Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Squeezing physics out of formal deformation quantizations

+ 9 like - 0 dislike
196 views

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 in Theoretical Physics by Alex M. (45 points) [ no revision ]
retagged Dec 22, 2015

1 Answer

+ 2 like - 0 dislike

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 Arnold Neumaier (12,355 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOver$\varnothing$low
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...