Mathematical Quantum Field Theory - Quantization - Comments

[Urs Schreiber]

Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Quantization

Continue reading the Original PF Insights Post.

 

[TeethWhitener]

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.

 

[Urs Schreiber]

"gauge fixing" is misspelled as "gauge fdscixing."

Thanks! Fdscixed now.

 

[dextercioby]

You have this Part 14 of the series as Chapter 13, but also Chapter 13 as the next one called „Free Quantum Fields”.

 

[Urs Schreiber]

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.

 

[Greg Bernhardt]

There is an inconsistency because I considered the intro as Part 1.

 

[Urs Schreiber]

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.

 

[A. Neumaier]

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.

 

[Urs Schreiber]

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":

 

[Urs Schreiber]

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).

 

[A. Neumaier]

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.)

 

[Urs Schreiber]

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.

(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.

 

[Urs Schreiber]

A more conceptual explanation is in our arxiv.org/abs/1304.6292.

Actually in arxiv.org/abs/1304.0236 p. 5 of the introduction, and remark 3.3.16. But this points to arxiv.org/abs/1304.6292 for the computation relating this to the Poisson bracket.

 

[A. Neumaier]

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.

 

[Urs Schreiber]

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.

 

[A. Neumaier]

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.