# Squeezing physics out of formal deformation quantizations

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I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested in quantization on the cotangent bundle of a Riemannian manifold. Even more, I am curious about the success so far of the approach based on formal deformations.

Background

In Kontsevich's "Deformation Quantization for Poisson Manifolds" - IHES version, remark 1.5 claims that "Also, it is not clear whether the natural physical counterpart for the “deformation quantization” for general Poisson brackets is the usual quantum mechanics. Definitely it is true for the case of non-degenerate brackets, i.e. for symplectic manifolds manifolds, but our results show that in general a topological open string theory is more relevant.". (Interestingly, the (original, earlier) arXiv version of the same paper is much more modest, stating only "Also, it is not clear whether the “deformation quantization” is natural for quantum mechanics. This question we will discuss in the next paper. A topological open string theory seems to be more relevant.")

In the same spirit of unproven but seducing claims, Dito and Sternheimer write in a survey article from 2001 that "We stress that deformation quantization should not be seen as a mere reformulation of quantum mechanics or quantum theories in general. At the conceptual level, it is the true mathematical formulation of physical reality whenever quantum effects have to be taken into account. The above examples show that one can indeed perform important quantum mechanical calculations, in an autonomous manner, entirely within deformation quantization – and get the results obtained in conventional quantum mechanics. Whether one uses an operatorial formulation or some form of deformation quantization formulation is thus basically a practical question, which formulation is the most effective, at least in the cases where a Weyl or Wigner map exists. When such a map does not exist, a satisfactory operatorial formulation will be very hard to find, except locally on phase space, and deformation quantization is the solution. One can of course (and should in practical examples, as we have done here, and also for algebraic varieties) look for small domains (in $N$) where one has convergence. We can then speak of “strict” deformation quantization. In particular we can look for domains where pointwise convergence can be proved; this was done e.g. for Hermitian symmetric spaces [CGR]. But it should be clearly understood that one can consider wider classes of observables – in fact, the latter tend to be physically more interesting – than those that fit in a strict $\Bbb C ^∗$-algebraic approach." (remark 2.2.3.3).

To justify the above, the authors present (section 2.2.3) an "autonomous manner" [sic!] of obtaining numbers (i.e. spectra) in the framework of this approach, for instance using the "star exponential". This puzzles me like nothing else, because $\hbar$ is no longer trated as a formal parameter, but as what it is in physics, and the formal parameter is $\nu$; this is weird, since quantum mechanics was supposed to have $\hbar$ as the "deformation parameter".

The Question

Given all the above, is there any possibility to extract physics (i.e. numbers: spectra of observables - corresponding to the possible outcomes of measurements, means, mean square deviations, all the numerical stuff that is of interest to a physicist) from these formal deformations, and to recover the same numerical results as one obtains from the usual formulation of quantum mechanics? What made Kontsevich become certain of this possibility between those successive versions of his paper? What is paragraph 2.2.3.2 from Dito and Sternheimer's article trying to do? In short, could one entirely reformulate quantum mechanics in this new language, a perfect parallel of the traditional formulation? Or are its proponents aggressively advertising for a product that in reality is hard to sell? (At a superficial look, and from discussion with people working in connected areas, it seems to me that the hype around this subject that manifested in the '90s and early 2000s has faded, together with the optimism and great expectations that accompanied it. Is my perception correct?)

This post imported from StackExchange MathOverflow at 2015-12-22 18:44 (UTC), posted by SE-user Alex M.
asked Dec 15, 2015
retagged Dec 22, 2015

## 1 Answer

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There are two versions of deformation quantization, the formal one that treats $\hbar$ as a formal parameter in a power series, and the topological one (strict, Rieffel quantization) that that treats $\hbar$ as a small but finite number. The latter is much harder (and less developed), as the existence requirements are much more stringent, but it is fully equivalent to quantum mechanics (it gives a Hilbert space and everything else).

On the other hand, formal deformation quantization is much more versatile but only gives semiclassical expansions of quantities presumed (but not known) to exist - unless one has also alternative approaches. However, whenever one gets an expansion that - like any asymptotic expansion - makes sense numerically with the true value of $\hbar$, one gets results qualitatively comparable with what one gets in quantum field theory from the asymptotic expansions there (which in 4D are also not known to correspond to rigorously constructed quantities). [In particular, the loop expansion in QFT is a semiclassical expansion that can presumably be obtained also by deformation quantization.] In this sense, formal deformation quantization can deliver quantitative predictions.

In summary, I would view formal deformation quantization as a view on quantum physics that gives some insight but is to be used together with other techniques. But to call it an autonomous approach to quantum physics is an exaggeration - it cannot construct the Hilbert space needed for the interpretation of quantum mechanics.

answered Dec 25, 2015 by (15,608 points)

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