The most general procedure for quantization

Question

I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me):

A physical system is usually described in terms of states and observable. In the Hamiltonian framework of classical mechanics, the states form a symplectic manifold $(M,\omega)$ and the observables are functions on $M$. The dynamics of a (time invariant) system is a one parameter group of symplectic diffeomorphisms; the generating function is the energy or Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine symplectic space and the motion is by a one-parameter group of symplectic transformations. This general descriptions applies to any system that includes classical particles, fields, strings and other types of objects.

The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:

1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?

2. What is the general procedure for such a construction, given $M$ and $\omega$?

3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)

4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?

5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)

6. What is meant by a 'free' system as given above?

7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?

8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system?

I have asked a lot of long questions, so please answer as many as you can and link relevant articles.

This post imported from StackExchange Physics at 2014-04-12 19:04 (UCT), posted by SE-user dj_mummy

Answer 1

This is a surprisingly good question. "Good", because it is indeed true that there is this very general prescription for quantization; and "surprisingly" because, while the general idea has been around for ages, this has been understood in decent generality only last year!

Namely, on the one hand it is long appreciated in the context of quantum mechanics that what physicists sweepingly call "canonical quantization" is really this: the construction of the covariant phase space as a (pre-)symplectic manifold, and then the quantization of this by the prescription of either algebraic deformation quantization or geometric quantization.

In contrast, it has been understood only surprisingly more recently that established methods of perturbative quantization of field theories, especially in the guise of Epstein-Glaser's causal peruturbation theory (such as QED, QCD, and also perturbative quantum gravity, as in Scharf's textbooks) are indeed also examples of this general method.

For free fields (no interactions), this was first understood in

• J. Dito. "Star-product approach to quantum field theory: The free scalar field". Letters in Mathematical Physics, 20(2):125–134, 1990.

and then amplified in a long series of articles on locally covariant perturbative quantum field theory by Klaus Fredenhagen and collaborators, starting with

• M. Dütsch and K. Fredenhagen. "Perturbative algebraic field theory, and deformation quantization". In R. Longo (ed), "Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects", volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001.

Curiously, despite this insight, these authors continued to treat interacting quantum field theory by the comparatively ad hoc Bogoliubov formula, instead of similarly deriving it from a quantization of the (pre-)symplectic structure of the phase space of the interacting theory.

That last step, to show that the traditional construction of interacting peturbative quantum field theory via time-ordered products and Bogoliubov's formula also follows from the general prescription of deformation/geometric quantization of (pre-)symplectic phase space was made, unbelievably, only last year, in the highly recommendable thesis

• Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory arxiv:1603.09626

Just read the introduction of this thesis, it is very much worthwhile.

(I learned about this article from Igor Khavkine and Alexander Schenkel, for which I am grateful.)

In a similar spirit a little later appeared

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory arxiv:1612.09157

which disucsses the situation in a bit more generality than Collini does, but omitting the technical details of renormalization in this perspective.

Comments

Very nice references by Collini and Hawkins/Rejzner. However, these are not true quantizations like in geometric quantization (where a Hilbert space action is constructed) but only formal quantizations as (divergent) power series in $\hbar$. The full construction of interacting relativistic QFTs (i.e., with operators rather than formal series) is still completely open in 4D.

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Yes, strict (i.e., Rieffel) deformation quantizations are true quantizations, but they suffer from at least the same limitations as geometric quantization; the latter seems to me the more general framework. Of course, QFT is the quantization of infinite classical dimensions, where geometric quantization seems to have survived only as a collection of ad hoc constructions, at least up to now.

I'll comment your PF-article on that forum.

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Yes, geometric quantization needs re-thinking in field theory, but strict deformation quantization could at least provide a clean statement of the problem.  

(In an age where there are people claiming they have already non-perturbatively quantized gravity, it is good to at least state the problem description precisely. Also Jaffe-Witten's Millenium Prize "problem description" remains very vague on what the problem description really is.)

I believe (here) that geometric quantization in field theory will require passing attention to the "higher prequantum bundle" on the jet bundle of the field bundle; but this still needs more work before I can claim the Millenium Prize money :-)

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For the Yang-Mills Millennium problem some minimal requirements extracted from the Jaffe-Witten description are in my review here. I am working with Tom Hales on a fully precise description of what is asked for (including a view of the background needed, complementary to what Jaffe-Witten write); cf. his request here. There are many aspects to this problem that must be viewed (and answered) together....

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Yes, Tom Hales, who solved the 350 years old Kepler conjecture, is interested in getting a formally fully precise (i.e., machine-understandable) version of all Clay Millennium problems. See his lecture Big Conjectures at https://www.newton.ac.uk/event/bpr/seminars (video available).

To get a formalizable description of the Clay Millennium Yang-Mills problem, I think it will take about half a years work for a gifted PhD student who is already familiar with functional analysis and ordinary quantum mechanics - given the (20-30 page?) background outline that Hales and myself will hopefully have written in the near future. It is by all means not a simple task, as part of the work asked for is to provide a mathematical model defining what it means to have a rigorous model of a quantum gauge theory in 4D. To actually rewrite such a formalizable description as formal (i.e., machine understandable) mathematics may well take another year....

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I naively thought that we should quantize the observables, which are quantized in Nature. But I was wrong. The desire to quantize has its own life and we now quantize according to quite different criteria.

Similarly for the perturbation theory applications. I Naively thought that we should choose the initial approximation as close to the exact solution as possible and then the perturbative corrections would be small and reasonable. But I was wrong. The exact solution may not exist at all and the initial approximation may be completely weird, but still we apply the perturbation theory and then modify the obviously wrong solutions by hands to get the right results.

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@ArnoldNeumaier thanks for the pointer! I had missed your discussion of Dynin's articles. That's very useful.

Also, while I had seen the request about formalization of the QYM problem, I had missed that it was no one less than Prof. Tom Hales asking this. So you think you can provide him with a problem description which he will recognize as formal mathematics? That would be remarkable, please do alert me once you have something ready for sharing on this point.