Is MWI Self-Contradictory and Does Time Travel Need a New Approach?

[Varon]

If Many worlds has only wave function and no particles. Why didn't Schroedinger discovered it? He spent a lifetime believing only wavefunction exist, what makes him fail to propose the Many worlds where there are only waves?

Back in the 1920s when Schroedinger thought it's all waves and no particles. Henrik Lorentz made him realized that wavepacket spreads. Lorentz told Schroedinger "Wave packet will spread with time and your idea of representing particles completely in terms of the superposition of waves is invalid"

Now what I want to know is how is this wave packet spreading related to one of the eigenvalues (let's avoid Many worlds in this question to avoid complication).

 

[Dmitry67]

I know. It's just a bad choice of words when I said one of the eigenvalues is chosen. What I meant was our branch only experience one of the eigenvalues and all outcomes exist. Since all are wave, I'm asking what is the wavelength of the particle in this sense. So it's the same de Broglie wavelength? But this is based on wavelengh = h/momentum. Is there another formula that only inputs the particle existence without regards to momentum?

Why do you think that it should be different in MWI from what QM tells us? Note that the wavelength of particle has nothing to do with how small detector can be. For example, famous 21cm hydrogen line can excite hydgogen atoms which are 10-8cm in size. If we have a 'matrix' with step of 0.001cm, made of such atoms, we would see a 0.000001cm^2 'dot', created by 21cm lightwave (these results will be true in any Int, of course).

 

[Dmitry67]

If Many worlds has only wave function and no particles. Why didn't Schroedinger discovered it? He spent a lifetime believing only wavefunction exist, what makes him fail to propose the Many worlds where there are only waves?

Back in the 1920s when Schroedinger thought it's all waves and no particles. Henrik Lorentz made him realized that wavepacket spreads. Lorentz told Schroedinger "Wave packet will spread with time and your idea of representing particles completely in terms of the superposition of waves is invalid"

Now what I want to know is how is this wave packet spreading related to one of the eigenvalues (let's avoid Many worlds in this question to avoid complication).

I think there were 2 major reasons.
1. The idea was too crazy - they already had enough crazyness to deal with;
2. In the beginning of 20th century, there was a gap separating macro and micro, huge gap separating 'measurement devices' from QM. So there was a hope that QM weirdness would somehow 'fade' when approaching the macroscopic scale. Like how Bohr showed earlier in his model how Qm levels in atom transform into continuum of states on bigger scales.

 

[Varon]

I think there were 2 major reasons.
1. The idea was too crazy - they already had enough crazyness to deal with;
2. In the beginning of 20th century, there was a gap separating macro and micro, huge gap separating 'measurement devices' from QM. So there was a hope that QM weirdness would somehow 'fade' when approaching the macroscopic scale. Like how Bohr showed earlier in his model how Qm levels in atom transform into continuum of states on bigger scales.

You prefer MWI because of the collapse postulate seems to make the deterministic wave function inconsistent. It is not. Here comes the magic of the observers (either particles sensing other particles or bigger environment or system). It has got to do with Information Theory and information exchange between the quantum. This is actually quite elegant. Fra has even formulated almost a complete theory of it. Try it before delving into the MWI schizophenia. What's your criticism of Fra appraoch? (Note: Fra is the one without the Wolverine icon)

 

[Demystifier]

Maybe a particle in Copenhagen is also a localize wave just like in Many Worlds?

Yes. In fact, the only (relatively well-known) interpretation of QM in which "particles" really are particles - is the Bohmian interpretation.

 

[Demystifier]

If Many worlds has only wave function and no particles. Why didn't Schroedinger discovered it?

Because he did not understood the mechanism of wave-function splitting (now well understood through decoherence).

 

[Varon]

Because he did not understood the mechanism of wave-function splitting (now well understood through decoherence).

Any idea how to connect MWI with quantum field theory and particle physics?

 

[Fyzix]

Because he did not understood the mechanism of wave-function splitting (now well understood through decoherence).

It must be said that assuming wavefunction is real and all there is + decoherence does not give you many worlds.

The Bare Theory, which assumes this alone is not a many world interpretation.

Mitchell Porter also touches on this:

http://lesswrong.com/lw/19s/why_manyworlds_is_not_the_rationally_favored


Side note:
I'm pretty sure I read somewhere that Schroedinger realized MWI did not work because of botn probability and the fact that it would violate relativity.
Unfortunately I can't find the source of this claim right now.

 

[Demystifier]

Any idea how to connect MWI with quantum field theory and particle physics?

Why do you think there would be a problem?

 

[Varon]

Why do you think there would be a problem?

Many worlds are classical theory. When you have interacting fields and particles like quantum field theory.. i wonder how you could mix Many worlds in them. Has anyone tried modeling how Many worlds would appear in quantum field theory and whether it would still give the same experimental results and predictions? Most would state that since QM works in QFT, Many worlds should work too.. but it's a classical model. Won't you have trouble when you embed it into QFT or gauge theory?

 

[Demystifier]

Many worlds are classical theory. When you have interacting fields and particles like quantum field theory.. i wonder how you could mix Many worlds in them. Has anyone tried modeling how Many worlds would appear in quantum field theory and whether it would still give the same experimental results and predictions? Most would state that since QM works in QFT, Many worlds should work too.. but it's a classical model. Won't you have trouble when you embed it into QFT or gauge theory?

If you think of many worlds as a "classical theory", then you can extend the whole idea to a many world "classical theory" of QFT. In this theory, neither particles nor fields exist. What exists are certain generalized wave functions which depend on an infinite number of variables.

 

[Dmitry67]

I'm pretty sure I read somewhere that Schroedinger realized MWI did not work because of botn probability and the fact that it would violate relativity.
Unfortunately I can't find the source of this claim right now.

Note:

Fyzix fights some weird flavor of MWI, which is not based on pure unitary evolution, but adds some weird stuff about 'spacetime unzipping' or 'hard, definite and final world splitting'. Then he attacks MWI based on the fact, that the equation for the fictious 'border', which separates different 'worlds' is non covariant.

There is a good quote from Everett FAQ "The concept of "world" in the MWI is not a rigorously defined mathematical entity, but a term defined by us (sentient beings) in describing our experience".

 

[Dmitry67]

Note 2:

MWI is a 'bare theory' as is explicitly stated in wiki article:
http://en.wikipedia.org/wiki/Many-worlds_interpretation

MWI is distinguished by two qualities: it assumes realism,[16][17] which it assigns to the wavefunction, and it has the minimal formal structure possible, rejecting any hidden variables, quantum potential, any form of a collapse postulate (i.e., Copenhagenism) or mental postulates (such as the many-minds interpretation makes).

This quote also explains that if someone talks about 'hard picture of spacetime unzipping' etc - then it is not MWI.

 

[Fyzix]

No, I've already informed you that the Everett-DeWitt account of MWI requires violation of relativity.

And now I find myself ONCE AGAIN informing you about your favorite interpretation: BARE/PURE wavefunction realism yields BARE THEORY read Jeffrey Barrett or David Albert on this subject.

You got your own warped MWI where doctors collapse worlds, so stop talking as if you understand what you are talking about and everyone else is wrong.

 

[Dmitry67]

No, I've already informed you that the Everett-DeWitt account of MWI requires violation of relativity.

Informed about your opinion? Yes
provided any useful info?
We had only one quote about "unzipping the spacetime".
As there is no "unzipping" there is nothing we can discuss.
Do you have anything else?

From another side, as unitary evolution is covariant, MWI *must* be covariant. You did not reply to that either.

For the commonly accepted definitions of flavors of theories, I insist on using WIKI. One can publish an article claiming "here is MY understanding of MWI and only it is true!".And you will be positng that link from arxiv as a "proof". So please provide a WIKI prooflink for the definition of the BARE theory.

 

[Fyzix]

Informed about your opinion? Yes
provided any useful info?
We had only one quote about "unzipping the spacetime".
As there is no "unzipping" there is nothing we can discuss.
Do you have anything else?

From another side, as unitary evolution is covariant, MWI *must* be covariant. You did not reply to that either.

I have written probably 5 pages of information to you, when you have not comprehended the information, I have RE-written it.
When you have argued back, I have thought about it and even inquired others about it to give you as easy to understand explanatino as possible, after doing this 50 times, I have given up.
You believe in MWI because it is infact your religion.
You ignore EVERY obstacle MWI faces because like you said your self "I WOULD BE DEVASTATED IF MWI ISNT TRUE".
You have also admitted here on this forum several times that all problems will be solved by some futuristic theory of consciousness.

When you are willing to take such leaps of faith to defend a flawed hypothesis it is obvious(atleast to me) that a discussion with you will only be a waste of time.
Remember, your view of MWI is wrong.
Whether you go to the doctor or not, will not put you in a branch where you have a tumor or not, THIS IS what you are saying about MWI, this is so far removed, even from the Deutsch-Wallace approach, so yes, there is no way I can argue against your arguments because you have your own MWI interpretation and consciousnes theory that has nothing to do with reality.

 

[Fyzix]

And please don't lie again and say I don't give you sources.

I have quoted way more than just the stanford entry on Everett by Barrett.

 

[Dmitry67]

I have written probably 5 pages of information to you, when you have not comprehended the information, I have RE-written it.
When you have argued back, I have thought about it and even inquired others about it to give you as easy to understand explanatino as possible, after doing this 50 times, I have given up.
You believe in MWI because it is infact your religion.
You ignore EVERY obstacle MWI faces because like you said your self "I WOULD BE DEVASTATED IF MWI ISNT TRUE".
You have also admitted here on this forum several times that all problems will be solved by some futuristic theory of consciousness.

When you are willing to take such leaps of faith to defend a flawed hypothesis it is obvious(atleast to me) that a discussion with you will only be a waste of time.
Remember, your view of MWI is wrong.

I have admitted that it would be sad for me if MWI was false; however, I would accept the result if there would be proof. So it isn't a religion. You had asked me about it, and I had explicitly said that. Why are you giving false statements about what I have said? Do you need me to go thru my Sent box and give you the exact quotes?

The 5 pages you provided was similar to what you had said above, "clearly it is wrong" or "MWI has severe problems with relativity" or "there are tons of articles about...". It is like saying "you are wrong... no you are wrong..."

I tried to focus on something specific: compatibility of MWI with SR. Once again, you did not reply. Fyzix, this is simple: if unitary evolution is covariant, and MWI does not have any additional assumptions, then it is compatible with SR. If you have any additional assumptions, you should state them clearly before making conclusions.

When I ask to discuss it step by step, you switch discussion to other subjects, just to paphosly repeat the same claim "MWI has severe problems with relativity!" when new audience come into thread.

Are you ready to discuss this small subject (MWI and SR) and go thru it - step by step?

 

[Dmitry67]

UPD:
I decided that it is not scietific to discuss the emotional component or PM here. If you want to ask anything about our discussion in PM - send PM.

Here I will discuss *only* the technical stuff - if you are ready to discuss the MWI vs SR.

 

[Fyzix]

Dmitry67:

I have already told you that the Many Worlds account by Everett & DeWitt was incompitable with special relativity.
I never said that Deutsch-Wallace approach had the same problem!
So why are you bringing tihs up now?!

Because I said earlier in the thread that Schroedinger figured out there was a problem with probability and relativity when thinking of the wavefunction as real?
THIS IS TRUE!
Because Schroedinger was trying to make a bare theory where ontology actually existed!
Tim Maudlin brought this historical bit to my attention, but if you google it with Maudlin's name, I'm sure you can find it too.
I brought that up to inform Varon about history and why Schroedinger believed what he believed, not because we were arguing the DeutschWallace approach!


Now have you bothered to read up on Bare Theory?

 

[Dmitry67]

Fyzix,

You don't need to TELL me anything. You need to PROVE.
The ONLY link you provided to support your point of view was:

Anyways, let's move on to a more "serious" problem facing MWI.
Namely the relativity problem.
After further discussions with Jeffrey Barrett, I've realized that even the decoherence approach suffer from the same fate regarding relativity.
As Barrett himself explains in the Stanford entry.

Reread the passage where he explains exactly the technical difficulties regarding this and tell me how you get around this problem?
http://plato.stanford.edu/entries/qm-everett/

And based on a link:

Those who favor a decoherence account of splitting worlds sometimes seem to imagine some sort of “unzipping” of spacetime that occurs along the forward light cone of the spacetime region that contains the measurement interaction. While decoherence effects can be expected to propagate along the forward light cone of the region that contains the interaction event between the measuring device and the object system, and while there is no problem describing the decoherence effects themselves in a way that is perfectly compatible with relativity, there is a problem in imagining that such a splitting process somehow physically copies the systems involved. A strong picture of spacetime somehow unzipping into connected spacetime regions along the forward light cone of the measurement event, would not be compatible with special relativity insofar as relativity presupposes that all events occur on the stage of Minkowski spacetime. And if we give up this assumption, then it is unclear what the rules are for compatibility with special relativity.

So Barett has explicitly admits that MWI is compatible with relativity, unless you use extra assumptions (like "unzipping"). Later barret admits that"

If one understands Everett's talk of splitting as in some sense only metaphorical, then one may avoid the problems associated with a strong notion of physical splitting.

But it is Barrett's personal choice to stuck with the unzipping.
So the only reference you ppovided works AGAINST you, it proves MY point.
Lets no go any further until we discuss this subject.

 

[Fyzix]

Oh my ****ing nonexistent God, I USED BARRETT AS SOURCE FO THE DEWITT SPLITING PROBLEM, YES.
Which IS accurate!

Also what is this ******** about me having to PROVE things to you?
No, not at all, you are the claimant here, YOU need to prove.
You claim pure wavefuncton leads to MWI, I have given you sources that show otherwise, you just jump back to the MWI-DeWitt relativity problem, because it's the only argument you can counter, but you fail because it's accurate for the SPLITING ACCOUNT!

Now move on, adress my other criticism of the approach you adhere to!

 

[Dmitry67]

At first, please answer my questions.
1. Do you agree that Barrett admits that unitary evolution is covariant?
2. Do you agree that Barrett admits that pure MWI, without any additional assumptions, is covariant?
(above I am not asking about your point fo view, I am asking about how you interpret Barretts words)
3. Do you (personally) agree with #1?
4. Do you agree with #2?
5. Do you agree that QM (QFT) is covariant?
6. Do you agree that evolution of wavefunction in MWI is described by QM (QFT) and only by it?

Please answer Y or N.

 

[rkastner]

Hi, I just wanted to mention that Chris Fields (2010, on the arxiv--sorry, I can't get it to load right now so can't provide the link) has argued that decoherence fails to account for emergence of a classical world in the sense of "quantum Darwinism". I.e., to get classical behavior out, one must put it in by choosing what counts as 'system' and what counts as 'environment'. So claims that MWI can account for the 'emergence' of a classical world need to be more closely examined.

 

[Dmitry67]

to get classical behavior out, one must put it in by choosing what counts as 'system' and what counts as 'environment'.

Yes. But it is known: for the decoherence you need a basis. So the existence of the 'classical world' is not 100% objective in a naive sense: it is only objective when is mapped into some basis (an observer). If you chose a 'basis' which consist, say, of a hydrogen atom, you would get a nonsense. As hydrogen atom has very limited defrees of freedom, from its 'prespective' macroscopic objects can be in superposition!

I can add even more.
1. The observer's basis is a fuzzy concept - for example, if 'I' am a basis, should I add my hair to the basis? Or should I count only brains?
2. For continuus observations, basis is constantly redefined (as atoms move in my body). What is worse, basis is redefined in a previous-basis-dependent manner (as in my different branches my atoms can move differently)

It is very exciting. It is a huge unknown area, and it will remain so even when (if) TOE is discovered. So physics won't end when TOE will be discovered.

 

[Demystifier]

From another side, as unitary evolution is covariant,

I don't think it is. It is local, but not manifestly relativistic-covariant. If you still claim it is, can you provide an appropriate reference?

See also
http://xxx.lanl.gov/abs/1012.0992

 

[Dmitry67]

1. "Not manifestly covariant" does not mean "not covariant". Because a claim that QM is not compatible with SR is a very strong one, right?

2. The article you provided goes in the same direction as Barrett:

Then, one seems obliged to conclude about the following inconsistency in the very foundations of the Everett interpretation: the universally valid quantum mechanics
implies the ”splitting” can not occur.

Of course, "splitting" or "unzipping" can't occur in mathematical sense, it is not "final" or "complete" and interference terms never vanish to zero, but it occurs FAPP. Usually, even we have different assumptions, our logic is compatible (I can't say the same about Fyzix), so could you explain, what is the motivation for discussing this subject?

 

[Demystifier]

1. "Not manifestly covariant" does not mean "not covariant".

I agree. Nevertheless, I would really like to see a paper which shows that unitary time evolution of the quantum state in QFT or many-particle QM is covariant. It would help me a lot, not only for better understanding of MWI, but also for many other physical issues. In the meanwhile, I still think it isn't covariant.

Because a claim that QM is not compatible with SR is a very strong one, right?

It depends on what one means by "QM".

If QM are matrix elements, then it is covariant. The simplest way to see this is to calculate them in the Heisenberg picture and observe that matrix elements do not depend on the picture.

But if QM is the time-dependent wave function (which MWI claims), then I claim it is not covariant. The time-dependent wave function is the state in the Schrodinger picture. In that picture neither the operators nor the states are covariant, but their appropriate combination (matrix elements) are. In the Heisenberg picture both the operators and the states are covariant, but the state in the Heisenberg picture is a time-independent state, which is not supposed to represent reality in MWI.

Or let me formulate the problem in another form. Is there MWI in the Heisenberg picture? I don't think so.

 

[Dmitry67]

Is there MWI in the Heisenberg picture? I don't think so.

I agree with your "no", because in MWI on the fundamental level there are no "observables". So there is only Schrodinger picture, and you claim it is non covariant. Could you provide a counter-example to the covariance of such time evolution?

 

[Demystifier]

So there is only Schrodinger picture, and you claim it is non covariant. Could you provide a counter-example to the covariance of such time evolution?

OK, let x denote the space coordinate and t the time coordinate. Consider a 1-particle wave function psi(x,t). If it satisfies the Dirac equation, then it is covariant. Namely, x and t are treated on an equal footing.

Now consider a 2-particle wave function. It must have the form psi(x1,x2,t). However, there are 2 space coordinates (x1 and x2), but only one time coordinate (t). Therefore, space and time are not treated on an equal footing. Consequently, the theory describing it cannot be covariant.

A way out of this problem is to introduce a many-time wave function. For instance, for two particles the wave function has the form psi(x1,t1,x2,t2). Such a theory can be made covariant. But the problem is that now we have two time coordinates. Which one represents the "time-evolution" of the whole system? Neither, of course. If you insist on covariance, then the concept of time evolution should be abandoned, or more precisely, replaced with something more general and abstract. This is fine, even MWI can be based on it. But in that formulation of the theory, there is no concept of unitary evolution. In that sense, the initial claim that unitary evolution is not covariant - is still true.

 

[Dmitry67]

But how do you know that there are exactly 2 particles? You know it from the previous measurements only, telling you have many particles have been observed. Obviously, in that form the equation is already "spoiled" by the measurement problem, and has a strong smell of "observables" (even thay are not there directly) which, on the fundamental level, it should not have.

If some source gives you a sequence of particles, and you can somehow confine them in some area and make sure that there are only 2, then you look at a very specific 'slice' of the reality/universal wavefunction. Reality is a superposition of cases when you have 0,1,2,3,... particles + unknown (and hence not-decoherenced) non-integer quantity of 'particles' that had completely escaped the measurement.

Now the question is, can we formulate completely measurement-neutral picture and is that picture covariant?

P.S. And Unruh effect... the number of particles is observer dependent...

 

[Demystifier]

You are missing the point. It doesn't matter how many particles are really there. It's enough to know that the number of particles in not exactly 1. Whenever this is the case, you cannot write the equations of unitary evolution in a covariant form.

 

[Demystifier]

Now the question is, can we formulate completely measurement-neutral picture and is that picture covariant?

Yes we can, and unitary evolution of such a system is not covariant.

P.S. And Unruh effect... the number of particles is observer dependent...

Another way of saying it is that the number of particles is not covariant. There is a way to avoid this problem by not even talking about particles in QFT, but it does not change the fact that unitary evolution in QFT is not covariant.

 

[Dmitry67]

Now I am confused; how apparently covariant behavior comes from the non-covariant equation?

 

[Demystifier]

Now I am confused; how apparently covariant behavior comes from the non-covariant equation?

Non-covariance is closely related to non-locality (where "nonlocality" is meant in the interpretation-independent sense of nonlocal EPR correlations). In particular, in the classical limit nonlocality disappears and covariance restores.

But there is also another way of viewing it. One may say that non-covariance of the wave-function evolution is not important because the wave function by itself is not physical. What is physical are the matrix elements, which are covariant. Unfortunately (for you), such a view is not compatible with MWI.

 

[Dmitry67]

Non-covariance is closely related to non-locality (where "nonlocality" is meant in the interpretation-independent sense of nonlocal EPR correlations). In particular, in the classical limit nonlocality disappears and covariance restores.

So, if I insist that wavefunction is real, then covariance becomes not a fundamental property, but an emergent property at the classical limit (it must be proven as a theorem?) , exactly like the 'definite outcomes' and decoherence in general?

 

[Demystifier]

So, if I insist that wavefunction is real, then covariance becomes not a fundamental property, but an emergent property at the classical limit (it must be proven as a theorem?) , exactly like the 'definite outcomes' and decoherence in general?

Yes.

More precisely this is so if, by a wave function, you mean a single-time wave function. But personally I prefer the many-time wave function, which opens a possibility to have a fundamentally covariant wave function.

 

[Fyzix]

Dmitry67, I will have to wait until Barrett answers those questions as I won't speculate as to what he "believes".

It seems you ignored Ruth Kastner's post about the problem of emergnce...
This is why I get a little annoyed, you often just flat out ignore things.

Since you guys have gone into discussion of Heisenberg picture in MWI.
Here are two papers on it.

Ruth Kastner's paper:

http://arxiv.org/ftp/arxiv/papers/1011/1011.3078.pdf

and

David Wallace & Chris Timpson's paper:

http://arxiv.org/PS_cache/quant-ph/pdf/0503/0503149v1.pdf

 

[Dmitry67]

Thanks, I will be waiting for his reply, even some of the questions probably should be better reformulated in a light of our recent discussion with Demystifier, but on a high level there is no harm if I leave them as is. I will check your papers, thank you.

 

[Hurkyl]

Yes we can, and unitary evolution of such a system is not covariant.


Another way of saying it is that the number of particles is not covariant. There is a way to avoid this problem by not even talking about particles in QFT, but it does not change the fact that unitary evolution in QFT is not covariant.

A form of QFT that manifestly contradicts your claim: http://en.wikipedia.org/wiki/Algebraic_quantum_field_theory

 

[Dmitry67]

Hurkyl, Demystifier had already convinced me, and now you are saying he is wrong? :)
Looks like I have to hold my opinion, looking at the battle of titans.

 

[Hurkyl]

I just noticed the nLab has a page on this too:

http://ncatlab.org/nlab/show/Haag-Kastler+axioms

 

[Hurkyl]

Hurkyl, Demystifier had already convinced me, and now you are saying he is wrong? :)

Yes. However, he's probably right regarding a slightly narrowed viewpoint.


Briefly, the main features of AQFT that make covariance manifest is:

The notion of operator is expanded slightly to include a domain on which it's defined. (A domain could be defined as a bounded open subset of Minkowski space)

Poincaré transformations of Minkowski space also transform operators -- e.g. if T is a transformation and X is an operator defined on U, then T(X) is an operator defined on T(U).


The "primitive causality" axiom from the wikipedia article imply that if the regions U and V causally determine the same region of space-time, then every operator on U can be converted into an operator on V, and vice versa. This is unitary evolution in the analog of the Heisenberg picture. Converting to the associated state spaces, you would get the same statement in the analog of the Schrödinger picture.

On this last point you can't expect anything better. If you consider as a time translation, then you couldn't hope for the future state to be determined by the past state because the future region is not causally determined by the past region. In the opposite direction, if you think of time evolution from U as determining the state within some region in the future that is causally determined by U, you cannot hope for a unitary transformation because you are throwing away information about the region that is influenced by U but not determined by it.

 

[Demystifier]

A form of QFT that manifestly contradicts your claim: http://en.wikipedia.org/wiki/Algebraic_quantum_field_theory

I don't see that it has anything to do with unitary evolution. Can you explain?

To repeat, I am not claiming that QFT cannot be formulated in a covariant way. I'm sure it can. All I am saying is that unitary time evolution of the state in the Schrodinger picture cannot be formulated in a covariant way (except by generalizing it to a many-time formalism).

 

[A. Neumaier]

All I am saying is that unitary time evolution of the state in the Schrodinger picture cannot be formulated in a covariant way (except by generalizing it to a many-time formalism).

?

The covariant formulation of unitarity uses in place of noncovariant wave functions psi(t) with time arguments t satisfying the equation
i hbar partial_t psi = H psi
wave functions psi(x) with space-time arguments x satisfying the covariant equation
i hbar partial_x psi = P psi,
where p is the 4-vector of translation generators of the Poincare group.

In both cases, psi(t) resp. psi(x) is for all arguments t resp. x in the physical Hilbert space. The connection with the Heisenberg picture is through the equation
psi(t) = e^{-ix dot P}psie^{ix dot P}.
This is independent of a multi-time formulation, and applies for example to a single Dirac particle, and to all covariant few-particle systems discussed in the work by Polyzou and his collegues.

 

[Hurkyl]

All I am saying is that unitary time evolution of the state in the Schrodinger picture cannot be formulated in a covariant way (except by generalizing it to a many-time formalism).

If the open subsets U and V of Minkowski space causally determine the same subset of space-time, then the state spaces of the U-subsystem and the V-subsystem are isomorphic. Even more, the formalism picks out a specific unitary transformation as the isomorphism.

 

[Demystifier]

If the open subsets U and V of Minkowski space causally determine the same subset of space-time, then the state spaces of the U-subsystem and the V-subsystem are isomorphic. Even more, the formalism picks out a specific unitary transformation as the isomorphism.

OK, I see that it has something to do with unitarity, but I still don't see how exactly is it related to time evolution. U and V are 4-dimensional, right? By contrast, a notion of time evolution would require 3-dimensional spacelike objects of "constant time". Hence, it looks to me like a formalism that avoids the notion of time evolution (which I like), which is exactly why the covariance can be achieved.

 

[Hurkyl]

By contrast, a notion of time evolution would require 3-dimensional spacelike objects of "constant time".

I think you're being too narrow with the term. For example, in that setting, if we had a quantum state defined over the region (with c=1)

-5 < x,y,z < 5
0 < t < 1
​

then the formalism implies the ability to compute from this a new state that is defined over the region

-2 < x,y,z < 2
2 < t < 3
​

I think it reasonable to call that "time evolution". Of course, we can't expect this particular computation to be unitary because it loses information.
(Aside: I find it quite plausible that the formalism could work with three-dimensional slices rather than actual extended 4-dimensional objects, but I'm not in the mood to chug through the category theory to prove it)

If we are specifically interested in unitary transformations, then an example would be taking some state defined over

-5 < x,y,z < 5
0 < t < 1
​

and from that computing the corresponding state defined over the union of the three regions

• -5 < x,y,z < 5 and 0 < t < 1
• -6+t < x,y,z < 6-t and 1 < t
• -5-t < x,y,z < 5+t and t < 0

(this is the causal completion of the original set) Again I think it reasonable to call this time evolution, because it takes a state defined over some region, and "extends" (along timelike directions) the state to be defined over a larger region.


If a particular AQFT allows any unbounded open region as a domain, another example like my first one (but unitary this time!) would be taking a state defined over some region 0<t<1 and computing a new state defined over the region 7<t<9.
(Again, I suspect for purely category theoretic regions we can say something about slices, e.g. evolving a t=0 region into a t=8 region. But ATM I'm not in the mood to work it out)


Edit: I've thought through the category theoretic argument. The answer is "yes you can", but the assertion is almost entirely content free.

 

[Demystifier]

Thanks Hurkyl, that's very interesting. But this, in fact, is very similar to a variant of many-time formalism I've been talking about. Namely, as I discuss in
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
http://xxx.lanl.gov/abs/0905.0538 [Phys.Lett.B678:218-221,2009]
http://xxx.lanl.gov/abs/0912.1938
in this formalism states are better viewed as states on spacetime, rather than states on a spacelike slice. The concept of "time evolution" does not play any fundamental role. At best, the time evolution is emergent, just as it is emergent in classical "block-universe" view of relativistic mechanics. The same can be said for the formalism you are talking about.

Edit: I've sent you a PM.

 

[Fyzix]

Demystifier: did you check the two papers I posted on MWI in the Heisenberg picture?