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Tagged with deformation-quantizationquantization
17 questions
1vote
2answers
194views
Canonical vs Deformation Quantization
In canonical quantization, given any two functions $f$, $g$ on phase space, one quantizes the theory by demanding that the commutator of the operators $O_f$, $O_g$ associated to $f$, $g$ is given by ...
- 1,117
5votes
1answer
252views
Does geometric quantization work for arbitrary "particle with constraint + potential" systems?
I was struck by the following line in Hall's Quantum Theory for Mathematicians (Ch. 23, p. 484): In the case $N = T^*M$, for example, with the natural “vertical” polarization, geometric quantization ...
- 3,700
2votes
1answer
185views
Basic question in similarities and difference on quantizations
In physics, usually quantization means canonical quantization. i.e., which we treat classical objects to quantum operators. i.e., For the association $Q:f \mapsto \hat{f}$ from functions on the ...
- 3,732
2votes
1answer
299views
Physical motivation of quantization
I am a mathematics student recently looking into (geometric and deformation) quantization. I'd like to know more about their physical motivations. Here by "quantization" I mean any process ...
- 73
2votes
1answer
393views
A Hamiltonian with a potential depending on the momentum
Imagine we have a Hamiltonian, whose potential depends on velocities (and hence on the momentum), like, for example, $$ H= \frac{p^{2}}{2m}+ V(x,p)$$ then how can I quantize that?
- 4,847
3votes
2answers
178views
Quantum corrections in the phase space formulation
I'm trying to reconcile the following two statements: Quantum Mechanics gives physical predictions which are different than the predictions that are obtained in the $\hbar \rightarrow 0$ limit, that ...
- 16.5k
0votes
2answers
104views
Is it possible to minimize the number of axioms/rules of the canonical quantization?
In the standard canonical quantization procedure there are two rules. Transform all quantities to operators. Transform the Poisson bracket to a commutator. Of course it will be nicer to minimize the ...
- 1,563
6votes
1answer
196views
Operator traces in Kontsevich quantization
In quantization, one studies maps from functions on the phase space to operators acting on the Hilbert space. Let's fix one such map and call it $Q$. Deformation quantization is based on the idea that ...
- 16.5k
2votes
0answers
104views
Physical aspects of representations of $C^{*}$ algebras
Suppose I have a $C^{*}$ algebra $\mathcal{A}$ of quantum observables. I could have used deformation quantization to obtain it from the classical Poisson manifold, or I could've just guessed it – for ...
- 16.5k
1vote
0answers
142views
Constructing Quantum Theories without Semiclassical Quantization
This question builds off of this previous question particularly the excellent answer by @Cosmas Zachos and the this document which he attached. Quantization whatever form it takes always seeks to ...
- 435
1vote
0answers
88views
Wigner's function in geometric quantisation
Let $\overleftarrow{a}$ and $\overrightarrow{a}$ represent the action of the operator $a$ in arguments to the left and to the right of it, respectively. Define, then, $$\star := \exp \left \{ \frac{i ...
- 4,153
8votes
1answer
986views
Quantum systems without a classical analogue? [closed]
I am now reading the quantum mechanics textbook by Dirac (chap. 4, $\S21$, p. 88). He says that his quantization procedure does not include all possible systems in quantum mechanics and there are ...
- 508
1vote
0answers
231views
References on deformation quantization
I'm looking for books or introductory review papers or lecture notes on the topic of deformation quantization. (And preferably, geometric quantization as well.) I'm mainly interested in the ...
2votes
0answers
472views
What new does geometric or deformation quantization give to physics? [closed]
What new does geometric quantization or deformation quantization give to physics? For example: prediction of new physical phenomena or just better tool for quantization. What can these schemes do in ...
1vote
0answers
182views
How functions become operators in quantum mechanics? [duplicate]
What used to be functions in the context of classical mechanics like position, linear momentum, angular momentum, etc in quantum mechanics are operators (these operators act on the state to get ...
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