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- #37

LittleSchwinger

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

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- #39

LittleSchwinger

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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".

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- #41

LittleSchwinger

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Agreed, basically students are increasingly asking why can't the total closed system also be described with a CPTP evolution.Only the total closed system, particle+heat bath, is described by a unitary time evolution.

- #42

Couchyam

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@vanhees71: nowhere in my post did I mention time reflectionTime 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.

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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- #43

bhobba

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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

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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- #44

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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.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

- #45

gentzen

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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.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

- #46

LittleSchwinger

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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.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.

- #47

bhobba

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Whoops. Sorry guys - fixed now.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.

Thanks

Bill

- #48

bhobba

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This book had a strong effect on me.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.

Thanks

Bill

- #49

LittleSchwinger

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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.Whoops. Sorry guys - fixed now.

Thanks

Bill

- #50

bhobba

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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

- #51

bhobba

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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

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- #53

bhobba

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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

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