Questions tagged [deformation-quantization]
A description of quantum mechanics in phase space a common ambit with classical mechanics, through the Wigner map from Hilbert space. May be used to address Quantum Mechanics in phase space, the star product binary operation controlling composition of observables, and Wigner, Husimi, and other distribution functions in phase space.
74 questions
5votes
2answers
135views
Linkage of a classical canonical transformation to a quantum unitary transformation
Suppose I have an classical Hamiltonian $H(q, p)$ and the quantum equivalent $\hat{H}(\hat{q}, \hat{p})$. We can apply linear canonical transformation defined by a type-II generating function $S_2(q,p^...
- 489
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
0votes
0answers
75views
Do generalized uncertainty principles violate the triviality of second Hochschild cohomology group of Heisenberg algebra?
The second Hochschild cohomology group of Heisenberg algebra is trivial and it means that we can not deform $[x,p]=i$ further, but there are tons of papers on generalized uncertainty principles which ...
- 63
1vote
1answer
123views
"Deriving" Poisson bracket from commutator
This Q/A shows that deriving P.B.s from commutators is subtle. Without going into deep deformation quantization stuff, Yaffe manages to show that $$\lim_{\hbar \to 0}\frac{i}{\hbar}[A,B](p,q)=\{a(p,q),...
- 822
0votes
1answer
83views
On the Born-Jordan quantization being an equally weighted average of all operator orderings
On my way studying quantization schemes, I came across the expression saying that the Born-Jordan quantization rule is the equally weighted average of all the operator orderings and that the Weyl's ...
- 193
3votes
1answer
230views
$ \hbar^2$ Correction to the Bohr-Sommerfeld Quantization Condition
We can get the Bohr-Sommerfeld quantization from the WKB method as answered. Since we use approximation, there should be an error in the system, I know this is not right all the time; in some ...
- 185
0votes
1answer
130views
Operating with the Weyl transform on a wave function
I'm very new to studying quantum mechanics in phase space, so I'm trying to demonstrate some results that I see in books to get used to the formalism. I recently got stuck when i applied the Weyl map $...
2votes
2answers
274views
Why is $[ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}$?
If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by $$ [ \hat{A},\hat{B} ] \...
- 1,150
0votes
1answer
165views
Quantization of $x$ and $p$ through the Weyl transformation
I have a question about the development of the integral for calculating the quantization of the classical variables $x$ and $p$ using the Weyl transformation method. The notation that the textbook I'm ...
1vote
1answer
229views
Weyl Quantization Integral
I have some doubts when calculating the integral for Weyl Quantization symbol. If I understand correctly, quantization using the Weyl symbol takes a function in phase space and takes it to an operator ...
2votes
2answers
242views
What does it mean for an operator to depend on position or momentum?
While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
- 1,045
0votes
1answer
125views
Lie algebra with $\sim \!N^3$ generators [closed]
Is there a Lie algebra whose number of generators scales as $N^3$, or in general $N^p$ with $p$ an arbitrary positive integer? All the familiar examples, such as $\mathrm{U}(N)$ or $\mathrm{SU}(N)$ or ...
- 604
0votes
1answer
170views
Elastic potential energy formula
From the Wikipedia page on elastic energy, we can find a bunch of formulas to describe it. For example, in the continuum section it talks about energy per unit of volume (density?): $U=\dfrac{1}{2}C_{...
- 751
1vote
1answer
319views
Classical limit of Moyal bracket in integral representation
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$ ...
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
