varepsilon}_{M_s} = P^s \sigma^{\varepsilon}_{M_s - 1} P^s/ \sigma^{\varepsilon}_{M_s -1}(P^s). \eqno{(8.2)}$$ We shall also suppose that, for $s = 1, \dots, S$, $$\sigma^{\varepsilon}_{M_s -1}(P^s B) = \sigma^{\varepsilon}_{M_s -1}(B P^s) \quad \text{for all $B \in \B(W_{M_s}^\varepsilon)$}. \eqno{(8.3)}$$ Even if the assumption of many simultaneous parallel channels is inaccurate for some particular type of observation, the macroscopic and thermal nature of the brain will make it reasonable to assume decoherence on a given switch between one determination and the next. With the same justification, we shall also assume that, for $s = 1, \dots, S$, the projections $P^s$ all commute. For $s = 1, \dots, S$, set $R_s = P^1 \cdot P^2 \cdots P^s$. The commutativity of the $P^s$ implies that the $R_s$ are projections and are independent of the ordering of their components. (\link{8.2}{8.2}) and (\link{8.2}{8.3}) imply that, for $B \in \B(W_{M_s}^\varepsilon)$ $$\eqalignno{ \sigma^{\varepsilon}_{M_s -1}(B) &= \sigma^{\varepsilon}_{M_s -1}(P^s B) + \sigma^{\varepsilon}_{M_s-1}((1-P^s) B) \cr &= \sigma^{\varepsilon}_{M_s -1}(P^s B P^s) + \sigma^{\varepsilon}_{M_s -1}((1-P^s) B (1-P^s)) \cr &= \sigma^{\varepsilon}_{M_s-1}(P^s)\sigma^{\varepsilon}_{M_s}(B) + \sigma^{\varepsilon}_{M_s -1}((1-P^s) B (1-P^s)). \cr}$$ By (\link{5.3}{5.3}), this implies that \name{8.4} $$\app{\B(W_{M_s}^\varepsilon)}{\sigma^{\varepsilon}_{M_s}}{ \sigma^{\varepsilon}_{M_s -1}} = \sigma^{\varepsilon}_{M_s -1}(P^s). \eqno{(8.4)}$$ A significant aspect of (\link{8.4}{8.4}) is that the projection $P^s$ is localized to the region $s$. Because of this, it is not unreasonable to assume that the expected value $\sigma^{\varepsilon}_{M_s -1}(P^s)$ does not depend on the irrelevant determinations. The state at the beginning of the ``glance'' is $\sigma^{\varepsilon}_{M_0}$. Denote this state by $\rho$. Set $\rho = \rho_0$ and, mimicking (\link{8.2}{8.2}), define \name{8.5} $$\rho_s = R_s \rho R_s/ \rho(R_s), \eqno{(8.5)}$$ for $s = 1, \dots, S$. In (\link{8.5}{8.5}), changes in $\sigma^{\varepsilon}_{M_0}$ due to irrelevant determinations are omitted and only the local changes defined by (\link{8.2}{8.2}) are kept. The commutativity of the $P^s$ implies that $$\rho(R_s) = \rho(R_{s-1} P^s) = \rho(R_{s-1} P^s R_{s-1}) = \rho_{s-1}(P^s) \rho(R_{s-1})$$ and hence, by induction, that \name{8.6} $$\rho(R_s) = \rho_{s-1}(P^s) \rho_{s-2}(P^{s-1}) \cdots \rho_0(P^1). \eqno{(8.6)}$$ Now we shall use the locality of the $P^s$ to justify replacing $\sigma^{\varepsilon}_{M_s -1}(P^s)$ in (\link{8.4}{8.4}) by $\rho_{s-1}(P^s)$. Making similar replacements for $i = 1, \dots, s$ in (\link{8.6}{8.6}), allows us to model the situation by supposing that $$\displaylines{ \sigma^{\varepsilon}_{M_0}(R_s) = \rho(R_s) = \sigma^{\varepsilon}_{M_s -1}(P^s) \sigma^{\varepsilon}_{M_{s-1} -1}(P^{s-1}) \cdots \sigma^{\varepsilon}_{M_1 -1}(P^1) \hcr = \app{\B(W_{M_s}^\varepsilon)}{\sigma^{\varepsilon}_{M_s}}{ \sigma^{\varepsilon}_{M_s -1}} \app{\B(W_{M_{s-1}}^\varepsilon)}{\sigma^{\varepsilon}_{M_{s-1}}}{ \sigma^{\varepsilon}_{M_{s-1} -1}} \cdots \app{\B(W_{M_1}^\varepsilon)}{\sigma^{\varepsilon}_{M_1}}{ \sigma^{\varepsilon}_{M_0}}. \cr}$$ Using the localization and decoherence of determinations yet again, allows us to change the moment at which the set of observables $\B(W_{M})$ is defined. In particular, we may suppose that, for $M = M_s, M_s +1, \dots, M_{s+1} - 1$, \name{8.7} $$\displaylines{ \sigma^{\varepsilon}_{M_0}(R_s) = \app{\B(W_M^\varepsilon)}{\sigma^{\varepsilon}_{M_s}}{ \sigma^{\varepsilon}_{M_s -1}} \app{\B(W_M^\varepsilon)}{\sigma^{\varepsilon}_{M_{s-1}}}{ \sigma^{\varepsilon}_{M_{s-1} -1}} \hcrh \cdots \app{\B(W_M^\varepsilon)}{\sigma^{\varepsilon}_{M_1}}{ \sigma^{\varepsilon}_{M_0}}. \qquad \qquad \llap{(8.7)} }$$ (\link{8.7}{8.7}) is the conclusion of the first step towards the goal of this section. It equates a product of a priori probabilities to the expected value of a composite projection. The product is a factor in the a priori probability of the $\varepsilon$-manifestation $((\sigma^{\varepsilon}_m)^{M}_{m=1}, W_{M}^{\varepsilon})$ of the switching structure $S(M, N_M, [d_M, \varphi_M])$ for $M = M_s, M_s +1, \dots, M_{s+1} - 1$. All the other terms in the a priori probability are independent of the outcome being observed. Thus, setting $\varepsilon = 0$, \name{8.8} $$\app{}{S(M, N_M, [d_M, \varphi_M])}{\omega} = \sigma^{\varepsilon}_{M_0}(R_s) w^{nd}_1(a, S(M, N_M, [d_M, \varphi_M]), \omega) \eqno{(8.8)} $$ where $w^{nd}_1$ is some factor which does not depend on the conventional quantum probability ($\rho(P^{\B}_a)$) of the outcome ($a$) which is observed, although it may depend on the nature of the outcome; for example, it will depend on the number ($s$) of determinations already made, and on the length of time over which those determinations have been made. In this situation, we return to the calculation of the normalization constant $\xi(S(M, N_M, [d_M, \varphi_M]))$ defined by (\link{eq8.1}{8.1}). For $M = M_s, M_s +1, \dots, M_{s+1} - 1$, there are two types of term in the sum. One type involves successors $S(M', N', [d', \varphi'])$ in which the new determinations are irrelevant to the observed outcome. For these terms an approximate result of the form $$\app{}{S(M', N', [d', \varphi'])}{\omega} = \sigma^{\varepsilon}_{M_0}(R_s) w^{nd}_1(a, S(M', N', [d',\varphi']), \omega)$$ will continue to hold. There will also be successors $S(M', N', [d', \varphi'])$ in which the new determinations are relevant. Now $\app{}{S(M', N', [d', \varphi'])}{\omega}$ will have a factor like, for example, $\sigma^{\varepsilon}_{M_0}(P^{s+1})$ and will not be of the same form. However, in the sum \newline $\xi(S(M, N_M, [d_M, \varphi_M]))$ there will be far more terms of the first type than of the second (for example, a factor of $10^7$ more with the numbers guessed at above). This means that terms of the second type can essentially be ignored, and that it is reasonable to write \name{8.9} $$\xi(S(M, N_M, [d_M, \varphi_M])) = \sigma^{\varepsilon}_{M_0}(R_s) w^{nsd}_2(a, S(M, N_M, [d_M, \varphi_M]), \omega) \eqno{(8.9)}$$ where $w^{nsd}_2$ is some factor which is not strongly dependent on the conventional probability. In the ordinary operation of the brain, the total number of terms in the sum (\link{eq8.1}{8.1}) is huge. The suggested number of determinations made within a millisecond, already shows this, and in calculating (\link{eq8.1}{8.1}), one would also have to take account of the number of minor differences in the way in which any given determination might arise. With (\link{8.4}{8.4}) indicating that, for $S(M', N', [d', \varphi'])$ an immediate successor of $S(M, N, [d, \varphi])$, the ratio $$\app{}{S(M', N', [d', \varphi'])}{\omega}/\app{}{S(M, N, [d, \varphi])}{\omega}$$ can be approximated by the conventional probability of some possible impending neural event, this huge number of terms will ensure that, except in exceptional circumstances, we shall have \name{8.10} $$\xi(S(M, N, [d, \varphi])) \geq \app{}{S(M, N, [d, \varphi])}{\omega} \eqno{(8.10)}$$ with the consequence, according to \link{G7}{G8}, that there will be no possibility of extinction. The sequence $(S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}$ is one possible path in the stochastic process which leads from the original switching structure to a structure in which a given outcome is known. The probability of that path is a product of the probabilities of the individual steps given by \link{G7}{G8}: \name{8.11} $$\eqalignno{ \Pr((&S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S} | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) \cr &= \prod_{M = M_0}^{M_S -1} \app{}{S(M + 1, N_{M+1}, [d_{M+1}, \varphi_{M+1}])}{\omega} / \xi(S(M, N_M, [d_M, \varphi_M])) \cr &= \sigma^{\varepsilon}_{M_0}(R_S) w^{nsd}_3(a, (S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}, \omega) &{(8.11)} \cr & &\hbox{ by induction using (\link{8.8}{8.8}) and (\link{8.9}{8.9}).} \cr}$$ According to (\link{8.11}{8.11}), the probability of an individual path is given by the expectation in the initial state $\sigma^{\varepsilon}_{M_0} = \rho$ of a compound projection $R_S$ multiplied by a complicated geometrical and combinatorial factor. $\rho(R_S)$ is not necessarily equal to $\rho(P^{\B}_a)$. However, the projection $R_S$ does express sufficient information within the brain to determine the outcome of the observation. This means that, at least as far as the state $\rho$ is concerned, $R_S$ should be a subprojection of $P^{\B}_a$, or, in mathematical terms, that $\rho(P^{\B}_a R_S B) =\rho(R_S B)$ for all $B$ in some set sufficiently large to contain all the observables relevant to the present situation, for example, $\B(W_M^\varepsilon) \cup \{ A C : A \in \B, C \in {\cal C}(W_M^\varepsilon) \}$ (cf.~Definition \link{6.4}{6.4}). On such a set of observables, $R_S$ differs from $P^{\B}_a$ in that it fixes not only the outcome of the observation, but also the precise neural processing by which that observation is made. Independence between the internal processing and the cause of the external event makes it possible to write \name{8.12} $$\rho(R_S) = \rho(P^{\B}_a) w^{nd}_4(a, (S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}, \omega) \eqno{(8.12)}$$ where, once again, $w^{nd}_4$ is a factor which, while it may depend on the nature of the outcome, does not depend on its conventional quantum probability. Substituting (\link{8.12}{8.12}) into (\link{8.11}{8.11}) gives \name{8.13} $$\displaylines{ \qquad \quad \Pr((S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S} | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) \hcrh = \rho(P^{\B}_a) w^{nsd}_5(a, (S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}, \omega). \hfill \llap{(8.13)} }$$ \medskip There are many many paths like $(S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}$ which lead to structures in which outcome $a$ is known. To calculate the total probability \newline $\Pr(a| S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega)$ of outcome $a$ given the initial structure, we must sum over all such distinct paths. Following (\link{8.13}{8.13}), the result will take the form \name{8.14} $$ \Pr(a | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) = \rho(P^{\B}_a) w_6(a, S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega). \eqno{(8.14)}$$ This states that the total probability of outcome $a$ is equal to the conventional probability of that outcome multiplied by some factor $w_6$ which depends on the number of ways in which that $a$ can be observed. $w_6$ combines many terms, none of which is strongly dependent on the conventional probability of $a$, but this does not allow us to conclude that the combination also does not depend strongly on the conventional probability. Here is a caricature of the situation: \pdfproclaim{example 8.15}{}{8.15} \noindent A) \quad Consider a discrete Markov chain $(X_n)_{n\geq0}$ with three states $a$, $b$, and $o$. $X_0 = o$. $a$ and $b$ are sink states on which the process terminates. At each $n$, the probability of passing to $a$ (respectively $b$) is proportional to $p$ (resp.~$q$). The probability of staying at $o$ is proportional to $x$. The constant of proportionality is determined by normalizing the probabilities. $a$ and $b$ model the outcomes of the observation. Let $F_n(a)$ (resp.~ $F_n(b)$) be the probability of the process terminating at the $n^{th}$ step at $a$ (resp.~$b$). We are interested in calculating the net probability $F(a) = \sum_{n=1}^\infty F_n(a)$ (resp.~$F(b) = \sum_{n=1}^\infty F_n(b)$) of the process terminating at $a$ (resp.~$b$). Staying at $o$ is a caricature of making an irrelevant determination, so we shall suppose that $x \gg p, q$. $$F_n(a) = {p \over p + q + x} \left({x \over p + q + x}\right)^{n-1},$$ so that $F_n(a) \sim p/x$ for $x/(p + q) \rightarrow \infty$. (This caricatures (\link{8.11}{8.11}).) $$F(a) = \sum_{n=1}^\infty {p \over p + q + x} \left({x \over p + q + x}\right)^{n-1} = {p \over p + q}.$$ If $p$ corresponds to $\rho(P^{\B}_a)$ and $q$ to $\rho(P^{\B}_b)$ then we would have $p + q = 1$, and it would be the case that $F(a)$ would be proportional to $p$ with a constant of proportionality ($1$) independent of $p$. We would also have $F(a)/F(b) = p/q$. A variation in the example, however, shows that this is, in general, too simple a conclusion. \smallskip \noindent B) \quad Suppose that there are two sink states $a_1$ and $a_2$ which model distinct ways in which the outcome $a$ might be observed. Suppose that the probability of passing to either of these states is proportional to $p$ (with the same constant of proportionality) Now we have $F(a) = \dsize{2p \over 2p + q}$. In this case, $F(a)/p = \dsize{2 \over p + 1}$ is strongly dependent on $p$ and $F(a)/F(b) = 2 p / q$. \smallskip \noindent C) \quad For a slightly more realistic model, suppose that there are many distinct states $a_i$, $i = 1, \dots, N_a$, corresponding to outcome $a$ and many distinct states $b_j$, $j = 1, \dots, N_b$, corresponding to outcome $b$, and suppose that different weights are allowed for each state, so that the probability of passing from $o$ to $a_i$ (resp.~to $b_j$, to $o$) is $w(a_i) p/W$ (resp.~$w(b_j) q/W$, $x/W$) where $W = \sum_i w(a_i) p + \sum_j w(b_j) q + x$. This gives $F(a)/p = \dsize{W_a \over W_a p + W_b q}$ and $F(a)/F(b) = W_a p / W_b q$ where $W_a = \sum_i w(a_i)$ and $W_b = \sum_j w(b_j)$. \hfill $\blacksquare$ \endproclaim \bigskip Returning to (\link{8.14}{8.14}), we have $${ \Pr(a | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) \over \Pr(b | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega)} = {\rho(P^{\B}_a) w_6(a, S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) \over \rho(P^{\B}_b) w_6(b, S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega)}.$$ We wish to argue that \name{8.16} $$\displaylines{ { \Pr(a | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) \over \Pr(b | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega)} = {\rho(P^{\B}_a) \over \rho(P^{\B}_b)} }$$ and this of course would follow if $$w_6(a, S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) = w_6(b, S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega). \eqno{(8.16)}$$ (\link{8.16}{8.16}) will hold if there is a correspondence between the ways in which outcome $a$ can be observed, and the ways in which outcome $b$ can be observed. For example, a sufficient condition for (\link{8.16}{8.16}) is that there should be a bijection $\lambda$ between sequences $(S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}$ leading to observation of $a$ and sequen\-ces $\lambda((S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S})$ leading to $b$ such that $$\displaylines{ w^{nsd}_5(b, \lambda((S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}), \omega) \hcrh = w^{nsd}_5(a, (S(M, N_M, [d_M, \varphi_M]))_{M = M_0}^{M_S}, \omega). }$$ The construction of (\link{8.13}{8.13}) suggests that this is not an unreasonable assumption; although it will be open to refutation in specific cases by detailed analysis of the precise modes of observation of the possible outcomes. In section \link{sec 6}{6}, it was noted that the hypothesis of this paper differs from the version of \link{psoo}{Donald (1995)}, in a way designed to make such an ``indifference assumption'' more plausible. We are investigating a situation which, in conventional terms, has only two outcomes $a$ and $b$. This means that, to a high degree of accuracy, $$\Pr(a | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) + \Pr(b | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) = 1.$$ It follows from this, and the fact that $\rho(P^{\B}_a) + \rho(P^{\B}_b) = 1$, that, if (\link{8.16}{8.16}) holds, then \name{8.17} $$\displaylines{ \Pr(a | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) = \rho(P^{\B}_a) \, \cr \rlap{\hbox{and}}\hfill \Pr(b | S(M_0, N_{M_0}, [d_{M_0}, \varphi_{M_0}]), \omega) = \rho(P^{\B}_b). \hfill \llap{(8.17)} }$$ (\link{8.17}{8.17}) is the result we have been aiming at. It is an exact form of thesis \link{6.5}{6.6}. It shows that under suitable circumstances, the stochastic process proposed in the hypothesis can define ``private'' probabilities which are equal to the expected values of appropriate projections, and it illustrates the claim that the hypothesis is a generalization of conventional quantum mechanics which provides a complete theory within which the probabilistic role of such ``public'' expected values can be explained. However, this section has also shown that the detailed analysis of neural processing in terms of the hypothesis is far from simple. As a result, there are many ways in which (\link{8.17}{8.17}) may break down. Reactions to such a breakdown might include abandoning the hypothesis, trying to modify it, or searching for experimental verification. And yet even if such verification were to be found, with the sort of bizarre properties described in section \link{sec 6}{6}, it would not necessarily be anything more than a demonstration of an ``observer-effect'' with a quite conventional interpretation. For example, no matter how often it has been placed in the infernal apparatus, no cat will remember having heard the cynanide flask in Schr\"odinger's experiment break more than once. Even in conventional terms, the details of neural processing are extremely complex and unpredictable. Subtle observer-effects, explicable in terms of sensory limitations or attention to the biologically significant, are inevitable. As individuals, we rarely conduct our own personal tests of quantum theory; certainly not in the conditions and with the repetitions required for the collection of significant statistics. Thus what may be most important about the arguments for (\link{8.17}{8.17}) is that they make it plausible that the hypothesis correctly reflects the rough probabilities of everyday experience (``X always/ usually/ sometimes/ seldom/ never happens''). In particular, this applies to the high probabilities needed in section \link{sec 6}{6} to confirm thesis \link{6.5}{6.5}. \pdfproclaim{9. Physics without any physical constants.}{}{sec 9} \endproclaim The hypothesis proposes an abstract characterization of the set of all possible structures for an observer, together with a definition of a stochastic process on that set, defined in terms of an a priori probability of existence for each structure. In this section, we shall consider the nature of reality in the light of this proposal, and reflect on the possibility of changing the definition of a priori probability so as to provide a formalism for a physics in which physical constants are not fixed a priori, but are indeterminate until observed. It will turn out that such a formalism will only be satisfactory if the initial conditions of the universe can be given a simple description, but as will be discussed, this requirement does not seem incompatible with current observational evidence. The idea that the physical constants we measure are observed from a range of possibilities is not new, as can be seen from discussion and references in \link{BDV}{Barrow and Tipler (1986} -- in particular, \sect 4.6), but the formalism proposed here gives a detailed technical framework for the idea. It has always been the case that quantum theory has suggested ways in which our beliefs about the nature of reality might be altered. A central purpose throughout the present work has been the investigation of the possibility, at the level of serious theoretical physics, of some of the more radical of these suggestions. The fundamental set of entities considered in this paper is the set of minimal switching structures. Denote this set by $\SM$. Any possible observer at any moment corresponds to a unique element $s = S(M, N, [d, \varphi]) \in \SM$, and \link{G7}{G7} defines the a priori probability $\app{}{s}{\omega}$ of existence of that element. Any future which can be experienced is the possible future of some observer, so if we know how to compute the probabilities of all possible futures of all possible observers, then, at least as far as prediction is concerned, we have a complete physical theory. In the present theory, conventional physics is certainly involved in the definition of the a priori probabilities. The definition of $\app{}{s}{\omega}$ is in terms of the ``universal state'' $\omega$ and fundamental physical quantities such as Lorentz transformations and local von Neumann algebras. These are defined by, or in terms of, some postulated underlying universal quantum field theory. If that field theory changes then so will the proposed a priori probabilities. For example, the sets of sequences of quantum states which satisfy the definitions in parts \link{E}{E} and \link{F}{F} of the hypothesis, will change with changes in the action of time-translations on those states. In this section, the dependence of a priori probability on the underlying field theory will be made explicit in the notation by writing $\F$ for a possible field theory and replacing $\app{}{s}{\omega}$ with $\app{}{s}{\F, \omega}$. For the sake of discussion, it will be assumed in this section that $\F$ is some sort of gauge theory; something like the standard model or a grand unified theory, with whatever scalar fields are required to drive cosmological inflation. In order to discuss possible constraints on $\omega$, cosmological arguments involving curved spacetimes will be invoked. Of course this is inconsistent with the assumption, in the hypothesis, of Minkowski spacetime, but this inconsistency will be ignored here. There is no doubt that the present hypothesis is incompatible with general relativity. Nevertheless, allowing observer-dependent field theory, and therefore observer-dependent time propagation may be a first step towards the possibility of observer-dependent spacetime geometry. I have considered ways of modifying the hypothesis in order to allow for such a possibility, but, for the moment, this is unfinished speculation. I expect that we shall have to be able to encompass some such possibility before we can make sense of the ultimate true theory of quantum gravity. Leaving this ambition aside, the framework of this section suggests that, even in the ultimate theory, it may not be necessary for physical constants to be determined. According to the present theory, everything we believe about the world, we have deduced from the pattern of information represented in our unique physical structure $s$. That pattern contains our entire neural history. Everything we have taken in from what we have heard or read exists within that history, and the meaning which we see in it tells us about our ``reality'' -- the ``world'' which we ``observe''. In that world, we see physical events and repetitions of such events, and we see our colleagues seeing and observing and reporting on physical events. From all this information, it becomes possible for us to estimate probabilities for individual events -- there will be rain before tomorrow, the Tories will not win the next election -- and to construct and learn about physical theories which are compatible with some or most or ultimately all of what we see. Nevertheless, according to the theory presented here, the ``world'' is essentially an illusion; a mental construction. From the very beginning of quantum theory, the conventional view of reality has been called into question. It was learnt that a particle cannot be seen to have, at the same time, both an exact position and an exact momentum. The idea arose that the properties of a particle might depend on the method by which that particle was observed. Everett's formalism went further in denying the conventional view. It suggested, in the first place, that, although there is an observer-independent physical reality subject to observer-indepen\-dent physical laws, that reality was not a ``world'' but a much more complex and abstract structure. To see this, recall Everett's original many-worlds argument (\link{Butterfield}{DeWitt and Graham (1973)} pp 65--68) in its simplest form (cf.~(\link{7.1}{7.1})). Everett imagines a universe consisting of an observer with wave function $\psi$ observing a system with wave function $\varphi$. The total wave function of the universe is then a sum of tensor products of $\psi$'s and $\varphi$'s. If, at the beginning of a measurement, with the observer in some fixed initial wave function $\psi$, the system is in an eigenstate $\varphi_a$ (respectively $\varphi_b$) of the operator being measured, then the observer at the end will be in some definite corresponding wave function $\psi_a$ (resp. $\psi_b$) and the final total wave function will be $\psi_a \otimes \varphi_a$ (resp. $\psi_b \otimes \varphi_b$). On the other hand, simply by the linearity of the Schr\"odinger equation, if the initial wave function of the system is a superposition $\lambda\varphi_a + \mu\varphi_b$, then the final total wave function must be \name{9.1} $$\Psi = \lambda \psi_a \otimes \varphi_a + \mu \psi_b \otimes \varphi_b. \eqno(9.1)$$ Such a superposition does not represent a unique fixed ``world'' outside the observer, but two ``worlds'' ($\varphi_a$ and $\varphi_b$) each relative to possible information gained by the observer. What is left of observer-independent ``physical reality'' in this picture is, firstly, the global Hamiltonian which provides the time-dependence for $\Psi$ -- this corresponds in the present theory to a choice of global quantum field theory $\F$. Secondly, physical reality determines the probability of the observer seeing ``world'' $\varphi_a$ given that he has seen the experiment set up (this probability is essentially $|\lambda|^2$). Finally, the determination of possible physical structures for observers; in other words, the determination that $\Psi$ should be split as in equation (\link{9.1}{9.1}), and not by any other possible splitting, is also part of observer-independent physical reality. This aspect was not considered by Everett but is fundamental in the present theory. Once we have begun on this path, it becomes possible to go further, without needing to deny that our observations are shaped by physical laws. One further step has already been taken in the current hypothesis with the proposal that the possible physical structure of an observer corresponds at an instant not to a unique wave-function but to a set of quantum states. Crucial in taking this step is the idea of defining probability by taking a supremum over possibilities between which the observer cannot distinguish. This idea pervades the definitions of probability in the present theory, not only in \link{G5}{G5} and \link{G7}{G7}, but also, and most fundamentally, in \link{G2}{G2c}, in the very definition of the a priori probability function. Suppose that we accept that the set $SM$ does characterize possible observers and that our lives are characterized by elements $s \in SM$. Suppose also that the laws of physics are given by something like some sort of gauge quantum field theory; by some $\F$. The physical constants which we deduce for such a theory (which we shall take to be part of the definition of $\F$), and the probabilities that it predicts by \link{G7}{G7} and \link{G7}{G8} are mutually dependent. Leaving aside the ambitions of string theorists, there appear to be many different plausible quantum field theories $\F$ which would allow $s$ to exist. For example, it seems possible that the fine structure constant might be exactly ${1\over 137.03601}$ or that it might be exactly ${1\over 137.03602}$. The conventional assumption is that, whether or not we can discover it, there is a true, absolute, real, and unique value for the fine structure constant (or for a corresponding free parameter in the ultimate theory of everything). We estimate that value by assuming that our world is reasonably probable. Ultimately, we attach most credence to the estimate which is {\it most likely }given all the available evidence. However, a revision to the hypothesis will show that this conventional assumption also can be denied. The question of whether the fine structure constant is a rational number, for example, simply may not have an answer. We may allow merely that our a priori probabilities are determined by a class of quantum field theories. In this case, $\app{}{s}{\F, \omega}$ is not the absolute a priori probability of the observer defined by $s$ but instead gives the a priori probability for that observer to observe the universe with the particular set of physical constants given by $\F$. In order to make the revision, suppose that we can identify some suitable class of quantum field theories; specified, for example, merely by choice of gauge group or set of possible gauge groups, but with the constants, and even perhaps with the number of elementary fields, left free. Suppose also that for each element $\F$ of the class, we can choose one or more appropriate ``universal states'' $\omega$. Denote the set of such pairs $(\F, \omega)$ by $\V$. In the hypothesis, $\A(\Lambda)$, $\tau_{(x,L)}$, ${\cal C}(W)$, $\B(W)$, ${\cal N}(W, E)$, and ${\cal N}(W)$ will all depend on $\F$ and hence so will $GSO(M, N, d, \varphi)$ and $\app{}{W}{\omega}$, which we shall write as $GSO_{\F}(M, N, d, \varphi)$ and $\app{}{W}{\F, \omega}$. Now replace \link{G7}{G7} and \link{G7}{G8} by \medskip \name{Gprime} \noindent G7$'$) Define the a priori probability for the minimal switching structure \newline $S(M, N, [d, \varphi])$, given the field theory $\F$ and the universal state $\omega$, to be $$\displaylines{ \app{}{S(M, N, [d, \varphi])}{\F, \omega} = \sup\{ \app{}{W}{\F, \omega} : W \in GSO_{\F}(M, N, d', \varphi') \crh \text{where } SO(M, N, d', \varphi') \in S(M, N, [d, \varphi])\}. }$$ \noindent G8$'$) Define the a priori probability for the minimal switching structure $S(M, N, [d, \varphi])$ given the class $\V$ of field theories and states, to be $$\app{}{S(M, N, [d, \varphi])}{\V} = \sup\{ \app{}{S(M, N, [d, \varphi])}{\F, \omega} : (\F, \omega) \in \V \}. $$ \noindent G9$'$) Use G8$'$ to define a classical discrete Markov process on the space of minimal switching structures $S(M, N, [d, \varphi])$. $$\text{Set }\xi = \sum \{\app{}{S(M', N', [d', \varphi'])}{\V}: S(M', N', [d', \varphi']) \in \Xi(M, N, d, \varphi)\}.$$ Define the probability of moving from $S(M, N, [d, \varphi])$ to an immediate successor $S(M', N', [d', \varphi'])$ $$\displaylines{ \text{to be } \hfill \app{}{S(M', N', [d', \varphi'])}{\V}/\xi, \hfill \text{if } \xi \geq \app{}{S(M, N, [d, \varphi])}{\V}, \cr \text{and to be } \hcr \app{}{S(M', N', [d', \varphi'])}{\V}/ \app{}{S(M, N, [d, \varphi])}{\V}, \hfill \text{ if } \xi < \app{}{S(M, N, [d, \varphi])}{\V}. }$$ Define the probability of extinction to be $0$ if $\xi \geq \app{}{S(M, N, [d, \varphi])}{\V}$, and to be $1 - \xi/\app{}{S(M, N, [d, \varphi])}{\V}$ otherwise. \medskip The essence of this modification is to add a choice of $\F$ to the definition \newline $((\sigma_m)_{m=1}^M, W)$ of an individual manifestation of a switching structure. With the modification, the theory answers the question of what the most likely values are for the coupling constants and the particle masses, given the observer's experience, in exactly the same way in which it answers the question of what will be the most likely future experiences of the observer. \link{Gprime}{G8$'$} simply broadens the supremum over observationally-indistinguishable possibilities. In particular, this means that there is no need to postulate a prior distribution on $\V$. The conceptual advantage to this approach is the potential disappearance of arbitrary initial conditions and arbitrary parameters. In a deterministic theory, all the complexity of everything which will happen has to be present from the beginning of time. In the proposed revised theory, it is possible to imagine that the set $\V$ might have a comparatively simple description -- for example, that there is a gauge group and a thermal equilibrium state -- in which case there will be very little constraint on what could have happened, and the constraints on what can happen will come almost entirely from what has been experienced. Without a complete theory of quantum cosmology, it is impossible to be precise about $\V$. Nevertheless, $\V$ cannot be arbitrary. \link{Gprime}{G7$'$}, \link{Gprime}{G8$'$}, and \link{Gprime}{G9$'$}, and \link{G7}{G7} and \link{G7}{G8} represent different proposals about the fundamental physical laws. Each different choice of $\V$ also corresponds to a different proposal and these proposals have empirical consequences. For a given plausible quantum field theory $\F$, $\V$ cannot contain all pairs of the form $(\F, \omega)$, because, if it did, the most likely choice for $\omega$ would be highly time-dependent; highly dependent on the observations which have been made; and the a priori probabilities defined by the hypothesis would not agree with those predicted from previous observations. This is best demonstrated by an elementary example: \pdfproclaim{example 9.2}{}{9.2} Suppose that an observation with two possible outcomes ($a$ and $b$) has been made on many similar systems all of which we expect to be in identical states, and that we wish to predict probabilities for the outcomes of the next observation. Suppose that after $T$ observations (with $T$ large) $a$ has been seen around 0.2T times and $b$ around 0.8T times. A simple quantum mechanical model proposes that an operator of the form $u|\varphi^a\>\<\varphi^a| + v|\varphi^b\>\<\varphi^b|$ with $\varphi^a$ orthogonal to $\varphi^b$ and $u \ne v$ is being measured. An elementary model for the relevant aspects of $\omega$ would involve a set of independent Hilbert spaces $\H_{sys^1}$, $\H_{sys^2}$, \dots $\H_{sys^T}$ external to the observer, one for each observation, (cf.~(\link{7.15}{7.15}) and example \link{6.7}{6.7}). Then we might expect that \name{9.3} $$\displaylines{ \omega\big|_{\B(\H_{sys^1}) \otimes \B(\H_{sys^2}) \otimes \dots \otimes \B(\H_{sys^T})} = \sum_{r^1=1}^{2}\sum_{r^2=1}^{2} \dots \sum_{r^T=1}^{2} p_{r^1} p_{r^2}\dots p_{r^T} \hcrh |\varphi_{r^1}\>\<\varphi_{r^1}| \otimes |\varphi_{r^2}\>\<\varphi_{r^2}| \otimes \dots \otimes |\varphi_{r^T}\>\<\varphi_{r^T}| \qquad \qquad \llap{(9.3)} }$$ where, for each $t$, if $r^t = 1$ then $p_{r^t} = 0.2$ and $\varphi_{r^t}$ corresponds to $\varphi^a$ while if $r^t = 2$ then $p_{r^t} = 0.8$ and $\varphi_{r^t}$ corresponds to $\varphi^b$, so that, in the language of example \link{6.7}{6.7}, the right hand side of (\link{9.3}{9.3}) is equal to $(0.2|\varphi^a\>\<\varphi^a| + 0.8|\varphi^b\>\<\varphi^b|)^T$. This is satisfactory both as a model for the first $T$ observations, and, by extension, for subsequent prediction. In the framework of \link{Gprime}{G7$'$} -- \link{Gprime}{G9$'$}, a modification of definition \link{6.1}{6.1} would allow us to define an $\varepsilon$-manifestation of a structure $S(M, N, [d, \varphi])$ as a sequence \newline $((\sigma^\varepsilon_m)^M_{m=1}, W^\varepsilon, \F^\varepsilon, \omega^\varepsilon)$ which comes sufficiently close to attaining all the relevant suprema. If $S(M, N, [d, \varphi])$ is sufficiently complex that, for all sufficiently small $\varepsilon$, $\F^\varepsilon$ is well-characterized, and if, given $\F$, the choice of $\omega$ is highly restricted, then (\link{9.3}{9.3}) may be a plausible model for restrictions of $\omega^\varepsilon$ to previously entirely unobserved systems, such as we might imagine sometimes observing in astronomy. In these circumstances, the suggestion is that, in the language of a suitably modified form of \link{6.4}{6.4}, the predicted state on each system $\B(\H_{sys^t})$, prior to its observation, will be close to $\omega^\varepsilon|_{\B(\H_{sys^t})}$, and our suppositions that the initial states of the systems are all identical and that outcome $a$ has probability 0.2, would be inductively-confirmed cosmological hypotheses. However, if, for given $\F$, a supremum over arbitrary $\omega$ were allowed, then (\link{9.2}{9.2}) could be replaced by \name{9.4} $$\omega\big|_{\B(\H_{sys^1}) \otimes \B(\H_{sys^2}) \otimes \dots \otimes \B(\H_{sys^T})} = |\varphi_{r^1}\>\<\varphi_{r^1}| \otimes |\varphi_{r^2}\>\<\varphi_{r^2}| \otimes \dots \otimes |\varphi_{r^T}\>\<\varphi_{r^T}|. \eqno{(9.4)}$$ where in this case $\varphi_{r^t}$ would correspond to $\varphi^a$ if $a$ had been the outcome of observation $t$ and $\varphi^b$ if $b$ had been the outcome. If, as in this elementary model, it is possible for $\omega$ to shadow each new piece of information, then the supremum in \link{Gprime}{G8$'$} would be unity, independent of the actual observations. With the normalization of probabilities, \link{Gprime}{G9$'$} would then imply that the outcome of the next observation would be $a$ with probability $0.5$ and $b$ with probability $0.5$. This would be absurd. \hfill $\blacksquare$ \endproclaim The idea of this section is to introduce a formalism which allows the field theory $\F$ to adjust to match the observation of physical constants. However, each observation of a given constant will be probabilistically constrained, in a sense precisely defined by \link{Gprime}{G9$'$}, by all previous observations. Anthropic arguments show how such constants can often effectively be observed, and to quite surprising degrees of accuracy, by our mere existence. On the other hand, example \link{9.2}{9.2} indicates that if we were to permit $\omega$ to adjust to match the result of each new observation, then \link{Gprime}{G7$'$} -- \link{Gprime}{G9$'$} would produce probabilities in conflict with observed statistics. This problem can be avoided if, for each choice of $\F$, we can define a suitable limited set of fixed possibilities for $\omega$. However, unless this set can be given a simple and satisfying definition, there would be little point in introducing \link{Gprime}{G7$'$} -- \link{Gprime}{G9$'$} just to avoid the arbitrariness of an exact definition of constants. Simple definitions which avoid the problem raised by \link{9.2}{9.2} will correspond to initial conditions which are by and large purged of information. The most attractive proposal for $\V$ is that $\omega$ should be the (or a) ground state of $\F$ (\link{Peacock}{Tryon (1973)}). \link{Albert}{Albert (1988)} made such a proposal in the context of a many-minds interpretation but according to his theory the proposal was without empirical consequence. The present interpretation provides a much more sophisticated analysis of probability, and now the proposal is an empirical postulate about cosmology. It is about cosmology, because to discover $\omega$ we need to undo each observed ``wave-packet collapse''; we need to go back to the state of the universe prior to any observation. Although the present interpretation is based on the idea that only the observations of each separate observer are relevant, so that $\omega$ is only observed during the lifetime of an observer, this does not imply that $\omega$ can be modelled by the sort of state which we would conventionally assign to the universe at the moment of an observer's birth. Such a conventional assignment refers only to the ``observed world'' constructed by the observer, and many of the observed events in it, like supernovae explosions, are represented as being in the distant past. To undo the ``collapse'' by which we see a supernova explode at some particular instant in some particular galaxy, we need to go back in our observed world to before the observed event occured. The conclusion of this process is that the state assigned by cosmologists to the very early universe should be a satisfactory model for $\omega$. At least in inflationary cosmology, the idea that the initial state of fields other than gravity might be some sort of vacuum is certainly taken very seriously: ``a non-vacuum initial state contradicts the whole spirit of the maximally symmetric initial state of the Universe which lies at the heart of the inflationary scenario'' (\link{LPS}{Lesgourgues, Polarski, and Starobinsky (1997)}). Quite general maximally symmetric states might also be considered. For example, assuming a background spacetime with a sufficiently large symmetry group (e.g.~de~Sitter space -- \link{BorB}{Borchers and Buchholz (1999)}), one could consider the class of homogeneous and isotropic thermal equilibrium states for $\F$. It is of course not required, in the present theory, that $\omega$ should be a pure state, nor even that it should be ``ergodic'' in the sense of quantum statistical mechanics (\link{Peacock}{Ruelle (1969)}, \sect 6.3). On the largest visible scales, the universe does seem to be remarkably homogeneous. Indeed, explaining this apparent homogeneity is one of the central motivations for inflationary cosmology (\link{Peacock}{Peacock (1999)}, chapter 11). Accepting a homogeneous initial state, however, then calls for an explanation of the inhomogeneities which are so apparent all around us. This can be done on a descending hierarcy of scales. Scales larger than the visible universe may be relevant in theories of quantum gravity or of chaotic inflation. On these scales, it is possible to assume, if necessary, that our observations give us a symmetry-broken part of some homogeneous, or otherwise simple, total state. On the scale of the visible universe, inflation attempts to explain the existence of galaxies as quantum vacuum fluctuations ``frozen in'' by an early exponential expansion. Finally, on subgalactic scales, it is permissible to assume homogeneity even at the end of the inflationary era. This is because, in a quantum mechanical analysis without collapse, for almost all plausible states at that time, the current state of a galaxy, at a fixed radial distance from the centre, will be close to a homogeneous mixture and will need to be broken by observation into a specific pattern of stars, planets, life-forms, and events. This means that there is no advantage in not assuming that the initial state itself was homogeneous. The conventional historical description of the path which has lead from the early density fluctuation which gave rise to our galaxy to our existence on this particular planet involves many processes in which observed outcome prediction would have been impossible under a global quantum theory. These include fluid dynamical processes involving fragmentation and shock waves in unstable gas clouds, nuclear processing in turbulent stellar interiors, and the dispersal of the resulting nuclei by supernova explosions. Global quantum mechanical descriptions of all these processes would involve local decoherence, and a final state which, locally, contains almost no trace of any initial state, with the exception of large-scale thermodynamic parameters and the radial mass distribution. Implicit in a conventional account is a continual symmetry-breaking sequence of ``wave-packet collapses'' which keeps individual atoms well-localized. Decoherence by itself does not break symmetry; only decoherence plus collapse or observation. In a many-minds theory, apparent symmetry breaking can be a result of the requirement that an observer have a specific type of structure. Thus we can suppose that we see the specific constellations that we do, for the same reason that we see a particle in a bubble chamber move in a specific direction, even if that particle has arisen in a spherically-symmetric decay process. When cosmologists try to calculate whether, for example, observational evidence about the spectrum of inhomogeneities in the cosmic microwave background is compatible with particular hypotheses about the quantum field theory of the universe and its initial state, what they are doing can be interpreted as attempting to constrain the choice of $\V$. If cosmology were to provide clear evidence that a homogeneous initial state was unlikely to give rise to the sort of observations that we make -- for example, if the microwave background temperature had turned out to be strongly direction-dependent -- then it might be difficult to find a choice of $\V$ which would be both compatible with observation and simple to describe. At present however, there does not seem to be any such evidence. Given a suitable choice of $\V$, the conceptually radical change from \link{G7}{G7} and \link{G7}{G8} to \link{Gprime}{G7$'$}, \link{Gprime}{G8$'$}, and \link{Gprime}{G9$'$} seems unlikely to have directly observable consequences. This is because, under both scenarios, we describe our present observations in terms of the quantum field theory and the initial conditions which best describe our observations. In my opinion, \link{Gprime}{G7$'$} -- \link{Gprime}{G9$'$} constitutes the simpler and more attractive theory. In this section, the choice of $\V$ has been discussed under the assumption of the sort of fixed and symmetrical background spacetime in which equilibrium states can be defined. This assumption would surely fail in a complete quantum gravity theory. Nevertheless, as long as cosmology can provide us with a time in the early universe when the universal state can be given a simple description, it will remain plausible that a simple description may also be available for the initial states of any complete theory. The possibility of a simple description in the full theory may also be indicated by the apparent low entropy of the geometry of the early universe (\link{Peacock}{Penrose (1979)}). If \link{Gprime}{G7$'$}, \link{Gprime}{G8$'$}, and \link{Gprime}{G9$'$} hold, then the value of, for example, the fine structure constant is not determined, and, indeed, the fine structure constant does not have a value. There is no single unique quantum field theory $\F$ which governs the physical structure of every observer. Quantum field theory is observer-dependent. In fact, even given an observer, the supremum in \link{Gprime}{G8$'$} need not be attained at a unique $\F$. The physical universe as we know it has disappeared. It has become a projection from our experiences, rather than being the arena in which our experiences occur. Our experiences are possibilities allowed by a set of rules. They still give us a localized view of ``what is really happening'', but this view is much more parochial than we had led ourselves to believe. Not everything is illusory. A given human has a switching structure $s \in \SM$ and the possible futures of that structure are governed by probabilistic laws. $\SM$ and the probabilistic laws are discovered by trying to make sense of the history presented by $s$. To do this, we have to assume, as discussed in section \link{sec 6}{6}, that, given the probabilistic laws, $s$ is ``reasonably typical'' as an observer of its type. Although it is inevitable that such an assumption cannot be precisely defined, and although it is the nature of probability that it need not be true; nevertheless, the apparent success and consistency of the deduced laws of physics in explaining our observations does indicate why it would be reasonable to take such a theory seriously. \pdfproclaim{10. Conclusion.}{}{sec 10} \endproclaim An attempt has been made in this paper and its predecessors to present a technically complete and consistent version of a many-minds interpretation of special-relativistic quantum field theory. This attempt has revealed some of the issues which may be relevant to any such project. At present quantum theory would seem to be our best handle on the weirdness of reality. It is almost certainly a mistake to expect that that weirdness to be tamed with yet another search for those hidden variables which would return us once and for all to the classical physics of our innocence. But it would also be a mistake to assume that reality is so far beyond our comprehension that all speculation is idle. Nor should we give up trying to understand how the world is, merely because we might be faced with a range of different possibilities, between which we cannot decide. It is important to know what the options are and to test each separate option as far towards its destruction as our abilities allow. The empirical verification of an ever-widening range of aspects of quantum theory has continued for many decades. But a theory can be tested by other means than direct observation. If, for example, the incompleteness and inconsistency of Copenhagen quantum mechanics does not count towards its refutation, then it is hard to see what could. Of course we can wait indefinitely for something to turn up to explain the problems, but then we can also wait indefinitely for an explanation for any adverse experimental result. This paper is about progress in a many-minds interpretation. It demonstrates that the many-minds idea is not empty metaphysical speculation, by showing how the careful analysis of a detailed theoretical program can refine an interpretation and lead to deeper questions and wider issues. For example, the refinement of the intrepretation through a precise definition for individual probabilities requires the discussion of the many probabilistic notions differentiated in section \link{sec 6}{6}, and leads to the consideration of the details of neural processing in section \link{sec 8}{8}. Another example is the resolution of the trimming problem, referred to in section \link{sec 3}{3}. This requires the explanation, again in section \link{sec 8}{8}, of the existence of a ``present moment''. Finally, the attempt, in section \link{sec 9}{9}, to extend the breadth of the interpretation leads to contact with issues in cosmology. Many further issues are bound to arise, before full compatibility with a theory of quantum gravity can be attained. However, here the progress may go both ways, as the development of such a theory will almost certainly depend on the simultaneous development of a compatible interpretation of quantum theory. Everett's many-worlds idea has always been taken fairly seriously because it seems to fit quite naturally into the mathematics of elementary quantum mechanics. The current work tests the idea. The test has not been failed, and progress has been made. \bigskip \noindent{\bf Acknowledgements } I am grateful to Guido Bacciagaluppi, Katherine Brading, Jeremy Butterfield, Talal Debs, Doreen Fraser, Peter Morgan, Oliver Pooley, Simon Saunders, and Christopher Timpson for useful conversation and comments. \bigskip \vfill \eject \parindent = 0pt \def\dent{\leavevmode\hbox to 20pt{}} \pdfproclaim{Appendix \hfill A Hypothesis. \hfill \hphantom{Appendix}}{}{hypothesis} \endproclaim \bigskip \dent The structure of a mind, at a given moment, can be described by a minimal switching structure $S(M, N, [d, \varphi])$. \bigskip \name{A} {\bf A}\quad A minimal ordered switching structure $SO(M, N, d, \varphi)$ is given by: \medskip A1) Two positive integers $M$ (the number of determinations of switch status) and $N$ (the number of switches). A2) An $M$-component ascending docket $d$. (This defines the spacetime relations between determinations.) \dent (A docket is a geometrical structure in spacetime defined as an equivalence class of ordered sequences $(A_i)_{i=1}^M$ of suitable spacetime sets. Two such sequences $(A_i)_{i=1}^M$ and $(B_i)_{i=1}^M$ will have the same docket if they have the same spacetime, or causal, arrangement -- in other words, if, for every pair $i$, $j$, $B_i$ is in the past of/spacelike to/in the future of $B_j$ exactly when $A_i$ is in the past of/spacelike to/in the future of $A_j$ -- and if one sequence can be continuously deformed into the other while the arrangement is essentially unaltered. A docket $d$ is ``ascending'' if and only if $(A_i)_{i=1}^M \in d$ and $i < j$ implies that $A_i$ is not in the strict timelike future of $A_j$.) A3) A function $\varphi$ from $\{1, \dots, M \}$ onto $\{ \pm 1, \dots, \pm N\}$. ($|\varphi(m)| = n$ is to be interpreted as meaning that the $m^{th}$ determination is a determination of the status of switch $n$. The two possible statuses of that switch are represented by the sign of $\varphi(m)$.) A4) Write $|\varphi|^{-1}(n) = \{j_n(k) : k = 1, \dots, K_n\}$, where $j_n(1) < j_n(2) < \dots < j_n(K_n)$. (Determination number $j_n(k)$ is the $k^{th}$ determination of the status of switch $n$. We shall write $j(\varphi)_n(k)$ in place of $j_n(k)$ in B to show the dependence on $\varphi$.) \name{A5} A5) For each $n \in \{1, \dots, N\}$, $K_n \geq 4$ and there exist $k_1, k_2, k_3, k_4$ with $1 \leq k_1 < k_2 < k_3 < k_4 \leq K_n$ such that $\varphi(j_n(k_1)) = -\varphi(j_n(k_2)) = \varphi(j_n(k_3)) = -\varphi(j_n(k_4))$. (A switch must open and close at least twice if all the constraints imposed below are to be brought into play.) \bigskip \name{BB} {\bf B}\quad Given $M$, $N$, $d$, and $\varphi$ as in A, define a minimal ordered switching structure $SO(M', N', d', \varphi')$ to be an immediate ordered successor of $SO(M, N, d, \varphi)$ if and only either B1 or B2 holds: B1) $N' = N$ and $M' = M + 1$ (there is a single new determination of status on an existing switch). There exists an order-preserving map $f : \{1, \dots, M\} \rightarrow \{1, \dots, M'\}$ ($m_1 < m_2 \implies f(m_1) < f(m_2)$) such that, for $m \leq M$, $\varphi'(f(m)) = \varphi(m)$ ($f$ preserves switch numbers and their statuses), and there exists a sequence $(A_i)_{i=1}^{M+ 1} \in d'$ such that $(A_{f(i)})_{i=1}^{M} \in d$ (the geometric relations of the existing determinations are unchanged). B2) $N' = N +1$, $M' = M + 4$, and there exists an order-preserving map $f : \{1, \dots, M\}$ $\rightarrow \{1, \dots, M'\}$ and a sequence $(A_i)_{i=1}^{M'} \in d'$ such that, for $m \leq M$, $\varphi'(f(m)) = \varphi(m)$ and $(A_{f(i)})_{i=1}^{M} \in d$. (In this case, a new switch is introduced.) \pdfeject \name{B3} B3) The minimal switching structure $S(M, N, [d, \varphi])$ is defined by $$\displaylines{ S(M, N, [d, \varphi]) = \{SO(M, N, d', \varphi') : d' \text{ is ascending and there exist permutations} \hcrh \text{ $\pi$ on $M$ elements and $\pi'$ on $N$ elements } \crh \text{ such that $d' = d^\pi$, $\varphi' = \pi' \circ \varphi \circ \pi$, } \crh \text{ and $\pi(j(\varphi')_{\pi'(n)}(k)) = j(\varphi)_n(k)$ for each $n$ and $k$.} }$$ ($d^\pi$ is the docket defined by $(A_i)_{i=1}^{M} \in d \iff (A_{\pi(i)})_{i=1}^{M} \in d^\pi$. B3 identifies structures which differ only in the labels assigned to switches or to determinations.) B4) Define a minimal switching structure $S(M', N', [d', \varphi'])$ to be an immediate successor of $S(M, N, [d, \varphi])$ if and only if there exist $SO(M, N, d, \varphi) \in S(M, N, [d, \varphi])$ and $SO(M', N', d', \varphi') \in S(M', N', [d', \varphi'])$ such that $SO(M', N', d', \varphi')$ is an immediate ordered successor of $SO(M, N, d, \varphi)$. \dent Let $\Xi(M, N, d, \varphi)$ denote the set of immediate successors of $S(M, N, [d, \varphi])$. \bigskip \name{C} {\bf C}\quad The geometrical manifestations of $SO(M, N, d, \varphi)$ comprise the set \newline $GSO(M, N, d, \varphi)$ of all sequences $$W = (x, \Lambda, \theta, (T_n, (t_{nk})^{K_n}_{k=1}, (t'_{nm})_{m = m_n^i}^{m_n^f}, x^n(t), L^n(t), P_n, Q_n)^N_{n=1})$$ such that, for $n = 1, \dots, N$, C1) $x \in \Lambda \subset \Minkowski$ (Minkowski space). $\Lambda$ is a spacetime retract. (At any moment, any switch occupies a Poincar\'e transform of the set $\Lambda$. The requirement that $\Lambda$ be a spacetime retract is a weak restriction on $\Lambda$ related to the definition of the docket $d$.) C2) $0 \leq t_{n 1} < t_{n2} < \dots < t_{n K_n} \leq T_n$. We shall write $S_n$ for $t_{n 1}$. (The parameter on the path of switch $n$ runs from $0$ to $T_n$. $t_{n k}$ is the parameter time on that path of the $k^{th}$ determination of the switch status. The switch is only ``active'' from $S_n$ but the path is extended back to parameter $0$ to allow comparison between switches.) C3) $1 \leq m_n^i \leq m_n^f$. $S_n = t'_{n m_n^i} \leq t'_{n (m_n^i+1)} \leq \dots \leq t'_{n m_n^f} \leq T_n$. Set $t'_{n(m_n^f + 1)} = T_n$. (For $m = m_n^i , \dots, m_n^f$, the state of switch $n$ is $\sigma_{m}$ from parameter time $t'_{nm}$ until $t'_{n (m +1)})$). \name{C4} C4) The $x^n(t)$ are continuous paths in $\Minkowski$ defined for $t \in [0, T_n]$ and with $x^n(0) = x$. ($x^n(t)$ defines the translational motion of switch $n$.) C5) The $L^n(t)$ are continuous paths in ${\cal L}^\uparrow_+$ (the restricted Lorentz group) defined for $t \in [0, T_n]$, having a right derivative $L^n\,'(t^+)$ for $t \in [0, T_n)$, and with $L^n(0) = 1$. ($L^n(t)$ defines the rotations and boosts of switch $n$.) C6) For $m = m_n^i , \dots, m_n^f$ and $t \in [t'_{nm}, t'_{n(m+1)}]$, $$x^n(t) = x^n(t'_{nm}) + \int_{t'_{nm}}^t L^n(s) u^n(t'_{nm}) ds \text{ where $u^n(t'_{nm})$ is a four-vector.}$$ (This implies that $\dsize{{dx^n\over dt}(t) = L^n(t) u^n(t'_{nm})}$.) C7) The $u^n(t'_{nm})$ are timelike, future directed, and $(u^n(t'_{nm}))^2 = -1$. (It follows from C6 that $u^n(t) = \dsize{{dx^n\over dt}(t)}$ has the same properties and that the path $x^n$ is timelike, future directed, and parametrized by proper time $t$.) \pdfeject \name{C8} C8) For $m = m_n^i , \dots, m_n^f$, $x^n(t'_{n m})$ is in the closure of the causal future of at least $m$ members of $\{x^{n'}(t_{n'k'}) : n' = 1, \dots N, k' = 1, \dots, K_{n'}\}$. (The ``collapses'' of the quantum state are ``caused'' by the determinations and ordered by those causes.) C9) Set $\Lambda_n(t) = \{ x^n(t) + L^n(t)(y - x) : y \in \Lambda \}$ for $t \in [0, T_n]$. Set $A_{j_n(k)} = \Lambda_n(t_{nk})$. \dent Then $(A_m)^M_{m=1}$ has docket $d$. ($\Lambda_n(t)$ is the spacetime set occupied by switch $n$ at parameter time $t$, and $A_{j_n(k)}$ is the spacetime set which it occupies at the moment of its $k^{th}$ determination.) C10) For $1 \leq m', m'' \leq M$, if $y' \in A_{m'}$ and $y'' \in A_{m''}$ are spacelike separated, then there is a spacelike path from $y'$ to $y''$ in $\{ y \in \Lambda_{n'}(t) : n' = 1, \dots, N, \ t \in [S_{n'}, T_{n'}] \}$. (Individuals are spatially connected.) C11) For any $t \in [S_n, T_n]$, the number of elements of $$\{n' : n' \ne n \text{ and for some } t' \in [S_{n'}, T_{n'}],\ \Lambda_n(t) \cap \Lambda_{n'}(t') \ne \emptyset \}$$ (the number of switches whose paths hit $\Lambda_n(t)$) is bounded by a constant $C$ -- the ``contact number''. $C = 13$. C12) $\theta : \{1, \dots, N\} \rightarrow \{(n', k', k''): n' = 1, \dots, N, 1 \leq k' < k'' \leq K_{n'}\}$ such that writing $\theta(n) = (n', k', k'')$, we have $n' \ne n$ if $N > 1$, and $\varphi(j_{n'}(k')) = -\varphi(j_{n'}(k''))$. Then, for some $ t' \in [S_{n'}, T_{n'}]$, $A_{j_{n}(1)}$ is neither in the strict future nor the strict past of $\Lambda_{n'}(t')$. (In \link{F}{F5}, the states of switch $n$ will be required to be similar to those of switch $\theta(n)$.) \name{C13} C13) If $k_4, k_5 \in \{1, \dots, K_n\}$ with $k_4 < k_5$ and $\varphi(j_n(k_4)) = \varphi(j_n(k_5))$ then there exist $k_1, k_2, k_3 \in \{1, \dots, K_n\}$ with $k_1 < k_2 < k_3$ and $\varphi(j_n(k_1)) = -\varphi(j_n(k_2)) = \varphi(j_n(k_3))$ such that $t_{n{k_5}} - t_{n{k_4}} \geq {1\over 2}(t_{n{k_3}} - t_{n{k_1}})$. (A status on a given switch cannot be redetermined until after the elapse of a proper time at least as large as half the minimum cycle time.) C14) $P_n$ and $Q_n$ are projections in $\A(\Lambda)$ with $P_n$ orthogonal to $Q_n$. \bigskip \name{D} {\bf D} D1) Let ${\cal C}(W)$ be the von Neumann algebra generated by $$\displaylines{ \{\tau_{(x^n(t_{n k})-L^n(t_{n k})x, L^n(t_{n k}))}(P_n), \tau_{(x^n(t_{n k})-L^n(t_{n k})x, L^n(t_{n k}))}(Q_n) \hcrh : k = 1, \dots, K_n , n = 1, \dots, N\}. }$$ (This is the algebra of correlations of switch projections experienced by the observer. If $(x, L)$ is a Poincar\'e transformation which acts to send the spacetime set $\Lambda$ to $(x, L) \Lambda = \{x + Ly : y \in \Lambda\}$, then $\tau_{(x,L)}$ is the corresponding transformation on observables, so that, for $A \in \A(\Lambda)$, $\tau_{(x,L)}(A) \in \A((x,L)\Lambda)$. Transformations on states $\sigma$ will be defined so that $\tau_{(x,L)}(\sigma)(\tau_{(x,L)}(A)) = \sigma(A)$, implying that if $\sigma$ is a state on $\A((x,L)\Lambda)$, then $\tau_{(x,L)}^{-1}(\sigma)$ is a state on $\A(\Lambda)$.) D2) Let $\B(W)$ be the norm closure of the linear span of $$\{ A_1 C_1 + C_2 A_2 : A_1, A_2 \in \A(\Lambda_n(t)), C_1, C_2 \in {\cal C}(W), t \in [S_n, T_n], n= 1, \dots, N\}. $$ (This is the set of all observables accessible to the observer. Elements of ${\cal C}(W)$ are correlated with local observables along the paths of the switches.) D3) A quantum state $\rho$ on $\B(\H)$ -- the set of all bounded operators on a Hilbert space $\H$ -- is a positive linear functional of unit norm on the set $\B(\H)$. (Thus $\rho(A A^*) \geq 0$ for all $A \in \B(\H)$ and $\rho(1) = 1$. A density matrix $\rho = \sum_{n=1}^\infty r_n |\psi_n\>\<\psi_n|$ defines a state (referred to as a ``normal'' state) on $\B(\H)$ by $\rho(A) = \tr(\rho A) = \sum_{n=1}^\infty r_n \<\psi_n| A |\psi_n\>$.) For $\B \subset \B(\H)$, a state $\rho$ on $\B$ is the restriction to $\B$ of some state $\rho'$ on $\B(\H)$. (This is written $\rho = \rho'|_\B$.) \bigskip \dent E defines the set of sequences of states for which $x^n(t)$ is the path along which change of state is locally minimized. E1 requires that the states be such that the initial conditions $u^n(t'_{nm})$ and $L^n\,'(t'_{nm}{}^+)$ are optimal and E2 requires that the continuation at parameter $t$ is optimal. \medskip \name{E} {\bf E}\quad ${\cal N}(W, E)$ is the set of all sequences of restrictions to $\B(W)$ of sequences of quantum states $(\sigma_m)^M_{m=1}$ which satisfy the following requirements for each $n \in \{1, \dots, N\}$ and each $m$ such that $m_n^i \leq m \leq m_n^f$, E1) Set $$ \eqalign{ X^{nm} = \{(L,& v) : L \text{ is a $C^1$ path in ${\cal L}^\uparrow_+$ on some interval} \cr &[t'_{nm}, t'_{nm} + \varepsilon) \text{ with $\varepsilon > 0$ and with $L(t'_{nm}) = L^n(t'_{nm})$, and} \cr &v \text{ is a future-directed four-vector satisfing } (v)^2 = -1\}. \cr}$$ For $(L, v) \in X^{nm}$, define $f_{nm}(s, L, v) = \tau^{-1}_{(y^{nm}(s, L, v), L(s))}(\sigma_m)|_{\A(\Lambda)}$ where $$y^{nm}(s, L, v) = x^n(t'_{nm}) + \int_{t'_{nm}}^s L(s') v ds' - L(s) x.$$ Then we require that $f_{nm}(s, L^n, u^n(t'_{nm}))$ has a right derivative at $s = t'_{nm}$ and that $$\inf\{\ \limsup_{h\rightarrow0^+}||( f_{nm}(t'_{nm}+h, L, v) - f_{nm}(t'_{nm}, L, v))/h\ || : (L, v) \in X^{nm} \}$$ is attained when $L'(t'_{nm}{}^+) = L^n\,'(t'_{nm}{}^+)$ and $v = u^n(t'_{nm})$. E2) For each $t \in (t'_{nm}, t'_{n(m+1)})$, set $$ \displaylines{ X^n_t = \{L : L \text{ is a $C^1$ path in ${\cal L}^\uparrow_+$ on some interval $[t, t +\varepsilon)$ with}\cr \varepsilon > 0, \text{ and } L(t) = L^n(t)\}. }$$ \dent ($X^n_t$ and $X^{nm}$ could be replaced by finite dimensional sets defined in terms of the Lie algebra of ${\cal L}^\uparrow_+$.) For $L \in X^n_t$, define $f_t(s, L) = \tau^{-1}_{(y^n_t (s, L), L(s))}(\sigma_{m})|_{\A(\Lambda)}$ where $$y^n_t(s, L) = x^n(t) + \int_t^s L(s')u^n(t'_{nm}) ds' - L(s)x.$$ Then we require that $f_t(s, L^n)$ has a right derivative at $s = t$ and that $$\inf\{\ \limsup_{h\rightarrow0^+} ||(f_t(t+h, L) - f_t(t, L))/h\ || : L \in X^n_t \}$$ is attained when $L'(t^+) = L^n\,'(t^+)$. \bigskip \dent F1 -- F4 are the formal expression for the idea that ``A switch is something spatially localized, the quantum state of which moves between a set of open states and a set of closed states, such that every open state differs from every closed state by more than the maximum difference within any pair of open states or any pair of closed states.'' F5 is a requirement of ``homogeneity''; allowing only for gradual change between different switches. \medskip \name{F} {\bf F}\quad Set $$\sigma_{nk} = \tau^{-1}_{(x^n(t_{nk})-L^n(t_{nk})x, L^n(t_{nk}))}(\sigma^n(t_{nk}))|_{\A(\Lambda)} $$ where $\sigma^n(t)$ is the state of switch $n$ at parameter time $t$, which is defined to be $\sigma_{m^n(t)}$ for $m^n(t) = \sup\{m' \leq m_n^f: t \geq t'_{nm'}\}$. Then ${\cal N}(W)$ is the subset of ${\cal N}(W, E)$ consisting of sequences $(\sigma_m)^M_{m=1}$ such that, for each $n \in \{1, \dots, N\}$ and for $k$, $k' \in \{1, \dots, K_n\}$, \smallskip F1) $\sigma_{nk}(P_n) > {1\over 2}$ \vadjust{\kern1pt} for $\varphi(j_n(k)) > 0$. (``a set of open states'') F2) $\sigma_{nk}(Q_n) > {1\over 2}$ for $\varphi(j_n(k)) < 0$. (``a set of closed states'') F3) $|\sigma_{nk}(P_n) - \sigma_{nk'}(P_n)| > {1\over 2}$ and $|\sigma_{nk}(Q_n) - \sigma_{nk'}(Q_n)| > {1\over 2}$ for all pairs $k$ and $k'$ such that $\varphi(j_n(k)) \varphi(j_n(k')) < 0$. (``every open state differs from every closed state'') F4) There is no triple $(P,k,k')$ with $P \in \A(\Lambda)$ a projection and $k$ and $k'$ satisfying $\varphi(j_n(k)) \varphi(j_n(k')) > 0$ such that $|\sigma_{nk}(P) - \sigma_{nk'}(P)| \geq {1\over 2}$. (``by more than the maximum difference within any pair of open states or any pair of closed states.'') F5) If $\theta(n) = (n', k', k'')$, then there are no projections $P \in \A(\Lambda)$ such that either $|\sigma_{n1}(P) - \sigma_{n' k'}(P)| \geq {1\over 2}$ or $|\sigma_{n2}(P) - \sigma_{n' k''}(P)| \geq {1\over 2}$. \bigskip \name{GG} {\bf G} G1) The set of manifestations of the minimal switching structure $S(M, N, [d, \varphi])$ is $$\displaylines{ \{((\sigma_m)^M_{m=1}, W) : W \in GSO(M, N, d', \varphi') \text{ and } (\sigma_m)^M_{m=1} \in {\cal N}(W), \hcrh \text{ for } SO(M, N, d', \varphi') \in S(M, N, [d, \varphi])\}. }$$ \name{G2} G2) For $\sigma$ and $\rho$ quantum states on a set of operators $\B$, a function $\app{\B}{\sigma}{\rho}$ which gives ``the probability, per unit trial of the information in $\B$, of being able to mistake the state of the world on $\B$ for $\sigma$, despite the fact that it is actually $\rho$'' is defined by $$\app{\B}{\sigma}{\rho} = \exp\{ \ent{\B}{\sigma}{\rho} \}$$ where $\ent{\B}{\sigma}{\rho}$ (the relative entropy of $\sigma$ with respect to $\rho$ on $\B$) is the unique function satisfying $$\displaylines{ \rlap{a)}\hfill \ent{\B(\H)}{\sigma}{\rho} = \tr(-\sigma \log \sigma + \sigma \log \rho) \hfill \text{for $\sigma$ and $\rho$ normal states on $\B(\H)$.} }$$ $$\displaylines{ \rlap{b)} \hfill \ent{\B(\H)}{\sigma}{\rho} = \inf\{ F(\sigma, \rho) : F \text{ is w* upper semicontinuous, concave, and given} \crh \text{ by a) for $\sigma$ and $\rho$ normal} \}. }$$ (This is a natural extension of a) to the closure of the set of density matrices in the $w^*$-topology.) $$\ent{\B}{\sigma}{\rho} = \sup\{ \ent{\B(\H)}{\sigma'}{\rho'} : \sigma'|_\B = \sigma \text{ and } \rho'|_\B = \rho \}. \leqno{c)}$$ G3) The a priori probability of a sequence $(\sigma_m)^M_{m=1}$ of states on a set $\B$, given an initial state $\omega$, is defined by $$\app{\B}{(\sigma_m)^M_{m=1}}{\omega} = {\textstyle\prod\limits^M_{m=1}} \app{\B}{\sigma_m}{\sigma_{m-1}} \qquad \text{where $\sigma_0 = \omega$.}$$ G4) For $m = 1, \dots, M$, define $${\cal N}^m(W) = \{(\sigma_i)^m_{i=1} : \exists (\sigma_i)^M_{i=m+1} \text{ with } (\sigma_i)^M_{i=1} \in {\cal N}(W)\}.$$ \pdfeject \name{G5} G5) Define, by induction on $m$, the following a priori probabilities. Start with $$\mathop{\rm app}({\cal N}(W), \B(W), 1, \omega) = \sup\{ \app{\B(W)}\sigma\omega : \sigma \in {\cal N}^1(W)\}.$$ Then, for $1 < m+1 \leq M$, set $$\displaylines{ \mathop{\rm app}({\cal N}(W), \B(W), m+1, \omega) = \sup\{ \limsup_{n\rightarrow\infty}\, \app{\B(W)}{(\sigma^n_i )^{m+1}_{i=1}}{\omega} : ((\sigma^n_i )^{m+1}_{i=1})_{n\geq1} \text{ is} \hfill \cr \hfill \text{a sequence of elements of } {\cal N}^{m+1}(W) \text{ and, for }1 \leq k \leq m, \cr \hfill \app{\B(W)}{(\sigma^n_i )^{k}_{i=1}}{\omega} \rightarrow \mathop{\rm app}({\cal N}(W), \B(W), k, \omega) \}. }$$ G6) Define the a priori probability $\app{}{W}{\omega}$ of existence of an individual geometric manifestation $W \in GSO(M, N, d, \varphi)$ by $$\app{}{W}{\omega} = \mathop{\rm app}({\cal N}(W), \B(W), M, \omega).$$ \name{G7} G7) Define the a priori probability for the minimal switching structure $S(M, N, [d, \varphi])$ to be $$\displaylines{ \app{}{S(M, N, [d, \varphi])}{\omega} = \sup\{ \app{}{W}{\omega} : W \in GSO(M, N, d', \varphi') \hcrh \text{ where } SO(M, N, d', \varphi') \in S(M, N, [d, \varphi])\}. }$$ (This takes account of the re-labellings allowed by \link{B3}{B}.) G8) Use G7 to define a classical discrete Markov process on the space of minimal switching structures $S(M, N, [d, \varphi])$. $$\text{Set }\xi = \sum \{\app{}{S(M', N', [d', \varphi'])}{\omega}: S(M', N', [d', \varphi']) \in \Xi(M, N, d, \varphi)\}.$$ \dent Define the probability of moving from $S(M, N, [d, \varphi])$ to an immediate successor $S(M', N', [d', \varphi'])$ $$\displaylines{ \text{to be } \hfill \app{}{S(M', N', [d', \varphi'])}{\omega}/\xi, \hfill \text{if } \xi \geq \app{}{S(M, N, [d, \varphi])}{\omega}, \cr \text{and to be } \hcr \app{}{S(M', N', [d', \varphi'])}{\omega}/ \app{}{S(M, N, [d, \varphi])}{\omega}, \hfill \text{ if } \xi < \app{}{S(M, N, [d, \varphi])}{\omega}. }$$ \dent Define the probability of extinction to be $0$ if $\xi \geq \app{}{S(M, N, [d, \varphi])}{\omega}$, and to be $1 - \xi/\app{}{S(M, N, [d, \varphi])}{\omega}$ otherwise. \bigskip \vfill \eject \pdfproclaim{References.}{}{Ref} \endproclaim \frenchspacing \parindent=0pt \everypar={\hangindent=0.75cm \hangafter=1} \name{Albert} Albert, D.Z. (1988) ``On the possibility that the present quantum state of the universe is the vacuum.'' pp 127--133 of {\sl Proceedings of the 1988 Biennial Meeting of the Philosophy of Science Association, Vol. 2,} ed. A. Fine and J. Leplin. (P.S.A.). \name{BDV} Bacciagaluppi, G., Donald, M.J., Vermaas, P.E. (1995) ``Continuity and discontinuity of definite properties in the modal interpretation.'' {\sl Helv. Phys. Acta \bf 68}, 679--704. Barbour, J.B. (1994) ``The timelessness of quantum gravity: I, II.'' {\sl Class. Quant. Grav. \bf 11}, 2853--2897. Barrow, J.D. and Tipler, F.J. (1986) {\sl The Anthropic Cosmological Principle.} (Oxford) \name{Bell} Bell, J.S. (1975) ``On wave packet reduction in the Coleman-Hepp model.'' {\sl Helv. Phys. Acta \bf 48}, 93--98. Reprinted, pp 45--51 of Bell (1987). Bell, J.S. (1976) ``The measurement theory of Everett and de Broglie's pilot wave.'' pp 11--17 of {\sl Quantum Mechanics, Determinism, Causality, and Particles,} ed. M. Flato et al.~(Reidel). Reprinted, pp 93--99 of Bell (1987). Bell, J.S. (1981) ``Quantum mechanics for cosmologists.'' pp 611--637 of {\sl Quantum Gravity 2,} ed. C. Isham et al.~(Oxford). Reprinted, pp 117--138 of Bell (1987). Bell, J.S. (1987) {\sl Speakable and Unspeakable in Quantum Mechanics.} (Cambridge) \name{BorB} Borchers, H.-J. and Buchholz, D. (1999) ``Global properties of vacuum states in de Sitter space.'' {\sl Ann. Inst. H. Poincar\'e, Phys. Th. \bf 70}, 23--40. {\sl gr-qc/9803036} \name{Butterfield} Butterfield, J. (1996) ``Whither the minds?'' {\sl Brit. J. Phil. Sci. \bf 47}, 200--221. DeWitt, B.S. and Graham, N. (1973) {\sl The Many-Worlds Interpretation of Quantum Mechanics.} (Princeton) \name{otre} Donald, M.J. (1986) ``On the relative entropy.'' {\sl Commun. Math. Phys. }{\bf 105}, 13--34. \name{qtab} Donald, M.J. (1990) ``Quantum theory and the brain.'' {\sl Proc. R. Soc. Lond. \bf A 427}, 43--93. \name{appalo} Donald, M.J. (1992) ``A priori probability and localized observers.'' {\sl Foundations of Physics, \bf 22}, 1111--1172. \name{psoo} Donald, M.J. (1995) ``A mathematical characterization of the physical structure of observers.'' {\sl Foundations of Physics \bf 25}, 529--571. \name{ommi} Donald, M.J. (1997) ``On many-minds interpretations of quantum theory.'' \newline {\sl quant-ph/9703008}. Dowker, F. and Kent, A. (1996) ``On the consistent histories approach to quantum mechanics.'' {\sl J. Stat. Phys. \bf 82}, 1575-1646. {\sl gr-qc/9412067}. \name{GJKKSZ} Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., and Zeh, H.D. (1996) {\sl Decoherence and the Appearance of a Classical World in Quantum Theory.} (Springer) Griffiths, R.B. (1998) ``Choice of consistent family, and quantum incompatibility.'' {\sl Phys. Rev. \bf A 57}, 1604. {\sl quant-ph/9708028}. Haag, R. (1992) {\sl Local Quantum Physics.} (Springer) Hartle, J.B. (1968) ``Quantum mechanics of individual systems.'' {\sl Amer. J. Phys. \bf 36}, 704--712. Hepp, K. (1972) ``Quantum theory of measurement and macroscopic observables.'' {\sl Helv. Phys. Acta \bf 45}, 237--248. \pdfeject \name{LPS} Lesgourgues, J., Polarski, D., Starobinsky, A.A. (1997) ``Quantum-to-classical transition of cosmological perturbations for non-vacuum initial states.'' {\sl Nucl. Phys. \bf B497}, 479--508. {\sl gr-qc/9611019}. Lockwood et al.~(1996) Symposium: ``Many-minds interpretations of quantum mechanics.'' {\sl Brit. J. Phil. Sci. \bf 47}, 159--248 and {\sl Brit. J. Phil. Sci. \bf 47}, 445--461. Mermin, N.D. (1998) ``What is quantum mechanics trying to tell us?'' {\sl Amer. J. Phys. \bf 66}, 753--767. {\sl quant-ph/9801057}. Namiki, M., Pascazio, S., and Nakazato, H. (1997) {\sl Decoherence and Quantum Measurements.} (World Scientific) Papineau, D. (1995) ``Probabilities and the many minds interpretation of quantum mechanics.'' {\sl Analysis, \bf 55,} 239--246. \name{Peacock} Peacock, J.A. (1999) {\sl Cosmological Physics.} (Cambridge) Penrose, R. (1979) ``Singularities and time-asymmetry.'' pp 581--638 of {\sl General Relativity: An Einstein Centenary Survey,} ed. S.W. Hawking and W. Israel. (Cambridge). Ruelle, D. (1969) {\sl Statistical Mechanics.} (Benjamin) Saunders, S. (1996) ``Reply to Michael Lockwood.'' {\sl Brit. J. Phil. Sci. \bf 47}, 241--248. Tryon, E.P. (1973) ``Is the universe a vacuum fluctuation?'' {\sl Nature }{\bf 246}, 396--397. Wheeler, J.A. (1957) ``Assessment of Everett's `relative state' formulation of quantum theory.'' {\sl Rev. Mod. Phys. \bf 29}, 463--465. Reprinted, pp 151--153 of DeWitt and Graham (1973). \end