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Results tagged with mp.mathematical-physics
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user 12482
Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
18
votes
Applications of symplectic geometry to classical mechanics
The list will be long, very long indeed. But to start:
Questions about dynamics of Hamiltonian systems are at the heart of symplectic topology, symplectic capacities are precisely introduced for that …
- 52.4k
78
votes
The Planck constant for mathematicians
Let's give it a try. Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. But if I understand correctly, you want to gain intuition ;)
The first observ …
- 111
4
votes
Deformation quantization of a closed Riemann surface with genus >1
One should definitely take a look at the work of Bordemann, Meinrenken, and Schlichenmaier: they provide a Berezin-Toeplitz inspired deformation quantization for all compact quantizable (i.e. the Kähl …
- 8,098
3
votes
graded generalization of the Moyal–Weyl product
Yes, it's just putting signs correctly. Martin Bordemann has a preprint from the 90s where he adapted Fedosov's construction in the graded setting. If you are only interested in the flat situation thi …
- 8,098
6
votes
Accepted
Formal series convergence in deformation quantization and $C^*$-condition
OK, let me give a try on this question. There are several problems hidden underneath which one has to address.
First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constan …
- 8,098
13
votes
Is the quantum algebra unique (up to isomorphism) in deformation quantization ?
In deformation quantization there is a full classification available: let us first focus on the symplectic case which is easier. If $(M, \omega)$ is a symplectic manifold (like the $\mathbb{R}^2$ in y …
- 8,098
17
votes
Accepted
Can a sphere be a phase space?
Of course, the spheres are compact while cotangent bundles are noncompact (unless in dimension 0). Nevertheless, a bit more interesting is the question whether the even dimensional spheres can be phas …
- 8,098
9
votes
Dimensional Analysis in Mathematics
Dimensional analysis can be viewed as the study of graded objects in algebra. The grading then corresponds to "counting the units" in a precise way. There are of course many examples and I believe tha …
- 8,098
4
votes
Accepted
Open symplectic embeddings and deformation quantization
Hi Igor,
there is a quite elementary way to see that star products restrict to open subsets: it's essentially part of the definition of a star product. Here, I will focus on the case of smooth (sympl …
- 8,098
28
votes
Accepted
Quantum mechanics formalism and C*-algebras
In addition to what has already been said I would like to add some more comments. I completely understand your suspicion that the passage from unbounded operators to bounded ones is at least tricky. F …
- 8,098