Second Quantization and Field Operators

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back to the original question...i have been studying these things for the past few days and wrote a latex document on the subject. nothing really new here, but i tried to focus on pedagogy and examples so that someone with a basic knowledge of QM can understand 2nd quantization. take a look at my paper and let me know what you think!

http://www.physics.ucsd.edu/~mkrems/research-physics/secondquantization.pdf [Broken]
 
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  • #27
reilly
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back to the original question...i have been studying these things for the past few days and wrote a latex document on the subject. nothing really new here, but i tried to focus on pedagogy and examples so that someone with a basic knowledge of QM can understand 2nd quantization. take a look at my paper and let me know what you think!

http://www.physics.ucsd.edu/~mkrems/research-physics/secondquantization.pdf [Broken]
Nice job. I went through very quickly, and it seems to me that you have it right. Many texts cover the same material, often in the same fashion as yours. Congrats.
Regards,
Reilly Atkinson
 
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  • #28
strangerep
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My question was and is pretty simple minded.
I guess I don't understand the question... :-(

The unitary transformation to which I referred is that between the Hilbert space for an infinite number of particles (or oscillators, say in a small oscillation system with an infinite number of oscillators. ) In this case, so-called 2nd quantization is just a unitary map away from the particle representation.
No,... wait... 2nd quantization is the passage from 1 degree of freedom (1-particle Hilbert
space) to infinite degrees of freedom (many-particle Hilbert/Fock space). That's not a
unitary mapping.

Further, this type of mapping is ubiquitous -- see Boguliubov's treatment of superfluidty, see the work done on superconductivity. The basic version goes like: q and p <-> a and a*, with q = a + a*, p-> i (a*-a), apart from constants. q,p <-> a*,a is clearly acomplished by a unitary transformation.
What you wrote is just a mapping between 1-particle operators. However, the Bogoliubov
transformations act (in general) between inequivalent (disjoint) Fock spaces (i.e., between
distinct Fock spaces, each constructed cyclically from different orthogonal vacua).

(I'd better not attempt any further response to the rest of your post now until the
question becomes clearer.)
 
  • #29
Fra
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reilly said:
My question was and is pretty simple minded. The unitary transformation to which I referred is that between the Hilbert space for an infinite number of particles (or oscillators, say in a small oscillation system with an infinite number of oscillators. ) In this case, so-called 2nd quantization is just a unitary map away from the particle representation. Further, this type of mapping is ubiquitous -- see Boguliubov's treatment of superfluidty, see the work done on superconductivity. The basic version goes like: q and p <-> a and a*, with q = a + a*, p-> i (a*-a), apart from constants. q,p <-> a*,a is clearly acomplished by a unitary transformation.
I'm not sure I understood your reflection in this context either.

As I see it the Boguliubov transformation is unitary, but doesn't directly have anything two do with the quantization procedure as such or maybe you see a connection that I'm missing?

I rather rather see it as a transformation between observers within the same quantization so to speak. Indeed this is also interesting but I don't see the direct connection to the discussion of quantization, relative to a specific observer? But it seems this could be a different discussion which I think of as appearing "later" in the reconstruction I have in mind?

But then I think there is different ways of attacking these things, and I can only speak for how I prefer to attack it but I don't expect everyone to agree, and I could certainly change my mind.

IMO, the notion of "particles" and the problems how to maintain a consistent particle interpretation in higher quantisations, does not have anything directly to do with the formal second quantization as I see it at least. That's why I personally try to not use those words. I see that almost as an "interpretational" kind of issue.

The only reason I said fock space above is to compare it to the standard interpretation. But IMO it's just a hilbert space in the same way as the original hilbert space - though anothre hilbert space, not the same one.

I think in terms of relative to the observer, distinguishable states of a distinguishable microstructure. This is why indistinguishable events can only be counted and correlated with other events and thereby - indirectly - be distinguished as always appearing correlated to other events, that effectively becomes a label.

If we call these events particle detections of detections or something else is IMO not the interesting part. "Particle" is IMO just a label for these things. I have no preconceptual ideas of properties particle may have. And just because of this, using the particle word brings more confusion that it resolves IMHO.

I think in terms of what we can observere and distinguish. And the question is, how can we best organized and exploit this data to predict what we will observe next? If we know this well, we can reorganized ourselves to be optimally fit in this environment. If we fail, the chance is that we(the observer) is destabilised. Relabeling the names of things doesn't change anything.

This is a more abstract view, but IMHO it gets cleaner, and there is no need to bring in visual elements from classical mechanics. It doesn't help me at least. It's only a diversion that found more desctructive than constructive. But that's just me.

Perhaps someone else can give some more sensible comments on particle interpretations specifically. I really don't want to corrupt my brain to even think about it. I go get headache and it's not leading me anywhere :)

The way I see quantization is how one ideally distinguishable states of a microstructure, now for whatever reason now has become to uncertain and unpredictable that it destabilises the support of the microstructure itself! and here the observations are focused in the observations of the first state. So that we know ask not what is this state, but rather what do we know about this state in a sense that our confidence is quantified, beyond the simplistic {true,false}. So we are inflating a boolean state space to a continuum probability [0,1]. So what this does is that it increases the degrees of freedom in the microstructure. And in my personal thinking, this microstructure lives in the structure that defines the observer. This is why the infinite quantization does not make sense. A finite observer can not support that structure.

Then we can parametrise this continuum. So that our information of the first state (which wasn't definite) is again definite. Now if for whatever reason, the observer finds that this still doesn't make sense, he can again consider the information of the information. Second quantization.

And so on - but WHY does this make sense? What distingusishes this from any arbirary scheme?

But in this view I ignored the dynamics, and the "trick" that suddently a superposition can be treated as real state in further quantizations. The complete understnading is IMO still not in my head at least. But in short I think that superpositioned states ARE real. They are realised in the observers microstructurs as dynamical correlations. And these correlations can ba rated. Here I see the issue of inertia coming in. A superposition can be assigned inertia which can explain it's stability.

I'm still thinking about this. I think one needs to bring some structure into this. There is so much semiclassical thinking going on that to speak for myself at least, it's easy to loose track of your own reasoning and where assumptions to in and where conclusion come out. I have started to form my own personal structure of this and I am reasonably clear on some things, but the entire picture is complex. The most hairy parts is the dynamics and how the microstructure evolves. Creating and collapsing of microstructres in the way I use the word I see related to the origin of inertia. When the microstructure looses it's inertia it collapses, and similarly the inertia of deviations somehow DRIVES the formation of new structures.

This is more interesting, but also more complex. And would need it's own thread. And I probably have more questions than answers anyway. Perhaps someone else who knows more can share their views.

So in my thinking, I'm questioning the formalism of quantum field theory, which is really no different in principle from QM, except for the space-time part, which I consider to be a separate problem - though still entangled. If we try to analyse this from scratch, one might want to release oneself from what we think we know, and requestion in, and along the way see what's good and what's not good. It's what I try to do as far as my tiny head allow me to :uhh:

/Fredrik
 
  • #30
Fra
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transformation between observers
About this, I even think of these transformations as non-deterministic. The point IMHO is that, one observer can not just transform his state and somehow figure out with certainty everybody elses opinon. The flaw in this is IMO that the transformation itself contains information! And this can sit nowhere else but in the observers microstructure, and it's dynamical just as everything else.

The knowledge of this transformation is physical information IMO. This is part of the problem with GR + QM IMO. And i think different people do have different views here. And my view is probably not the most common one, but I never made a poll on it.

This view makes everything float I know. But I stick to it because I think it's the way things are. I'm trying to acknowledge the limits and make the best possible predictions given these constraints.

/Fredrik
 
  • #31
reilly
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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 infra-red 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
 
  • #32
strangerep
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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.
 
  • #33
reilly
Science Advisor
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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
 
  • #34
reilly
Science Advisor
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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
 
  • #35
Fra
3,097
144
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
 

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