A Many-Minds Interpretation Of Quantum Theory

Matthew J. Donald

The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great Britain.

e-mail: mjd1014@cam.ac.uk

home page: http://people.bss.phy.cam.ac.uk/~mjd1014

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A Brief Summary

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Some Questions

Some Proposed Answers

Defining Patterns of Information

Probabilities

Time and Other Minds

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Some Questions

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“If they can get you asking the wrong questions, they don't have to worry about the answers.” (Thomas Pynchon)

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Quantum theory is highly successful in explaining properties of classes of systems:

e.g.

chemistry — molecular binding energies

optics — frequency-dependent susceptibilities

superconductivity — energy gaps

nuclear magnetic resonance — chemical shifts

particle physics — scattering cross-sections

cosmology — helium abundance

but many questions arise:

What does quantum theory tell us about the nature of reality?

Is quantum theory universally valid?

Can quantum theory describe individual events?

Can quantum theory be applied consistently at the macroscopic level?

Is an algorithmic treatment of measurement theory possible?

Is it possible to provide an interpretation of quantum theory which is compatible with special relativity/ general relativity/ quantum field theory/ this week's theory of everything?

Is the wave-function of the entire universe re-determined every time a quantum experiment is performed by one of the little green men who lives on planet Tenalp which lies 240 million light years away in the direction of Andromeda?

What time is that anyway?

Is it possible to provide a theory of measurement involving local rather than global state changes?

What becomes of causality in such a theory?

What parts of the universal wave-function are pure?

There are something like ten to the power twenty-five water molecules in a human brain. Is each separate molecule well-localized, and if so, how well, or equivalently, how often is it relocalized?

In a bubble chamber experiment, are we measuring particle positions or bubble positions? — or the state of our own visual cortex?

What is being measured when a brain observes itself?

What is an observer?

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Some Proposed Answers

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The central problem of the interpretation of quantum theory is to explain and characterize the existence, or apparent existence, of state “collapse”.

State collapse is the process by which, for example, an electron, despite going through a double slit as an extended wave, always appears to make a well-localized impact on a screen at only one of many possible places — the extended wave “collapses” to a localized state. State collapse is also referred to as “wave-packet reduction” or “von Neumann's process 1”.

A characterization of collapse should tell us what possibilities arise — “the preferred basis problem”. The characterization should be well-defined and unambiguous. It should be explained how one collapse leads on to the next. The probability of a given collapse should be well-defined.

Many-worlds theories assume the existence of a “universal wave-function” which does not collapse. The problem then is to explain the appearance of collapse, but the same problems of definite characterization arise.

In Everett's original many-worlds theory, each “world” was a way in which an “observer” could be correlated with the rest of the universe, but Everett did not analyse the nature of observers, so that his theory was incomplete.

I propose a complete many-worlds theory, which does provide an explicit analysis of the structure of observers.

It is a many-minds interpretation because an “observer” is taken to be something which has a “mind”, and different possibilities correspond to different experiences of individual observers.

For each observer, life is like a game of chance in which at any moment, a finite range of possibilities can occur; each possibility having its own associated probability. Finiteness is essential if this probabilistic structure is to make sense.

It is proposed that an observer is a processor of information and that each possibility for an observer is constituted by a definite pattern of information. However, an observer is not defined by an instantaneous pattern, nor by the instantaneous “dispositions” of his current structure, but rather by the total pattern which has developed over his entire life to date.

A major part of the work in developing the theory is concerned with how patterns of information are physically expressed. The central issue here is that it does not seem possible to solve the preferred basis problem directly. In particular, it seems most unlikely that any plausible characterization of a single observer could have a unique physical manifestation.

I deal with this problem by defining the life of an observer, up to a given moment, by a unique abstract pattern. The probability of that pattern is then defined through an uncountable set of possible physical manifestations in which continuous variations are allowed.

Thus, ultimately, I see physical reality just as that which provides the probabilities of our existence.

In much of my writing, there is a tension between a final fully-consistent but highly abstract analysis and a preliminary analysis posed in comprehensible terms. The preliminary analysis may describe, for example, a quantum state for a physical brain in the sort of physical world that we see around us. Ultimately, however, this is to be superseded by an understanding of such apparent worlds as mere appearance; merely a reflection of the abstract information-bearing structures which constitute our minds.

In philosophical terms, my final proposal is a form of idealism: minds exist and the probabilities of their development are determined by immutable laws, defined using quantum theoretical concepts.

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Defining Patterns of Information

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A pattern of information is expressed by a pattern of elementary localized events. Each of these events is a determination of the status (“open” or “closed”) of one of a finite number of switches.

In the preliminary analysis (as described above), these switches are physical. In the final analysis, they are abstract and they have many possible physical manifestations.

The paradigm of a pattern of information is, of course, the human observer.

As a first approximation, the essential functioning of the human brain can be seen as the processing of information through the pattern made by the switching of a finite number of neurons between two possible statuses: “firing” and “not-firing”.

A more precise version of a preliminary model of human information processing is

The Geometrical Neuronal Model: The information processed by a brain can be perfectly modelled by a three-dimensional structure consisting of a family of switches, which follow the paths of the brain's neurons, and which open and close whenever those neurons fire.

This model fails for two reasons:

Because too much detail is required to specify an exact spacetime path.

And because neurons are much too complex to be quantum mechanically elementary.

To attack the first of these problems, a pattern of elementary localized events is defined in terms of the spacetime relations between the events. This leads to the introduction of a set of finite structures (dockets) which have interesting geometrical and combinatorial properties.

To attack the second problem, the following abstract definition of a quantum mechanical switch is introduced:

A Quantum 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.

In this definition and elsewhere, “state” is taken to mean density matrix (rather than wave-function). This is because we always work with limited parts of the universe — “local systems” — such as human beings; and to allow for the warm wet environment of the human brain; and for compatibility with local relativistic quantum field theory.

Because of the level of abstraction at which observers are ultimately defined, in fact it becomes necessary to work with sets of density matrices rather than with individual states.

A Neural Quantum Switch is an entity in the human nervous system which satisfies the definition of a quantum switch and which has switching behaviour strongly correlated with the firing of some neuron.

Neural Quantum Switches exist.

Examples are parts of various ion channel molecules, and patches of neural and synaptic membrane.

Quantum coherence is not required by the definition of a quantum switch, and it is not required for the existence of quantum switches. What is required is a sequence of state changes and returns which satisfy specific mathematical constraints.

Now the neuronal model can be refined:

The Neural Switch Model: The information processed by a brain can be perfectly modelled by the pattern of spacetime events formed by a set of determinations of statuses of a family of neural quantum switches.

One instance of such a pattern over the life history of an individual up to a given moment constitutes the abstract structure of a “mind” in this “many-minds” theory.

Each pattern of determinations of switch status has a set of possible physical manifestations and each manifestation corresponds to a set of sequences of quantum states.

A complete mathematical definition of these sets is given. This involves, for example, defining what it means for a localized object to maintain its identity over time.

All the definitions are entirely in abstract terms with no explicit reference to statistical mechanical concepts such as temperature or information, or to biological concepts such as carbon atoms or any specific embodiment of the neural switches.

This means that they can be part of an abstract characterization of the physical structure of an observer.

A priori probabilities are defined for the patterns of determinations.

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Probabilities

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The calculation of probabilities.

We work in the Heisenberg picture so that quantum states do not change with time under the postulated universal Hamiltonian dynamics.

A sequence of quantum states corresponds to a sequence of (apparent) “collapses”.

Each individual “collapse” has its own a priori probability. These are calculated using an axiomatically defined function which generalizes the idea that, in appropriate circumstances, the weights in an orthogonal decomposition of a density matrix correspond to probabilities.

Treating successive “collapses” as independent gives the a priori probability for a sequence.

For a set of sequences, a priori probability is maximized over possibilities between which the observer cannot distinguish. By generalizing this maximization process, it is even possible to require that physical “constants” should be determined by observation and fixed only to the extent to which they have been observed.

Finally, using these a priori probabilities, the experience of an individual observer is modelled as the experience of observing a particular, identified, discrete stochastic process.

Constraints on probabilities:

  1. Agreement with experiment.
  2. With high a priori probability, humans possess the structures postulated for observers.
  3. The only entities which can with significant a priori probability possess such structures, at the human level of complexity, are entities which we would be prepared to believe might be physical manifestations of consciousness.

Agreement with experiment.

Decoherence theory gives us many models of state decomposition with negligible interference effects. These show that, following appropriate observations, the state of a macroscopic but localized observer will be close to a mixture of macroscopically distinguishable states weighted by measurable probabilities. In particular, square amplitudes of wave-functions of observed microscopic systems, appear as coefficients in such mixtures.

The calculated a priori probability is consistent with these models. From this, it can be argued that typical observers (as defined by the fundamental stochastic process) observe statistical probabilities in agreement with conventional quantum theory.

This means that the present theory is an interpretation of quantum theory and can annex all the empirical evidence which is normally taken to support the conventional theory.

(Of course, as the conventional theory is hopelessly incoherent, inconsistent, and incomplete, no empirical evidence can actually provide support for it, except as a ragbag of phenomenological models.)

With high a priori probability, humans possess the structures postulated for observers.

This certainly ought to hold in as far as the theory is constructed precisely in order to describe human observers.

A much deeper question, on the other hand, is whether, nevertheless, among all the possible ways in which stochastic processes can be defined on patterns of information, the physical laws postulated by the theory, despite their complexity, are actually as simple as any comparable set of rules making likely rich and meaningful patterns of information.

Also, the present theory is not a functionalist theory of mind, because mind-like behaviour is not assumed to be sufficient for the definition of an observer; rather “observers” are required to have a specific type of quantum mechanical structure. This means that it can be asked whether there are, or might be, significant classes of entities which apparently have observer-like behaviour but which the theory would imply did not correspond to “minds”.

The only entities which can with significant a priori probability possess such structures, at the human level of complexity, are entities which we would be prepared to believe might be physical manifestations of consciousness.

This claim cannot be proved and will always be open to falsification by example. Indeed, ruling out various potential falsifications was a significant method in the development of the theory. It can only be argued that it is difficult to imagine how else the definitions could be satisfied without continual rapid loss of a priori probability.

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Time and Other Minds

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According to the proposed theory, the time which appears to pass as we live, is not some absolute observer-independent time, but rather is an aspect of our structure as individual observers. Although our possible manifestations are constructs within a spacetime, our consciousness is not. The patterns of information by which we are characterized are geometrical patterns, but the geometry is abstract. The stochastic processes of which we are conscious are processes with their own time. The arrow of time is the arrow of these processes.

One consequence of this, which may be important in the development of quantum theories compatible with general relativity, is that although the present version of the theory is based on special relativistic quantum field theory, which does assume the existence of a fixed observer-independent background spacetime, that assumption is not fundamental.

In a many-minds theory, you can fantasize about the life you might be leading, had various stochastic events come out differently. The longer you live, the more of these possibilities you can contemplate. In the proposed theory, these correspond to genuine increases in the number of “your” minds. Going back in time, on the other hand, the proposals imply that all minds have the same unique null root. And this implies that any other person you choose to consider is leading one of the lives that you might have led had many stochastic events come out differently. Thus they are among “your” minds (and conversely).

Because we can only learn “the time” by observations — for example, by being told, or by looking at a clock, or a calendar, or the stars, or a newspaper — this applies as much to people who appear to be living in your past or in your future as it does to those apparently sharing your present. Of course the vast majority of your other possibilities are not represented directly in your apparent world and those which are, diverged from you very early in your mental life.

Now read on.