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2
votes
1answer
51 views

Heisenberg group deformation

I was wondering if there are some relations between deformations within small parameters and the Heisenberg group operation. We can think about the Heisenberg group $\mathbb{H_3}$ as $(\mathbb{R^3}, ...
8
votes
1answer
111 views

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 ...
7
votes
1answer
84 views

Angular Momentum Addition in Phase Space QM

In my very limited understanding of geometric quantization, we quantize spin by choosing as our phase space $S^2$ with a suitably normalized area form as the symplectic form. Depending on the ...
1
vote
0answers
41 views

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 ...
2
votes
0answers
117 views

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 ...
7
votes
1answer
335 views

Physical intuition for deformation quantization of Poisson manifolds

First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such ...
1
vote
0answers
100 views

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 ...
4
votes
2answers
303 views

Star product of two commuting spinors

Ok so this might be a very stupid and trivial question but I have spent a couple of hours on this little problem. I am trying to derive a simple formula in a paper. We have a real commuting spinorial ...
5
votes
1answer
474 views

Moyal Product in Non Commutative Quantum Mechanics

Can someone please explain me what is a Moyal product? Also, how does putting $$X_a(\psi) ~=~ x_a\star\psi$$ realise $$[X_a,X_b]=i\theta_{ab}{\bf 1}?$$ Ref: Quantum mechanics on non-commutative ...
8
votes
2answers
2k views

Weyl Ordering Rule

While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
21
votes
5answers
8k views

What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\}_{PB} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial ...