# Why unitary evolution?

Gold Member
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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.

bhobba
Gold Member
But "symmetry arguments" are usually considered "first principles reason".
How do you know there isn't a time-translation invariant CPTP evolution?

Gold Member
2022 Award
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.

Gold Member
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".

Gold Member
2022 Award
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
Gold Member
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?

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Mentor
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

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Couchyam, vanhees71, gentzen and 1 other person
Gold Member
2022 Award
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.

bhobba and LittleSchwinger
Gold Member
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.

bhobba, physika and vanhees71
Gold Member
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, gentzen and vanhees71
Mentor
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

gentzen and LittleSchwinger
Mentor
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
Gold Member
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.

vanhees71 and bhobba
Mentor
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

vanhees71 and LittleSchwinger
Mentor
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 and LittleSchwinger
Gold Member
2022 Award
But Dirac's book is excellent. What's the problem with it? Von Neumann is of course mathematically rigorous and Dirac is not.

LittleSchwinger and bhobba
Mentor
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

vanhees71 and LittleSchwinger