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.

Filter by
Sorted by
Tagged with
5
votes
2answers
170 views

What is a Borel subalgebra?

Borel subalgebra appears here https://arxiv.org/abs/hep-th/9508170 in the context of quantum double of $SU(2)$. I request a layman explanation of what a Borel subalgebra is.
0
votes
0answers
8 views

Resource recommendation: Relation between q-deformed groups and non-commutative field theory

I would like to know if non-commutative field theory is related to $q$-deformed groups in any way. There is a relation between non-commutative field theory and group field theory which I understand is ...
1
vote
2answers
93 views

Relationship between $\star$-products in phase-space QM and NC geometry

What exactly is the relationship between $\star$-products in phase-space quantum mechanics, i.e. $$ (f \star g) (x,p) = f(x,p) e^{\frac{i \hbar}{2} ( \overleftarrow{\partial_x} \cdot \overrightarrow{\...
1
vote
0answers
46 views

Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
3
votes
1answer
66 views

A question about $\delta(x) \star \delta(p)$

Using the Moyal product between two delta functions in $(x,p)$-space one gets $$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$ However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
2
votes
1answer
78 views

Non-commutative Fourier transform of an operator

Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative. $$ \hat{\rho} \xrightarrow[non-comm]{...
2
votes
3answers
591 views

Derivation question of WKB method

Quantum Mechanics (2nd Edition) by Bransden and Joachain contains the following passage: Substituting (8.176) into (8.171), we obtain for $S(x)$ the equation $$-\frac{i\hbar}{2m}\frac{\mathrm{d}^...
15
votes
3answers
669 views

What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid. However, in the phase space formulation ...
3
votes
2answers
159 views

Physical interpretation of differences between classical and quantum ensemble dynamics

The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. ...
2
votes
0answers
68 views

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 ...
6
votes
2answers
177 views

Symplectic reduction to moduli space in Chern-Simons theory

In QFT and the Jones polynomial, Witten claims that it is possible to perform symplectic reduction from the distributional Poisson bracket on the unconstrained phase space to a symplectic structure on ...
1
vote
0answers
112 views

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 ...
5
votes
1answer
448 views

On Groenewold's Theorem and Classical and Quantum Hamiltonians

I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping $$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
1
vote
1answer
109 views

Star Product and Poisson Brackets

I have the following definition of star product, \begin{equation} \star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\...
4
votes
2answers
198 views

Solving the *-genvalue equation of a free particle

The background I want to solve the $\star$-genvalue equation $$ H(x,p) \star \psi(x,p) = E~\psi(x,p),$$ where $\star$ denotes the Moyal star product given by $$ \star \equiv \exp \left\lbrace \...
4
votes
1answer
155 views

Measurements in the phase space picture of quantum mechanics

Suppose we are dealing with non relativistic quantum mechanics of point particles, that is we are in the realm of 'classic' quantum mechanics (no quantum fields ect.). In the Heisenberg picture, ...
4
votes
1answer
128 views

Proof of “non-existence” of marginals of the Husimi $Q$-function

There are many ways to consider the Husimi ($Q$) quasi-probability distribution function, e.g. as the expectation of the density operator in a coherent state or as the Weirstrass transform of the ...
18
votes
5answers
562 views

Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?

This answer by Qmechanic shows that the classical Liouville equation can be extended to quantum mechanics by the use of Moyal star products, where it takes the form $$ \frac{\partial \rho}{\partial t}~...
2
votes
1answer
129 views

Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics) [closed]

$\newcommand{\dd}{{\rm d}}$ In the paper "On the principles of elementary Quantum Mechanics" am trying to get from equation EQN 4.25 to EQN 4.27. I need help on exponential identities and integration ...
1
vote
4answers
413 views

Definition of symmetrically ordered operator for multi-mode case?

As I know, Wigner function is useful for evaluating the expectation value of an operator. But first you have to write it in a symmetrically ordered form. For example: $$a^\dagger a = \frac{a^\dagger ...
4
votes
2answers
186 views

Classical limit in deformation quantization

In deformation quantization, when dealing with the Moyal product, the classical limit is \begin{align} \lim_{\hbar\to0}\frac{1}{i\hbar}[f,g]_{\star_M}=\{f,g\}\, \end{align} which is just the Poisson ...
10
votes
3answers
594 views

Dequantizing Dirac's quantization rule

In this blog post, Lubos Motl claims that any commutator may be shown to reduce to the classical Poisson brackets: $$ \lim \limits_{\hbar \to 0} \frac{1}{i\hbar} \left[ \hat{F}, \hat{G} \right] =...
1
vote
0answers
65 views

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 ...
2
votes
1answer
99 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}, \...
9
votes
1answer
567 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 ...
6
votes
1answer
212 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 ...
0
votes
0answers
116 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
238 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 ...
1
vote
0answers
174 views

Tropical Geometry and Quantization

Recently I saw this question posted on Math Overflow asking about the motivations behind tropical geometry. The OP mentions that tropical geometry can be viewed as the classical limit of regular ...
7
votes
1answer
644 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
140 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
379 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 ...
8
votes
2answers
1k views

How to promote algebraic expressions to operators in quantum mechanics?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $...
6
votes
1answer
752 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 ...
14
votes
2answers
6k 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 $H(...
45
votes
5answers
19k views

What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\} ~:=~ \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 g}{\...