On the origin of probability in quantum mechanics
http://arxiv.org/abs/1110.0549
I give a brief introduction to many worlds or "no wavefunction collapse" quantum mechanics, suitable for non-specialists. I then discuss the origin of probability in such formulations, distinguishing between objective and subjective notions of probability.
Here's what I say in the conclusion.
Decoherence does not resolve the collapse question, contrary to what many physicists think. Rather, it illuminates the process of measurement and reveals that pure Schrodinger evolution (without collapse) can produce the quantum phenomena we observe. This of course raises the question: do we need collapse? If the conventional interpretation was always ill-defined (again, see Bell for an honest appraisal [1]; Everett referred to it as a "philosophical monstrosity''), why not remove the collapse or von Neumann projection postulates entirely from quantum mechanics?
The origin of probability is the real difficulty within many worlds interpretations. The problem is subtle and experts are divided as to whether it has been resolved satisfactorily. Because the wave function evolves entirely deterministically in many worlds, all probabilities are necessarily subjective and the interpretation does not require true randomness, thereby preserving Einstein's requirement that outcomes have causes.
[1] J.S. Bell's famous article Against Measurement.
27 comments:
Schrodinger evolution is unitary. Put very crudely, the various versions of the observer were already there if you write the initial state in the suggestive way that I did in the last equation.
However, the so-called "coarse grained" entropy can increase in time even in many worlds. That is the apparent entropy for an observer who can only see one decoherent branch at a time. This observer sees entropy added in each decoherence event: to keep track of his world history he has to add a bit to his memory record to keep track of the + or - outcome.
See my paper "what is the entropy of the universe" for more discussion (there are a couple of related blog entries).
simple question, if you follow the Schroedinger evolution backwards to smaller t do you get more or less branches than if you follow it forward as usual?
Excellent paper Steve.
Thanks! Looking at Deutsch's latest but no comment yet.
The same number. Look at equation 4.
Of course the branches are only decohered after the measurement. Before the measurement it's just a choice of basis.
i guess this is only a difference in semantics, to me branches are 'decohered' or in other words macroscopically distinguished.
so with your use of words, the number N of 'decohered branches' increases with t.
but this only brings us back to my original question, how can N increase with time if the Schroedinger evolution is unitary.
and i think the answer is that decoherence assumes that there is an environment of which we don't know the exact state and imho this is where this 'arrow of time' is smuggled in.
the obvious next question would be where this environment comes from if there was only the unitary Schroedinger equation of the initial state of the universe ...
a related question would be where the classical clocks come from which you implicitly assume to exist by using the Schroedinger evolution the the time parameter t ...
> the number N of 'decohered branches' increases with t.
but this only brings us back to my original question, how can N increase with time if the Schroedinger evolution is unitary. <
As I mentioned already, coarse grained entropy (counting of number of decohered branches) can increase in time even when unitarity (which is defined at a more microscopic or fundamental level -- i.e., the actual Hilbert space for all dof) holds.
Put another way, the dimensionality of the Hilbert space is not increasing due to or during decoherence. The whole point is that decoherence itself is a result of unitary evolution.
It might help if you try to keep track of all the degrees of freedom (e.g., electrons in the detector or memory device) to convince yourself that M+ and M- can be radically different states (i.e., decohered), yet both states were present from the beginning as possibilities in the original Hilbert space describing the apparatus.
You're not really asking a probability question, you are confused about MW and decoherence at a very basic level.
>> you are confused about MW
yes
If you look at the books in ref. [2] of the paper you can find some simple examples where a a state is decohered due to, e.g., scattering of air molecules. Perhaps that will help ...
i dont think my problem is with decoherence per se but how you want to use it.
the usual setting is an initial state |i> which evolves into a superposition |1> + |2>
you seem to suggest that an initial superposition |i1> + |i2> evolves into |1> + |2>
is this correct?
so your idea is that the Born probabilities for |1> and |2> are already contained in the initial state and in a sense all probabilities were already encoded in the initial state of the universe. this is quite unusual ...
i need to be more specific here: your idea is that the initial state |i1> + |i2> is fine tuned so as to give the appearance in each branch of the final state |1> + |2> that the Born rule holds (and eliminate the maverick states) .
*this* is quite unusual (the fact that the Born amplitude encodes the Born probabilities is of course not what is unusual).
Steve,
It might be that the probability of biology is unlikely in Maverick worlds. A world consistent with biology might be more conducive to forming atoms and having chemistry. If there were no consistency in dynamical rules of a decohered branch then scientists may never have evolved. I realize that this is an anthropic argument but it could be given weight if we tried to construct the resulting classical and semi-classical mechanics on non-Born branches to see if they could possibly support life. The ultimate amazing result would be if the probability to support life was exactly the inverse of the probability of a Maverick world.
Oops, in the second sentence I meant to type a world consistent with the Born rule...
As an aesthetic matter, all this suggests to me is that there must be non-local hidden variables -- some topologically ornate manifold we haven't yet been able to measure.
This briefly made me regret choosing computer science over physics as my major :\
(briefly :)
I think you are a little too hasty in your conclusions that decoherence is enough to give rise to "branches" in the modern MWI version.
This is disputed by most physicists as explained in the book "Many Worlds?"
It appears MWI also got a unresolved and almost not recognized problem with relativity.
Add to this the fact that neither of the current attempt to solve the Born Rule has been satisfactory and that the preferred basis problem isn't solved, MWI is just as bad if not worse than any collapse interpretation.
In regards to the decision-theoretic approach these papers show why it fails:
Alastair Rae - Everett and the Born Rule http://arxiv.org/abs/0810.2657
Jacques Mallah - Decision Theory is a Red Herring for MWI http://arxiv.org/abs/0808.2415
J. Finkelstein: Has the Born rule been proven? http://arxiv.org/abs/0907.2064
In regards to the preferred basis problem:
Louis Marchildon - Can Everett Be Interpreted Without Extravaganza?
http://arxiv.org/abs/1001.1926
In the last 2 weeks 2 new papers on 2 new problems with MWI has shown up too:
Quantum Structures of Model-Universe: Questioning the Everett Interpretation of Quantum Mechanics ( http://arxiv.org/abs/1109.6424 )
and
Many Worlds, the Cluster-state Quantum Computer, and the Problem of the Preferred Basis (http://arxiv.org/abs/1110.2514)
As Steven Weinberg said in a interview earlier this year, it's obvious we have to rethink QM.
Perhaps something like the original no-collapse interprettion (deBroglie Bohm) is on the right track, but we know that none of the current interpretations are 100% correct.
See link for an opinion poll of actual experts (people who work on quantum coherence and decoherence) on whether the dynamics of decoherence is as claimed: http://infoproc.blogspot.com/2010/09/on-origin-of-probability-in-quantum.html
See Weinberg's latest paper for his view on the status of qm: http://arxiv.org/pdf/1109.6462v1
Aside from the "extravagance" objection, he only mentions the origin of probability as an issue for MW. (So it might be fair to guess he would vote with the majority in the poll above.)
I don't (personally) know anyone who thinks relativity is an issue for MW.
Science isn't a democracy...
Besides there are countless of polls taken at similar conventions and conferences that end up being the exact opposite.
The last time I asked Weinberg about this (3-4weeks ago) he said he had grown uncomfortable with MWI and that a complete new way of thinking of QM was required.
Did you take a look at any of the links? there are atleast 4 unique problems facing the supposed "elegant" MWI, to solve these (most likely impossible) you would need a ton of postultes that would be way worse than any of the other interpretations...
>> anyone who thinks relativity is an issue
Well, MWI has a unique problem with (quantum) gravity.
If there is no collapse then quickly everything couples to everything and you cannot neglect quantum gravity (this argument is not mine but due to H.D.Zeh).
In other words, the initial state (which plays such an important role in your paper) can only be understood within quantum gravity
and since there is no collapse everything that follows depends on it too.
However, the picture of the branching worlds almost certainly is inadequate for quantum gravity (one reason is that there cannot be an evolution in classical time).
isn't Zeh a MWI'er?
>> isn't Zeh a MWI'er?
yes, sort of. but i think he is fully aware that several important questions are unanswered.
You do not have any rebutle to all the papers I gave you? Guess the only interpretation of that can be that you realize you were wrong, but dont want to accept it
I hope you are joking. Responding to your links to a few papers isn't exactly high on my list of priorities.
Oh I see so, the arguments and paper themselves are not enough? It's all about the authority behind the argument, I see. Now you are even more of a joke to me.
From your comments I suspect you don't really understand the subject, but it's possible I am mistaken.
To explain why I think the argument in a particular paper is wrong takes effort, and before I expend that effort I want to be sure I am not wasting my time. For example, if the author of one of those papers were to contact me for an opinion I would gladly reply to them. Posting a lengthy analysis just for your sake doesn't seem a particularly good use of my time.
Well "just for me" ? It's for your entire blog audience isn't it?
There are 3 papers showing exactly why it does not work and the consensus in the physic community is that it doesn't work...
Not to mention the problem with preferred basis, ontology and relativity...
But I guess your answer here is "decoherence" ?
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