Mathematical Quantum Field Theory - Quantization - Comments

In summary: It should. The link works for me.Actually in arxiv.org/abs/1304.0236 p. 5 of the introduction, and remark 3.3.16.
  • #1
Urs Schreiber
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Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Quantization
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In the link to the previous chapter, "gauge fixing" is misspelled as "gauge fdscixing." Ironically, the spelling of "fixing" needs fixing. Otherwise, I love the insights (even if I only barely understand most of it).

NB: this is probably the only opportunity I'll ever get to correct Urs Schreiber, so I'm reveling in it currently.
 
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  • #3
TeethWhitener said:
"gauge fixing" is misspelled as "gauge fdscixing."

Thanks! Fdscixed now.
 
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  • #4
You have this Part 14 of the series as Chapter 13, but also Chapter 13 as the next one called „Free Quantum Fields”.
 
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  • #5
dextercioby said:
You have this Part 14 of the series as Chapter 13, but also Chapter 13 as the next one called „Free Quantum Fields”.

Thanks! Fixed now.
 
  • #7
Greg Bernhardt said:
There is an inconsistency because I considered the intro as Part 1.

Right, this means that chapter ##n## in the series appears as the ##n+1##st article in the series. But apart from this I had had a typo at the top of this chapter here, where I was pointing to chapter "13. Free quantum fields" instead of chapter "14. Free quantum fields". I have fixed it.
 
  • #8
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] member: 186655 said:
Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group? The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group“, an object that seamlessly leads to the quantum mechanics of the system.
I don't think this is correct. The answer is well-known but different: The Lie group corresponding to the Poisson bracket Lie algebra is the group of classical canonical transformations. The group you refer to is a central extension of the latter. In the symplectic case, it is the group corresponding to the contact structure extending the symplectic structure in one dimension higher.
 
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  • #9
A. Neumaier said:
The Lie group corresponding to the Poisson bracket Lie algebra is the group of classical canonical transformations. The group you refer to is a central extension of the latter.

There is the Lie algebra of symplectomorphisms, locally the Hamiltonian vector fields. A central extension of that is the Poisson bracket, the extension arising from the choice of Hamiltonian for each Hamiltonian vector field.

E.g from Givental, "The nonlinear Maslov index":

GiventalQuantomorphism.png
 

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  • #10
A more canonical albeit less succinct reference is Brylinski 93, sections II.3 and II.4 The Poisson Lie bracket extension of the Hamiltonian vector fields is prop. 2.3.9, its Lie integration to the quantomorphism group extension of the group of Hamiltonian symplectomorphism is prop. 2.4.10 (with notation from prop. 2.3.17).
 
  • #11
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:
A central extension of that is the Poisson bracket, the extension arising from the choice of Hamiltonian for each Hamiltonian vector field.
The Poisson bracket and canonical transformations act on function on phase space. If ##X_f## denotes application of the Poisson bracket to ##g## by left operation with ##f## then ##e^{X_f}## is a canonical transformation.

On the other hand, the group described by Brylinski act on functions of the line bundle. (I don't have access to Givental.)
 
  • #12
A. Neumaier said:
The Poisson bracket and canonical transformations act on function on phase space. If ##X_f## denotes application of the Poisson bracket to ##g## by left operation with ##f## then ##e^{X_f}## is a canonical transformation.

On the other hand, the group described by Brylinski act on functions of the line bundle.

The underlying Hamiltonian vector fields act on functions. The lift of them to Hamiltonians (elements of the Poisson algebra) act on sections of the prequantum line bundle. The choice of Hamiltonian is exactly the choice of action on the fibers of the prequantum bundle. This yields the action of pre-quantum operators in geometric quantization (which becomes the action of actual quantum operators after restricting to polarized sections and to Hamiltonians whose pre-quantum operators respect the polarized sections).

This is all in Brylinki's two sections II.3 and II.4, though the account is a little long-winded. A more conceptual explanation is in our arxiv.org/abs/1304.6292.

A. Neumaier said:
(I don't have access to Givental.)

I had given the link to the copy in GoogleBooks, does it not work for you? But Givental doesn't dwell on this, he just quotes it as a standard fact.
 
  • #14
I still don't see it. Consider the Lie group SO(2,4) acting on the Hermitian symmetric spaces of noncompact type IV_4 (which is symplectic). This is the phase space of the classical Kepler problem. The corresponding Lie algebra so(4,2) acts as a Lie algebra of vector fields on the functions of the symmetric space and not on the line bundle. On the latter, we have the action of a central extension of SO(2,4) and so(2,4). Similarly, your group does not look like the Lie algebra given by the Poisson bracket but like the Lie algebra of a central extension of it.
 
  • #15
A. Neumaier said:
I still don't see it.

For ##(X,\omega)## a connected symplectic manifold, we have a central extension of Lie algebras

$$0 \to \mathbb{R} \overset{\mathrm{const}}{\longrightarrow} \mathrm{Pois}(X,\omega) \overset{p}{\longrightarrow} \mathrm{Ham}(X,\omega) \to 0$$

where

$$\mathrm{Ham}(X,\omega) \hookrightarrow \mathrm{Lie}(\mathrm{Sympl}(X,\omega))$$

is the Hamiltonian vector fields, i.e. those infinitesimal symplectomorphisms ##v## for which there exists some Hamiltonian ##H_v##, i.e. ##d H_v = \iota_v \omega##, and ##\mathrm{Pois}(X)## is the Lie algebra of the actual Hamiltonians ##H##, and ##p## sends ##H_v## to ##v##, thereby forgetting the freedom of adding a constant to ##H_v##.

Do you agree with this?

It seems to me that you keep thinking about ##\mathrm{Sympl}(X,\omega)## and/or its Lie algebra. This is what acts on functions. The lift to the qantomorphism group / Poisson Lie algebra is what acts on sections of any pre-quantum line bundle for ##\omega##; this is the definition of (pre-)quantum operators in geometric (pre-)quantization.
 
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[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:
thereby forgetting the freedom of adding a constant .
Do you agree with this?
OK, the shift of the Hamiltonian by a constant generates the central extension.
 
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What is mathematical quantum field theory?

Mathematical quantum field theory (QFT) is a branch of theoretical physics that combines the principles of quantum mechanics and special relativity to describe the behavior of subatomic particles and their interactions.

What is quantization in quantum field theory?

Quantization in quantum field theory is the process of converting a classical field, such as an electromagnetic field, into a quantum field. This allows for the description of particles as excitations of the quantum field.

What are the main differences between classical and quantum field theories?

Classical field theories describe particles as point-like objects with definite positions and momenta, while quantum field theories describe them as excitations of a quantum field with probabilistic behavior. Additionally, classical field theories do not consider the effects of Heisenberg's uncertainty principle, while quantum field theories do.

What are the mathematical tools used in quantum field theory?

The main mathematical tools used in quantum field theory include functional analysis, group theory, and differential geometry. These tools allow for the formulation and solution of the complex equations that describe quantum field behavior.

What are some applications of mathematical quantum field theory?

Mathematical quantum field theory has many applications in particle physics, condensed matter physics, and cosmology. It is used to study the behavior of subatomic particles, the properties of materials at a microscopic level, and the evolution of the universe.

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