A Why Unitary Evolution? QM Justification Ideas

[Couchyam]

@LittleSchwinger: Also, are you looking for a universality argument (i.e. something like 'unitarity is to be expected'), or explanations and didactic examples that would help build physical intuition?

 

[LittleSchwinger]

Probably more the former, but it doesn't matter too much. It's just that the usual argument of "probabilities must sum to one" isn't really sufficient.

 

[Couchyam]

Probably more the former, but it doesn't matter too much. It's just that the usual argument of "probabilities must sum to one" isn't really sufficient.

I thought the usual argument was that unitarity is to wave function (or Hilbert space?) dynamics what time reversibility (or maybe invertibility, and maybe also Liouville's theorem for numerical stability) is to classical dynamics: even if the laws of nature weren't invariant under time reversal, one could at least in principle infer the past (or future) of a wave function to arbitrary precision from its present value. One could concoct non-unitary maps that are reversible or invertible and that don't 'compress' Hilbert space weirdly and irreversibly, but those maps would probably inevitably break continuity (or violate the 'no distortion' condition.) That being said, wave function collapse is understood to be a relatively discontinuous process, and is certainly non-unitary, and so it could be that operations/transformations of comparable abruptness occur at comparably (in)frequent intervals. It might be interesting to look at ensembles of weak non-unitary transformations (small or infinitesimal deviations from the identity) and how they collectively influence the wave function: for example, would their combined effect result in a more severe non-unitary transformation, or would their non-unitarity somehow 'cancel out' so that they could be approximated by a unitary map? You might also consider the induced dynamics of a wave function with a trivial Hamiltonian (i.e. proportional to the identity) that is measured periodically or intermittently (at regular or random intervals) through observables that undergo a prescribed dynamics.

 

[vanhees71]

I thought the usual argument was that unitarity is to wave function (or Hilbert space?) dynamics what time reversibility (or maybe invertibility, and maybe also Liouville's theorem for numerical stability) is to classical dynamics: even if the laws of nature weren't invariant under time reversal, one could at least in principle infer the past (or future) of a wave function to arbitrary precision from its present value.

Time reflection symmetry has nothing to do with the unitary time evolution. In the parts of Nature (i.e., neglecting the weak interaction) it's an additional discrete symmetry, which must necessarily be represented as an anti-unitary transformation since the Hamiltonian must stay bounded from below under time-reversal. Although with the weak interaction the time evolution in Q(F)T is unitary.

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.

 

[LittleSchwinger]

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.

This is what makes unitarity less fundamental in QFT in curved spacetime, since there we do not have time-translation invariance.

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.

Coming back to non-Rel QM, this is basically what I was asking. Since one can have a CPTP evolution, why wouldn't you. A. Neumaier's argument that isolated systems do seem to undergo unitary evolution empirically is perfectly correct, but I was wondering if there was a more "first principles" reason.

 

[vanhees71]

But "symmetry arguments" are usually considered "first principles reason". Of course, if you have a non-static background spacetime, time-translation symmetry is gone, i.e., at least this argument is no reason for unitarity.

 

[LittleSchwinger]

But "symmetry arguments" are usually considered "first principles reason".

How do you know there isn't a time-translation invariant CPTP evolution?

 

[vanhees71]

This I don't know, but by definition a symmetry is realized in QT as a unitary (or antiunitary) ray representation. Analyzing the Galilei or Poincare groups for Newtonian or special-relativistic dynamics leads to unitary representations for the time translations, and that's why the time evolution of the system is described by a unitary transformation. The assumption of a symmetry is pretty restrictive in this sense.

 

[LittleSchwinger]

This I don't know, but by definition a symmetry is realized in QT as a unitary (or antiunitary) ray representation. Analyzing the Galilei or Poincare groups for Newtonian or special-relativistic dynamics leads to unitary representations for the time translations, and that's why the time evolution of the system is described by a unitary transformation. The assumption of a symmetry is pretty restrictive in this sense.

Let me give some context, because it's not really that I disagree in any way.

Quantum Information is now a common enough topic at universities, either as an upper undergraduate course itself, part of a quantum computing one or aspects of it are built into basic QM courses. As students become more familiar with its techniques they see CPTP maps and even CPTP evolutions more and more, so they could ask:
"Well why can't you represent time evolution with a CPTP evolution? Why does it have to be represented unitarily?"

My thinking is basically the same as yours. There are time-translation invariant CPTP evolutions, but they don't represent a symmetry. They won't preserve things like transition amplitudes. So this symmetry based argument you gave earlier is much better than the usual "probs should sum to one".

 

[vanhees71]

I must admit that I'm pretty ignorant about this very interesting topic of quantum information, but of course CPTPs that are not induced by a unitary time evolution are common for open quantum systems, because the fundamental symmetries must only be realized by "closed systems", i.e., if you have a system composed of two parts (e.g., a particle and a heat bath coupled to each) and you consider the particle alone, i.e., "trace out the heat bath" the time evolution of the reduced density operator of the particle is a CPTP, but not a unitary time evolution in the particle Hilbert space alone. Only the total closed system, particle+heat bath, is described by a unitary time evolution.

 

[LittleSchwinger]

Only the total closed system, particle+heat bath, is described by a unitary time evolution.

Agreed, basically students are increasingly asking why can't the total closed system also be described with a CPTP evolution.

 

[Couchyam]

Time reflection symmetry has nothing to do with the unitary time evolution. In the parts of Nature (i.e., neglecting the weak interaction) it's an additional discrete symmetry, which must necessarily be represented as an anti-unitary transformation since the Hamiltonian must stay bounded from below under time-reversal. Although with the weak interaction the time evolution in Q(F)T is unitary.

The symmetry argument for unitarity of the time evolution is that time-translation invariance as a continuous symmetry must be realized as unitary transformation.

@vanhees71: nowhere in my post did I mention time reflection symmetry. A process can be time reversible (i.e. there exists a way to infer the past from the present) without being symmetric under time reversal.

Also, regarding (continuous) symmetry as a basis for unitarity, (i) there exist an abundance of non-unitary representations of non-compact Lie groups such as the Galilean and Poincaré groups, (ii) how do you know that those symmetries aren't an emergent property at macroscopic (i.e. N of order Avogadro's number) scales, (iii) there are plenty of systems that exhibit unitarity (or "near" unitarity, to within one part in a million say) where those symmetries are absent (e.g. small molecules.) Are you arguing that unitarity applies at a 'global' level (i.e. for a wave function describing the universe) on the basis of those symmetries, and that the unitarity of systems for which those symmetries are spontaneously broken is "induced" in a way from the unitarity of the universe?

 

[bhobba]

What is it they said in Blazing Saddles - I like to keep my audience riveted. Lovely thread.

For a textbook reference, see Ballentine, page 64 in my edition, but is likely in the first couple of pages of Chapter 3 in any edition. Here he evokes Wigner's Theorem, which has already been mentioned.

My contribution is for those that may not have heard of it before; here is the statement and proof (as mentioned originally due to Weinberg):
https://arxiv.org/abs/1810.10111

Thanks
Bill

 

[vanhees71]

What is it they said in Blazing Saddles - I like to keep my audience riveted. Lovely thread.

For a textbook reference, see Ballentine, page 64 in my edition, but is likely in the first couple of pages of Chapter 3 in any edition. Here he evokes Wigner's Theorem, which has already been mentioned.

My contribution is for those that may not have heard of it before; here is the statement and proof (as mentioned originally due to Weinberg):
https://arxiv.org/pdf/1603.00353

Thanks
Bill

I don't know, where in this paper (no math!) is this important proof. The proof is of course due to Wigner and Bargmann. A very nice treatment is in the old edition in the QT textbook by Gottfried.

 

[gentzen]

My contribution is for those that may not have heard of it before; here is the statement and proof (as mentioned originally due to Weinberg):
https://arxiv.org/pdf/1603.00353

I guess you wanted to post a different reference. This is the same reference you also posted at the same time in another thread, and is unrelated to unitary evolution.

 

[LittleSchwinger]

For a textbook reference, see Ballentine, page 64 in my edition, but is likely in the first couple of pages of Chapter 3 in any edition. Here he evokes Wigner's Theorem, which has already been mentioned.

I only read Ballentine recently thanks to this forum. A really good text, wished I'd had it as an undergraduate. I loved the section on state tomography.

 

[bhobba]

I guess you wanted to post a different reference. This is the same reference you also posted at the same time in another thread, and is unrelated to unitary evolution.

Whoops. Sorry guys - fixed now.

Thanks
Bill

 

[bhobba]

I only read Ballentine recently thanks to this forum. A really good text, wished I'd had it as an undergraduate. I loved the section on state tomography.

This book had a strong effect on me.

Thanks
Bill

 

[LittleSchwinger]

Whoops. Sorry guys - fixed now.

Thanks
Bill

Thanks for that. What a lovely paper. Given the above and your like of Ballentine, I think you might enjoy Talagrand's new book on QFT where he really digs deep into representations of the Poincaré group. He treats even massive Weyl Spinors.

 

[bhobba]

Thanks for that. What a lovely paper. Given the above and your like of Ballentine, I think you might enjoy Talagrand's new book on QFT where he really digs deep into representations of the Poincaré group. He treats even massive Weyl Spinors.

Thanks for that. Just now got the book. Always on the lookout for QFT books for mathematicians because that is my background.

Thanks
Bill

 

[bhobba]

Got the e-book. Just skimming now. I love the remark: 'To top it all, I was buried by the worst advice I ever received, to learn the topic from Dirac’s book itself!'

It was the second serious book on QM I read and know the issue only too well. The first was Von Neumann's book which is excellent for mathematicians since it is just an extension of Hilbert-Space theory. I was confident when I went on to Dirac but became unstuck with that damnable Dirac Delta function. It led me on a sojourn in Rigged Hilbert Spaces that had nothing to do with physics. I came out the other end with the issues resolved - but at that stage of my QM journey, it was not a good move.

Thanks
Bill

 

[vanhees71]

But Dirac's book is excellent. What's the problem with it? Von Neumann is of course mathematically rigorous and Dirac is not.

 

[bhobba]

But Dirac's book is excellent. What's the problem with it? Von Neumann is of course mathematically rigorous and Dirac is not.

It is excellent, physically better Von-Neumann. The only issue, which has been 'fixed' and is no longer of any relevance, is the diatribe Von Neumann writes about it at the beginning of his book. It is easy for a math graduate to read Von Neumann after studying Hilbert Spaces, but Dirac is more problematic. I personally believe every math degree should include distribution theory because of its wide use in applied math. As part of that, a few paragraphs like that found in Ballentine is all that is needed. I am going through Talagrand's book and it has a more complete explanation. If that was done first, then Dirac is fine. Perhaps include it in a modern preface to both books - just an idea.

Thanks
Bill

 

[Demystifier]

Note that, despite for claims often to be read in textbooks, there's no need for parity or time-reversal symmetry for the detailed-balance relations to be valid.

Interesting, can you give a reference with more details?

 

[vanhees71]

Landau and Lifshitz vol. X! You find the argument also in my transport manuscript:

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

 

[Demystifier]

Landau and Lifshitz vol. X! You find the argument also in my transport manuscript:

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

So in classical physics, detailed balance is a consequence of the Liouville theorem, would you agree?

 

[A. Neumaier]

But Dirac's book is excellent. What's the problem with it?

It has an explicit collapse postulate!

 

[vanhees71]

If that were a problem, you couldn't recommend very many otherwise excellent QT textbooks. Indeed, physics students also have to learn to "not listen to their words but looking at their deeds" (as Einstein adviced concerning all writings of theoretical physicists). That the collapse postulate is at best superfluous and at worst simply a contradiction to causality in the relativistic context should be obvious (though obviously it isn't for many philosophy-inclined physicists who still think that it's needed), but all this is off-topic here in the scientific part of the QT forum!

 

[A. Neumaier]

physics students also have to learn to "not listen to their words but looking at their deeds" (as Einstein adviced concerning all writings of theoretical physicists).

This also applies to reading your lecture notes and your contributions to PF.. The collapse - not necessarily the von Neumann form but in the general POVM related version - is used informally almost everywhere, namely whenever one argues how a state is prepared.

 

[vanhees71]

For sure, it's not an instantaneous collapse affecting the physics at all points in space simultaneously. All our experiments are local, and nothing can violate the relativistic speed limit of signal propagation.