* * * * * * * * * * * * * *
| Dialogue One | “squark@my-deja.com” |
| Dialogue Two | Barry Adams |
| Dialogue Three | Jim Carr |
| Dialogue Four | Charles Francis |
* * * * * * * * * * * * * *
These dialogues are an edited version of my postings in a discussion during June and July of 2000 which stemmed from this question and this one of my FAQ. The replies are numbered, with material from the FAQ labelled (0), my exert from the first comment as (1), my response to that exert as (2), and so on.
MJD FAQ (0)
Decoherence is approximate. The decomposition into quasi-classical states is not unique. “Quasi-classical” may be an idea which we can understand for all practical purposes, but it is no less vague than the Copenhagen interpretation idea of a “measurement”. The identification of the fundamental subsystems, or the splitting between system and environment, is also left vague.
SQUARK (1)
Yes, but contrary to the Copenhagen idea of “measurement”, it is not a part of the fundamental model, but something we introduce when deriving macroscopic dynamics. In this sense, our model is no worse than any classical model - things ever get vague when we deal with big, complex systems.
MJD (2)
That's fine. The position I was arguing against was that decoherence combined with the many-worlds idea constitutes a fundamental interpretation of quantum theory all by itself. Of course, decoherence helps to derive the macroscopic quasi-classical dynamics of observed systems. But this leaves open the fundamental question of what we mean by “observed”.
SQUARK (3)
I may tell you what I think the MWI says about what is really “observed”: An observer consists of a system of commuting, Hermitian operators. These operators describe the notion of “perception” in a quantum language, i.e. an eigenvalue of these operators is a a certain state of the observer's mind. Thus, these mind observables contain information about the human (supposedly) brain structure. Now, given a general quantum state Psi, we may decompose it into mind eigenvectors and say what are the probabilities for the different observations in this case. The notion of “probability” is not a statistical one here, but rather a philosophical/metaphysical one, describing something like the “power of perception” of a certain reality by the observer's mind.
MJD (4)
Okay.
So is any system of commuting Hermitian operators an observer?
This seems unlikely as there are continuously many such systems.
Does a given observer always consist of the same system of commuting Hermitian operators?
(“same” defined using Heisenberg propagation of operators)
This seems unlikely, not only given the growth and decay over a lifetime of any human brain, but also given the extent to which Heisenberg propagation does not follow the “branching” of individual worlds.
So what systems of commuting Hermitian operators do correspond to observers and how does any such system change with time?
The papers on the home page of my web site are the fruit of a long effort which started by trying to answer exactly these questions and ended in answering modifications of them.
MJD FAQ (0)
The identification of the fundamental subsystems, or the splitting between system and environment, is also left vague.
SQUARK (1)
This I did not understand. What do you mean, precisely?
MJD (2)
Decoherence always involves looking at only part of the universe. According to decoherence, there is no global wave-function collapse. Instead information from a system is carried off by the environment. Entanglement does not disappear, it just moves away.
So which part of the universe is system and which is environment?
This is an aspect of the question of what we mean for a system to be “observed”. Unless we know to the nearest atom exactly where we draw the line between system and environment, we cannot expect to find an unambiguous way of defining significant decompositions of “observed” density matrices. It follows that the mere combination of decoherence with the many-worlds idea is insufficient to provide a foundation in terms of which we can explain the apparent temporal progression of our everyday world or to understand probability.
Lack of uniqueness really is a fundamental problem in the definition of probability. For the many-worlds interpretation, it's the problem of “how many worlds are there?”. And the answer to that is “well it depends on how fine a mesh you want to use”. With a finer mesh there are more worlds and each is less likely.
If you have a random walk where you keep varying the step size then the trajectories will be quite different from the trajectories of a random walk with fixed step size. The mere combination of decoherence with the many-worlds idea is like a random walk in which not only is the step size unspecified, but so are the dimensions of the state space and the directions of the co-ordinate axes.
MJD FAQ (0)
The eigenfunctions of realistic density matrices need not be quasi-classical.
SQUARK (1)
Why not?
MJD (2)
Here's one example.
Take the one-particle reduced density matrix of a one-dimensional ideal gas confined to an interval [0, L].
This has a decoherence length corresponding to the de Broglie thermal wavelength, so it is as “decoherent” a state as one could wish. The unique eigenfunctions however are of the form sin n pi x/ L and so are completely delocalized and utterly unclassical.
In general, density matrix eigenfunctions do not have the sort of properties that would be necessary if we wanted to base a fundamental theory on them. They need not be unique, they are not stable under perturbations of the density matrix, and, as the example above shows, individually, they often do not reflect physically-important properties of their density matrix.
SQUARK (3)
If I understand you correctly, you consider the density matrix exp(-beta*H) for an ideal gas. Well, I guess it's not as realistic as you might have thought!
MJD (4)
This is indeed the density matrix I am considering.
It is not realistic both because interactions are ignored and because it is an exact and perfect equilibrium state.
Two possibilities arise:
A) the realistic density matrix for a real sample of gas (say the air in this room) is quite like the ideal density matrix I was considering.
In this case, the realistic density matrix has lots of eigenvalues very close to degeneracy, and, just because of this, its eigenfunctions, although generically unique, are totally unpredictable. They will have no quasi-classical properties except in rare, accidental, and unstable cases.
B) eigenfunctions of the realistic density matrix are stable and have quasi-classical properties.
In this case, the realistic density matrix is a mixture of pure quasi-classical states with distinct probabilities. It is then nothing like an equilibrium Gibbs exp(-beta*H) state for any gas, in which case, either we are no long talking about many-worlds theory, or the whole of quantum statistical mechanics is deeply flawed.
MJD (2)
You can find more about eigenfunction decompositions in general in
“Continuity and discontinuity of definite properties in the modal interpretation,” by G. Bacciagaluppi, M.J. Donald, and P.E. Vermaas, Helvetica Physica Acta, Volume 68, pages 679-704 (1995). abstract, pdf (302K).
Note that a “definite property in the modal interpretation” corresponds essentially to an eigenfunction of a reduced density matrix.
I wrote an overview of the relevant part of this paper for the proceedings of a 1996 conference in Utrecht. This also discusses the ideal gas example mentioned above.
“Discontinuity and continuity of definite properties in the modal interpretation,” from “The Modal Interpretation of Quantum Mechanics”, pages 213–222, D. Diecks and P.E. Vermaas (eds.) Kluwer (1998). abstract, pdf (209K).
MJD FAQ (0)
There are many different decompositions of the resulting density matrix into quasi-classical states which each provide accurate descriptions of the system as we might see it at an instant.
SQUARK (1)
Yes, but there a simple criterion for determining the true quasi-classical states: the macroscopical observable need to be “near-eigen” (i.e. the state should be a near-eigenstate), and STABLY SO - this property must be stable under time evolution. I do not see how does it provide a serious problem for the interpretation i.e. gives rise to any paradox or lack of explanation
MJD (2)
The essential question here is what is meant by the true quasi-classical states.
Is there really some definitive set of quasi-classical states in terms of which our observed world is specified?
This would provide the foundation for a serious interpretation of quantum theory. But if such an interpretation were to be valid then the specification of the definitive set of quasi-classical states would have the status of physical law, because it would be the specification of a real and fundamental aspect of the physical world.
squark's “simple criterion” is nothing like a physical law; it is no more than a heuristic.
I agree that there are circumstances in which states do exist with the required properties. Nevertheless, for the reasons I spell out in my FAQ, they are never unique.
MJD (2)
The position I was arguing against was that decoherence combined with the many-worlds idea constitutes a fundamental interpretation of quantum theory all by itself. Of course, decoherence helps to derive the macroscopic quasi-classical dynamics of observed systems. But this leaves open the fundamental question of what we mean by “observed”.
Adams (3)
If you are prepared to settle for a thermodynamically irreversible observation as opposed to a physically irreversible observation, then observation may be defined (in the MWI+decoherence Inter), as a physical process that leaves a permanent record in some subsystem (the observer), that matches a physical property that was present in another subsystem, which has become thermodynamically irreversible due to information loss to a another subsystem (the environment). Now this is only a complete process up to a tiny but finite probability of reversibility. But note however if some of the information is in a photon radiated into empty space which reaches future null infinity then this process is also physically irreversible (barring a giant mirror in outer space).
MJD (4)
One can see very well from this exactly why the many-worlds + decoherence “interpretation” is not a fundamental theory; it's just a phenomenological description.
A complete fundamental many-worlds theory ought to provide us with a definitive list of possible future events and their probabilities of occurence; “either the cat gets it or it doesn't — both events have equal probability”.
But the description above gives us nothing like that. Among the phrases in it which are used but not defined are “the observer”, “matches”, “physical property”, “the environment”, “permanent record”, “irreversible”, and “reaches”.
We may all know “for all practical purposes” what these phrases mean. As external observers what we see is indeed that “either the cat gets it or it doesn't — both events have equal probability”, but many-worlds theory is supposed to be the theory without external observers.
At the level of splittings of local density matrices of the cat, however, we can change, by many orders of magnitude, how many “worlds” there are in which the cat dies, either by changing the resolution of our splitting or by changing the number of molecules in the cat's breath that we include as part of the cat. This is because the number of distinct orthogonal wavefunctions over which the density matrix of any thermal system can be expanded is of the order of exp(S/k) where S is the entropy of the system and k is Boltzmann's constant.
Adams (5)
The Many-worlds interpretion takes standard quantum theory with all the Hilbert space, and density matrices but throws away the collapse and observation axioms, and then uses relative states to show that a particular subsystem gains a relative state to other subsystems which is exactly analogous to what the Copenhagen interpation calls collapse. But you don't get preferred states here, expanding in terms of one set of eigenfunctions is as good as any other. But this doesn't mean MWI or Quantum is wrong, it could only mean we haven't made a good enough model of observation.
Decoherence takes the maths further again, and uses three subsystems to model observation, the observer, the environment and the observed. This time we do get preferred states. I'll put my neck out here and say, that as far as I've see in the literature, that this is a good model of observation.
MJD (6)
I agree that the main problem with the many-worlds interpretation does seem to be that we don't know which set of eigenfunctions we should choose to expand in. Indeed, I would even go so far as to say that when we do make an appropriate choice, the many-worlds interpretation itself becomes “a good model of observation”.
Nevertheless, “a good model of observation” given appropriate choices, is not the same as a fundamental theory.
As far as decoherence is concerned, my claim is simply that a choice of how we carve the world into “observer”, “environment”, and “the observed” also has to be made; that variations in this choice causes the details of the mathematics to vary; and that therefore many-worlds + decoherence is not a fundamental theory.
It is, of course, significant that the variations in the mathematical details are much smaller when decoherence is taken into account. This is why decoherence is so useful in practice.
Nevertheless, many-worlds + decoherence remains saturated with implicit observer-relative choices.
MJD (2)
Unless we know to the nearest atom exactly where we draw the line between system and environment, we cannot expect to find an unambiguous way of defining significant decompositions of “observed” density matrices.
Adams (3)
If that was true it would be impossible to get any results from the mathematics of decoherence: Fortunately it isn't. Decoherence superselects the same sets of eigenvalues every time for similar experiments no matter what the details of the heat sinks or size of the environment.
MJD (4)
We can get results from the mathematics of decoherence because we are always careful to resolve density matrices in ways which will be compatible with our observations, and because we can choose not to make radical changes within our calculations in the scale of resolutions which we use. Everything works fine, for all practical purposes, as long as we treat ourselves as external observers.
MJD (2)
the mere combination of decoherence with the many-worlds idea is insufficient to provide a foundation in terms of which we can explain the apparent temporal progression of our everyday world or to understand probability.
Adams (3)
Probability is almost completely defined by the MWI, decoherence is irrelevant here. However even here you ultimately require one axiom to get the probability interpretion, namely that: events with amplitude zero never happen. The proof of the probability interpretion from the above axiom appears in Everett's original paper.
MJD (4)
Everett's derivation of the probability interpretation is inadequate not because it requires that “events with amplitude zero never happen” but because it requires that events with amplitude close to zero almost never happen. Everett's argument does no more than confirm the consistency of the probability interpretation in the idealized situation to which he applies it.
In fact, probability is only defined by the MWI in Everett's sense to the extent to which it can be understood as a theory about decompositions of wavefunctions on tensor product spaces with respect to an orthogonal basis on one of the component spaces. The addition of decoherence is supposed to reduce the hopeless vagueness of this prescription, and reduce it it does, but it certainly doesn't eliminate it.
MJD (2)
Lack of uniqueness really is a fundamental problem in the definition of probability. For the many-worlds interpretation, it's the problem of “how many worlds are there?”. And the answer to that is “well it depends on how fine a mesh you want to use”. With a finer mesh there are more worlds and each is less likely.
Adams (3)
Many quantum systems have only a finite number of states, however position is normally defined as a continuum. Note however several promising theories of quantum gravity predict quantization of areas and volumes. These give a fundamental answer to “how many worlds are there?” by defining an ultimately fine mesh. However the answer is more practically defined by how much entropy you are prepared to create in order to measure the difference between these worlds.
MJD (4)
It is of course possible that quantum gravity theory may simultaneously solve the preferred basis problem, the problem of temporal progression, and the problem of time. In my opinion, however, it is more likely that quantum gravity will ultimately require the solution to the problem of time which is implicit in many-minds theory; to the effect that time is defined only relative to individual observers.
Any mesh provided by a quantum gravity theory would be extraordinarily fine. I suspect that a many-worlds theory based on such a mesh, even supposing it were both possible and unambiguous, would yield little understanding of the role that “observers” seem to play in quantum theory.
MJD (2)
Take the one-particle reduced density matrix of a one-dimensional ideal gas confined to an interval [0, L].
This has a decoherence length corresponding to the de Broglie thermal wavelength, so it is as “decoherent” a state as one could wish. The unique eigenfunctions however are of the form sin n pi x/ L and so are completely delocalized and utterly unclassical.
Adams (3)
Yes you're not measuring positions here. The macroscopic variables are pressure, and temperature, and so you'd expect a momentum state, which you indeed have.
MJD (4)
This is an interesting suggestion but it flies in the face of the conventional wisdom that decoherence theory picks out position as the “pointer observable” in systems with short decoherence length.
Moreover, in the high-temperature large-L (classical) regime, the density matrix referred to is asymptotically close (close for all practical purposes in other words) to a density matrix which has eigenfunctions defined by Hermite functions centered on x = L/2 (see my Utrecht paper, abstract, pdf (209K)).
MJD (2)
The position I was arguing against was that decoherence combined with the many-worlds idea constitutes a fundamental interpretation of quantum theory all by itself.
Carr (3)
I have a problem with the idea of a fundamental interpretation.
MJD (4)
Okay, there may be a linguistic problem here. By “interpretation”, I mean an answer to the question of what needs to be added to the mathematics of the Schroedinger equation (and its generalizations) in order to explain the appearance of our everyday world — we see individual electrons causing individual discrete marks on photographs; we hear individual clicks from Geiger counters; we see cats as either dead or alive; and so on.
By a “fundamental interpretation”, I mean an answer which is not only compatible with empirical evidence, but which is also sufficiently precise and complete that it could have the status of physical law. The Bohm interpretation, for example, might be such an answer, were it only compatible with the empirical evidence for special relativity theory.
For a correct fundamental interpretation, of course, the physical law would have actually to be true.
Carr (3)
Copenhagen introduces an arbitrary division into quantum system and classical observer that need not even be consistent from case to case. You might study the workings of a detector microscopically with a quantum theory, then use it as a “classical” device. It does this for convenience.
MJD (4)
Yes. And this is why the Copenhagen interpretation is not fundamental.
To make it fundamental, you would need to specify laws determining precisely which operator is being “measured” in any given situation and when the wavefunction of the universe “collapses” to one of the eigenfunctions of that operator.
Carr (3)
The idea of decoherence is to let the evolution of the many-body quantum system carry you to the point where statistical irreversibility takes over — a line that is necessarily vague.
MJD (4)
It looks as if the suggestion here is that statistical irreversibility is all that needs to be added to the many-body Schroedinger equation.
I think, on the contrary, that decoherence theory builds on the huge body of work on open quantum systems to show ways in which the mathematics of statistical irreversibility is already effectively in the many-body Schroedinger equation.
Decoherence theory tells us that locally (in other words when we “trace over environmental degrees of freedom”) the state of a macroscopic quantum system is *analogous* to the sort of state you get when you add a classical probabilistic structure to a quantum structure, and speak, for example, of a state as being wave-function Psi with probability p and wave-function Phi with probability q.
Classical probability is irreversible; there's no going back once the die is cast.
Unfortunately, the theory only provides a mathematical analogy, and anyway, it only works because you trace out the environment (otherwise the Schroedinger evolution would be unitary).
My claim is that for decoherence to underpin a “fundamental interpretation” one would need to invoke new physical laws to identify unambiguously a physically-real classical probabilistic structure.
MJD (2)
Decoherence always involves looking at only part of the universe.
Carr (3)
I thought it looked at all of it.
MJD (4)
Yes and no. Most of the universe is labelled as “environment”. The theory is centrally concerned with how the state of a subsystem of the universe is affected by interaction with its environment. The precise details of the probabilistic predictions of the theory change if one changes the division between environment and subsystem.
MJD (2)
Of course, decoherence helps to derive the macroscopic quasi-classical dynamics of observed systems. But this leaves open the fundamental question of what we mean by “observed”.
Carr (3)
What do you think closes that question, or would be needed to close that question?
MJD (4)
True physical laws which identify the structures which actually are observers.
I've been trying for a long time to understand what such laws might be like, were they to be valid. My current proposal as to what they might actually be constitutes the appendix to my paper “Progress in a many-minds interpretation of quantum theory”, quant-ph/9904001, abstract, pdf (463K).
You will see from that paper that what I am proposing is both complex and speculative. Nevertheless I believe that it comes closer to being, in the sense discussed above, a fundamental interpretation of quantum theory than any other proposal that I have yet seen.
Francis (3)
The very concept of the universe's wave function suggests a gross misunderstanding, not only of Copenhagen but of quantum mechanics itself. In Copenhagen we specifically segregate the system under study from the measurement apparatus. The implication is that the wave function is only expected to be an approximate description, in which the notion of a wave function does not apply to macroscopic objects such as a measurement apparatus, and certainly does not apply to the universe.
In a more modern orthodox interpretation the wave function does not have a fundamental interpretation as a physical entity. It is certainly not a description of state of matter, and in particular cannot be a description of the state of the universe. It is actually misleading to talk of wave functions, we should use Dirac notation and say that a ket is a label for our information about a state of matter, as given by the results of some measurement or set of measurement. Then the wave function is best seen as a relationship which can be used to calculate the probability of the results of a second measurement.
MJD (4)
Wavefunctions play a strange dual role in quantum theory. On the one hand, they are used to calculate probabilities, and on the other, they appear, when properly chosen, to provide accurate pictures of physical situations.
At the simplest level, the Bohr radius is a property of the ground state of the Schroedinger equation with 1 over r potential and that length is fundamental to any description of the hydrogen atom. More generally, any chemistry textbook shows atoms, molecules, and bonds directly through representations of electron wavefunctions. Then, because the mass of the nucleus is so much greater than that of the electron, it is possible to find nuclear wavefunctions which remain comparatively localized for long periods. So for ordinary matter we can build up wavefunctions which are accurate descriptions of the world we see about us. And finally we develop quantum theories for subatomic particles and interstellar matter and neutron stars. Everything looks like its wavefunction.
The only problem is that damned cat which keeps insisting that a wavefunction which is a good picture at one moment may not continue, under Schroedinger equation evolution, to be a good picture indefinitely.
Wavefunctions also provide us with the probabilities for particular breaks in the picture. A diffraction pattern forms with discrete marks; building up more where the wavefunction of the diffracting particles is large, and less where it is small. The probability of a transition is given by the absolute value squared of the transition amplitude.
So there is a duality.
How strange is it?
I think it is a bit strange in that the relation between “state” and probability is so simple, but one must expect that in an indeterministic theory there would be some functional relation between the two.
Can one treat wavefunctions just as “labels for information”?
I think that this is tempting but it raises some very hard problems.
What after all is information, and what is it information about?
What is “a state of matter” or a “measurement”?
If things are made of information, whose information are they made of?
What, in particular, are brains made of? Why does there appear to be such a close link between the detailed molecular structure of the brain (which can be described by a very complex quantum state) and the information in the corresponding mind?