Second Quantization and Field Operators

[reilly]

Fra and strangerep Quite honestly, I don't understand at all what you are saying.

Just for fun, I've gone to my library to see if, somehow I've misled myself into some dicey territory. So, Dirac(in Quantum Mechanics), Landau and Lifschitz (Quantum Mechanics; the Nonrelativistic Theory) March Young and Sampathar(The Manybody Problem in Quantum Mechanics.)Ian D. Lawrie(A Unified Grand Tour of Theoretical Physics) Mandel and Wolf(Optical Coherence and Quantum Optics with a particularly elegant discussion), Feynman in his groundbreaking paper, Theory of Positrons, Bloch and Nordsieck in Radiation Field of the Electron(1937), which solved the problem of the infrared divergence -- they all agree with me -- because that's where I learned about 2nd quantization.

And, my discussion of the unitary transformation is in total agreement with these authors. Second quantization i's basically pretty simple and straightforward, and has been a staple, an old chestnut, of much of physics throughout the history of modern QM. My discussion above could almost be a quote from any of the above references.

That is strangerep, all these authors agree that the mapping is as I suggested, Fock space is merely a representation of an infinite number of degrees of freedom. You have a very heavy job to demonstrate your claim, which is totally at odds with usual practice, and is not supported by many, many authors of significant stature.

This discussion provides a great example that physics has a history, and if you do not know about it, then you are very likely to make mistakes.

Fra - Bogulubov is taking a standard Fock space, and using a unitary transformation to another Fock space. Quantization? -- in the eye of the beholder. I you take his transformed states back to configuration space, you will find a different mix of states than you would get with the standard Fock transformation. That's by most definitions, a different quantization scheme.

So, you can find confirmation of my discussion in hundred(thousands?) of of books and articles and dissertations.
Regards,
Reilly Atkinson

 

[strangerep]

Fock space is merely a representation of an infinite number of degrees of freedom.

Yes, but that's not the whole story.

You have a very heavy job to demonstrate your claim, which is totally at odds with
usual practice, and is not supported by many, many authors of significant stature.

All I said was that 1st quantization is the passage from classical to single-particle (1-dof)
quantum. 2nd quantization is the passage from 1-dof quantum to inf-dof quantum.
There's nothing new in such a statement.

You seem to think I'm saying something non-standard, made up out of my own head.
But I'm not. The textbooks you mentioned don't talk much (if at all) about the issues
of inequivalent unitary representations in infinite-dimensions. Umezawa's "Thermofield
Dynamics..." (ch2, I think) explains some of the extra subtleties on a pedagogical level.
Haag's "Local Quantum Physics" also discusses it, though at a more demanding level.

However, I sense from your answer that I'm annoying you, so I'll shut up now until
I find out what the actual point of disagreement is here. I suspect we're currently
just talking past each other.

P.S: I don't understand Fra either. Too fuzzy.

 

[reilly]

Yes, but that's not the whole story.


All I said was that 1st quantization is the passage from classical to single-particle (1-dof)
quantum. 2nd quantization is the passage from 1-dof quantum to inf-dof quantum.
There's nothing new in such a statement.

You seem to think I'm saying something non-standard, made up out of my own head.
But I'm not. The textbooks you mentioned don't talk much (if at all) about the issues
of inequivalent unitary representations in infinite-dimensions. Umezawa's "Thermofield
Dynamics..." (ch2, I think) explains some of the extra subtleties on a pedagogical level.
Haag's "Local Quantum Physics" also discusses it, though at a more demanding level.

However, I sense from your answer that I'm annoying you, so I'll shut up now until
I find out what the actual point of disagreement is here. I suspect we're currently
just talking past each other.

P.S: I don't understand Fra either. Too fuzzy.

First let me say that I've got a lot of respect for your views. And your comments on inequivalent unitary representations are, I think, correct, and I need to review the topic, which I certainly will. Whether we agree, thus remains to be seen. But, in any event, I'll bet we would both write out the QED interaction and the typical potential energy for a second quantized system in much the same fashion. I suspect, as you apparently do, our differences are not as great as I might have once thought.

Annoyed? Not at all. Quite the contrary. Regards,
Reilly

 

[reilly]

Fra -- There are times when you are one hell of a thinker.And I'm sure there are some good nuggets in your posts here.You would be doing us all a favor if you could give a precis, a summary of your posts.
Thanks, Reilly

 

[Fra]

Reilly thanks for the nice words. Indeed I am fuzzy, but this is not easy stuff, it's a snapshot of the process that's fuzzy and the more clear direction and strategy I see may be subjective. Reality is fuzzy and that is stressful.

I think I'm slowly starting to see your point of view now, and I think that view is simplifying away some of the mystery without resolving it. I can say that when reading Dirac's classic : Principles of QM - I think I disagree with his reasoning very early in the book - and the first point is on the notion of probability. Sure, anything that fulfills the axioms of probability is a probability, but that's not the real problem at all. It's how that connects to real measuremnts, done by real observers subject to real constraints. He ignores a range of these issues. He seems content with imagining that "in principle". But that is a mistake IMHO. It's not that he is formally unclear, it's that he is physically unclear IMHO. Now, then he further builds onto this framework. I am going back to the first point of objection, and rework it from there. Now this doesn't mean redoing history, its not that bad at all.

It would be naive of me to even attempt a clear and precise description of my view here. At least now, easter coming up and all and this is a passion of mine, not a job. Meaning I don't have 8 hours / day to spend on this. But I will get back to it as soon as I have done more work myself.

Reilly, let me get back to this. I do think about these things and I do intend to produce some writings. That will explain my point of view, from start to finish. Sorry if I cause confusion. Sooner or later I'll probbly post more. I temporarily put my own work aside to read up on rovelli and penrose's view on foundational QM and QG, because I sensee a connection. but as soon as I've done that I'll resume where I left off.

/Fredrik