most recent 30 from http://mathoverflow.net 2013-05-20T13:28:20Z http://mathoverflow.net/feeds/user/17780 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107554/problem-of-quantization-state-of-the-art Sadiq Ahmed 2012-09-19T11:48:48Z 2012-09-21T11:02:16Z

<p><code>The "problem of quantization"</code>: </p> <blockquote> <p>Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate functions $p_j$ and $q_j$ $(j = 1, . . . , n)$, and a mapping $Q : f → Q_f$ from $Obs$ into self-adjoint operators on $L^2(R^n)$ such that (q1)–(q5)* are satisfied.</p> <p>(*Please refer to the paper for the conditions (q1) - (q5).)</p> </blockquote> <p>Ref: Quantization Methods: A Guide for Physicists and Analysts, pp. 2-3, [<a href="http://arxiv.org/abs/math-ph/0405065v1" rel="nofollow">math-ph/0405065</a>]</p> <p><code>To researchers in this area</code>:</p> <p>What is the current state-of-the-art in this area?</p>

http://mathoverflow.net/questions/107554/problem-of-quantization-state-of-the-art/107759#107759 Sadiq Ahmed 2012-09-21T10:55:16Z 2012-09-21T11:02:16Z

<p>Quantization is big area, so let me concentrate on some mathematical aspect which are close to me and somewhat related with representation theory and algebraic geometry. Undoubtedly this is biased and incomplete answer. Hope others would add more.</p> <p>Morally quantization is a bridge between commutative and non-commutative worlds. Let me say that the hope is to built such a fundamental bridge that everyone can put on a truck everything he wants on the commutative side and the truck can succsesfully go to the non-commutative side, while the current state of art that heavy trucks cannot go, and sometimes you should sit on a horse and be very experienced to reach the other side...</p> <p>1) Construction of non-commutative algebra.</p> <p>On the commutative side we have an algebra and Poisson bracket. We want to construct non-commutative algebra which is deformation quantization. Great achievement is due to M. Kontsevich 1998 here, which shows that smooth algebra of functions can be quantized in the sense of formal power series over formal parameter.</p> <p>However questions remains. a) We want not the formal power series, we want actual non-commutative algebra. This is not really understood (imho). Moreover I am not sure it is clear what kind of algebra we should get - C^*-algebra, fon-Neaumann algebra or what ? Nick Landsman worked on this, and at least he has some proposals what should we get in analytical setup, but it seems the goals are not achieved, yet.</p> <p>b) It is also not so clear about the uniqueness of the construction - ideally we want unique algebra up to isomorphism. However Kontsevich construction depends on two things 1) coordinate choice 2) choice of the "propagotor" (formality morphism) (and Galois-Teichmuller group expected to act on the space of quantizations). While the (1) is addressed, second seems to me obscure...</p> <p>2) Basic naive desire f-> \hat f</p> <p>To define quantum dynamical system from classical one, we need to "f" in commutative algebra associate "\hat f" in non-commutative algebra. E.g. for classical Hamiltonian to write down quantum Hamiltonian. Actually this desire is too naive, however in some sense it is very basic, all what follows would follow if such natural map exists.</p> <p>Different versions of quantization address this issue. However in all versions there are some additional choices and it is not clear how to deal with them. In deformation quantization naively this map can be taked as identity - since we just introduce the new product on the old algebra, however everything depends on the choice of the coordinates. Berezin-Toeplitz quantization needs to choose complex structure on symplectic manifold to processed. Geometric quantization needs choice of the polarization. Newly introduced "brane quantization" of Gukov and Witten, needs choice of the "complexifaction of the manifold".</p> <p>So undoubtly something non clear is here.</p> <p>The only thing which I believe that if "f" belongs to the Poisson center, then the correct "\hat f" is given by Duflo map (Duflo-Kirillov-Kontsevich) ( <a href="http://arxiv.org/abs/hep-th/0409005" rel="nofollow">http://arxiv.org/abs/hep-th/0409005</a> ).</p> <p>3) Now we can discuss correspondence between various structures on the commutative side with various structures on non-commutative side.</p> <p>3a) Consider classical integrable system: defined by some set of H_i : {H_i, H_j } = 0. We want to construct quantum integrable system [\hat H_i , \hat H_j ] = 0. The "practical experience" is that great number of integrable systems people know how to do it, but still there is not universal recipe and I do not know general results.</p> <p>3b) Generalization of the example above are Lagrangian and coisotropic manifolds, which means that {H_i, H_j } = F(H_k) i.e. ideal is closed with respect to Poisson bracket. One may want to contruct one sided ideal in non-comummuative algebra corresponding. Here is big progress by Felder, Cattaneo &amp;K however it might not be the last word.</p> <p>3c) Automorphisms. "quantization is a functor ? ". To each Poisson automorphism one may want to contstuct automorphism of the non-commutative algebra. Even in the case of R^2n this is very difficult conjecture related to the Jacobion conjecture - see papers by Kontsevich and Kanel-Belov in arXiv.</p> <p>3d) "All in one". Probably the most comprehensive point should be something like this. "All classical" data should be encoded in something like Fukay category on the commutative side. While on the non-commuative side we have "only" category of modules. The "hope" is that Fukay category is in some sense isomorphic to the category of modules of quantized algebra. See paper by Bressler-Soibelman.</p> <p>The list of objects on commutative and non-commutative side which should correspond to each other is not short. (Some years ago Picard group of Poisson manifold has been introduced and studied by analogy with the quantum Picard group; what should be an analogue of fundamental group in quantum case ? ; "quantum groups" - quantizing the group like structures; quantization of algebras given by quadratic relations, Koszul duality and so on; Boht-Sommerfeld quantization conditions and so on...)</p> <p>If it would be successful it should have various applications, let me mention that "orbit method" in representation theory is a particular case of quantization ideology - which hopes for each symplectic leaf to construct a representaion of the quantum algebra... Another application is to Langlands correspondence over complex numbers. The Hecke eigensheaves - should be understood as quantizations of the Liouville tori of the Hitchin integrable system and "eigen" property is more simple on the classical side.</p> <p>Huh, actually may be it is not good idea to cover in short words such a big theory, any way may be you can find something helpful...</p> <p>(from: <a href="http://mathoverflow.net/users/10446/alexander-chervov" rel="nofollow">Alexander Chervov</a>)</p>

http://mathoverflow.net/questions/2917/where-does-a-math-person-go-to-learn-quantum-mechanics/107690#107690 Sadiq Ahmed 2012-09-20T15:04:08Z 2012-09-20T15:04:08Z

<p><a href="http://www.amazon.com/Quantum-Mechanics-Field-Theory-Mathematical/dp/B00852Q1AI" rel="nofollow">Quantum Mechanics and Quantum Field Theory: A Mathematical Primer</a>.</p> <p>I haven't read it (yet), though, but, given the Preface, it was written specifically for math students seeking a mathematically-rigorous introduction to QM and QFT.</p>

http://mathoverflow.net/questions/107323/problem-of-quantization-state-of-the-art Sadiq Ahmed 2012-09-16T15:37:40Z 2012-09-17T08:21:55Z

<p>As the title suggests, I'm interested in finding out the state-of-the-art in the problem of quantization.</p> <p>Any suggestions and/or feedback would be greatly appreciated.</p> <p>Regards.</p>

http://mathoverflow.net/questions/81881/reference-request-introductions-to-current-mathematics-derived-from-related-to Sadiq Ahmed 2011-11-25T13:18:41Z 2011-11-25T17:55:01Z

<p>I was searching for introductions to current mathematics related to gauge theories.</p> <p>Can someone suggest some good references? </p> <p>E.g.</p> <p><em><a href="http://www.amazon.com/Topics-Physical-Mathematics-Kishore-Marathe/dp/1848829388" rel="nofollow">Topics in Physical Mathematics</a> by K. Marathe</em></p>

http://mathoverflow.net/questions/81883/reference-request-seminal-papers-in-gauge-theoretic-mathematics/81902#81902 Sadiq Ahmed 2011-11-25T16:26:29Z 2011-11-25T16:26:29Z

<p><a href="http://www.jstor.org/pss/37156" rel="nofollow">The Yang-Mills equations over Riemann surfaces</a> by M. F. Atiyah and R. Bott</p>

http://mathoverflow.net/questions/81883/reference-request-seminal-papers-in-gauge-theoretic-mathematics Sadiq Ahmed 2011-11-25T13:53:38Z 2011-11-25T16:26:29Z

<p>Following on from <a href="http://mathoverflow.net/questions/81881/reference-request-introductions-to-current-mathematics-related-to-gauge-theories%29" rel="nofollow">previous question</a> I was also searching for seminal papers in gauge theory.</p> <p>Would be greatly appreciative of references to such.</p>

http://mathoverflow.net/questions/75504/historical-basis-and-mathematical-significance-of-riemann-surfaces Sadiq Ahmed 2011-09-15T11:45:11Z 2011-09-15T19:16:05Z

<p>It is written in <em><a href="http://www.amazon.com/dp/0199606749" rel="nofollow">Riemann Surfaces</a></em> (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that:</p> <p>"[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus"</p> <p>Can someone please provide an articulated commentary on this statement.</p> <p>Specifically, the statement suggests, [or seems to suggest], that Riemann surfaces were the logical / mathematical outcome of many years of careful development and refinement of traditional calculus. But: (i) what was / were the major milestones(s) in this road? and (ii) when the author uses the word 'culmination' what specifically is it the culmination of, and what problems / issues did the introduction of Riemann surfaces help to solve / clarify / etc.?</p> <hr> <p>(This question was originally <a href="http://math.stackexchange.com/questions/64727/historical-basis-and-mathematical-significance-of-riemann-surfaces" rel="nofollow">posted</a> on Math SE, but I'm also posting it here because I'm seeking an expert's [in Riemann surface theory] feedback if possible.)</p>

http://mathoverflow.net/questions/107554/problem-of-quantization-state-of-the-art/107700#107700 Sadiq Ahmed 2012-09-20T17:24:08Z 2012-09-20T17:24:08Z

What would you suggest be a good primer / intro. to the mathematical foundations of quantum mechanics? - I found [this](<a href="http://www.amazon.com/Intermediate-Spectral-Dynamics-Progress-Mathematical/dp/3764387947" rel="nofollow">amazon.com/&hellip;</a>) helpful, but I'm wondering what you would suggest.

http://mathoverflow.net/questions/107554/problem-of-quantization-state-of-the-art/107700#107700 Sadiq Ahmed 2012-09-20T17:17:25Z 2012-09-20T17:17:25Z

Thank you sir. Much appreciated.

http://mathoverflow.net/questions/51395/a-soft-introduction-to-physics-for-mathematicians-who-dont-know-the-first-thing/51405#51405 Sadiq Ahmed 2012-09-20T15:25:37Z 2012-09-20T15:25:37Z

(Is there an electronic version available somewhere on the Net?)

http://mathoverflow.net/questions/107323/problem-of-quantization-state-of-the-art Sadiq Ahmed 2012-09-17T08:17:58Z 2012-09-17T08:17:58Z

@Alexander: Can you suggest any reading (book/s, paper/s, etc.) for a beginner to this area?

http://mathoverflow.net/questions/107323/problem-of-quantization-state-of-the-art Sadiq Ahmed 2012-09-16T16:33:42Z 2012-09-16T16:33:42Z

Essentially, updates on current research on the methods listed in <a href="http://arxiv.org/abs/math-ph/0405065" rel="nofollow">arxiv.org/abs/math-ph/0405065</a>.

http://mathoverflow.net/questions/81883/reference-request-seminal-papers-in-gauge-theoretic-mathematics Sadiq Ahmed 2011-11-29T19:21:38Z 2011-11-29T19:21:38Z

Maybe make the question into a CW. I don't have sufficient privilege for that though.

http://mathoverflow.net/questions/81881/reference-request-introductions-to-current-mathematics-derived-from-related-to Sadiq Ahmed 2011-11-25T14:30:51Z 2011-11-25T14:30:51Z

(ie. current pure mathematics inspired by / from &quot;gauge (field) theory&quot; in physics)

http://mathoverflow.net/questions/81881/reference-request-introductions-to-current-mathematics-derived-from-related-to Sadiq Ahmed 2011-11-25T14:28:24Z 2011-11-25T14:28:24Z

If I'm not mistaken &quot;gauge field&quot; in physics is equivalent to &quot;connections on vector bundles&quot; in maths. - In the context of your comment, I mean 'gauge theory' in the context of maths.

http://mathoverflow.net/questions/75504/historical-basis-and-mathematical-significance-of-riemann-surfaces Sadiq Ahmed 2011-09-16T06:39:59Z 2011-09-16T06:39:59Z

@Paul: &quot;... there is a well-known narrative which explains the passage from single variable calculus over C to the theory of Riemann surfaces. It is often described in basic complex analysis / algebraic geometry courses.&quot; -- Some references? - Regards

http://mathoverflow.net/questions/75504/historical-basis-and-mathematical-significance-of-riemann-surfaces Sadiq Ahmed 2011-09-15T18:05:47Z 2011-09-15T18:05:47Z

@Andy: @quid: Ok. Thanks for the responses. - Sadiq @Paul: ok. This is very useful. Thank you. - Sadiq

http://mathoverflow.net/questions/75504/historical-basis-and-mathematical-significance-of-riemann-surfaces/75508#75508 Sadiq Ahmed 2011-09-15T15:11:33Z 2011-09-15T15:11:33Z

This is more of a technical book. I was searching for an outline-type response.

http://mathoverflow.net/questions/75343/semi-off-topic-tangentially-related-post-an-mo-equivalent-site-for-theoretica Sadiq Ahmed 2011-09-15T02:17:50Z 2011-09-15T02:17:50Z

[clarification: &quot;b) helping to serve as a quasi-moderator to keep irrelevant and non-research questions and comments out of the site&quot;: posts that -*very*- obviously should not belong in the forum (- the inevitable predicament of any Internet open forums).]

http://mathoverflow.net/questions/75343/semi-off-topic-tangentially-related-post-an-mo-equivalent-site-for-theoretica Sadiq Ahmed 2011-09-15T02:09:26Z 2011-09-15T02:09:26Z

My Invitation letter: Dated: August 30 2011 &quot;Hello there: I'm writing to solicit your interest and feedback for the proposed-Stack Exchange forum 'Theoretical Physics'. If you've heard of 'Math Overflow', this site aims to be its counterpart for theoretical and mathematical physicists, i.e. &quot;[A] proposed Q&amp;A site for research level questions in any area of mathematical or theoretical physics.&quot; We would love to receive you feedback for the proposal. With warmest regards.&quot; ... Link: &quot;area51.stackexchange.com/proposals/23848/

http://mathoverflow.net/questions/75343/semi-off-topic-tangentially-related-post-an-mo-equivalent-site-for-theoretica Sadiq Ahmed 2011-09-15T02:02:29Z 2011-09-15T02:02:29Z

... left was because he didn't find a site on SE to his sufficient liking to want to remain a committed user). ... I then decided to help in reviving the Theoretical Physics proposal. -- Thanks. And no need to post on this topic again.

http://mathoverflow.net/questions/75343/semi-off-topic-tangentially-related-post-an-mo-equivalent-site-for-theoretica Sadiq Ahmed 2011-09-15T01:58:45Z 2011-09-15T01:58:45Z

... So, while I am generally in favour of such a view that you share, I will respectfully refer to the above as my response. (... Furthermore, I don't intend to regularly participate in the site, except to a) suggest Meta-level ideas once in a while (e.g. integrating Research and Review papers within the format of the site, etc.; and b) helping to serve as a quasi-moderator to keep irrelevant and non-research questions and comments out of the site -- something, sadly, is difficult to do, but necessary to maintain in an open-format forum. (ps: part of the reason people like Lubos Motl...