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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.

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"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),...
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Poisson Bracket and commutators in quantum mechancs [duplicate]

how did they reach the conclusion that quantization of the Poisson brackets $ (A,B) $ was equal to the commutator $ \frac{1}{i\hbar}[A,B] $ in quantum mechanics? so the quantum equations of motion ...
Jose Javier Garcia Morata's user avatar
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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 ...
user536450's user avatar
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$ \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 ...
Lady Be Good's user avatar
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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 $...
Wagner Coelho's user avatar
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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} ] \...
Nicolas Schmid's user avatar
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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 ...
Wagner Coelho's user avatar
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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 ...
Wagner Coelho's user avatar
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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}}{...
Lourenco Entrudo's user avatar
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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 ...
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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_{...
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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 $$\lim_{\hbar\rightarrow 0} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$ ...
Nicolas Medina Sanchez's user avatar
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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 ...
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Is there a relationship between the phase space path integral and phase space quantum mechanics?

I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space? I also ...
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Commutator Constant

I have seen a lot of commutators in quantum mechanics having a constant factor $i\hbar$. I have read about Dirac supplanting Poisson Brackets with commutators having a constant $i\hbar$. I want to ...
Principia Mathematica's user avatar
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Wigner transform, convolution, and poles

Let \begin{equation} \int\mathrm{d}z~ A(x,z) B(z,y) = \delta(x - y). \end{equation} Taking Wigner transform of both sides we readily obtain \begin{equation} A^W(X,p) \star B^W(X,p) = 1, \end{equation} ...
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Do we lose information about the state when we obtain the Wigner function by solving the eigenvalue equation?

It can be shown that $$H(q,p)\star W_{\psi}(q,p)=EW_{\psi}(q,p)$$ where $H(q,p)$ is the classicaly Hamiltonian function, $\star$ is the Moyal/Groenewold star product and $W_{\psi}(q,p)$ is the Wigner ...
Adrien Amour's user avatar
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Obtaining the star product from the Weyl quantisation of the product of two symbols

It can be shown (Groenewold 1946) that the Weyl quantisation of the product of two Weyl symbols is given by $$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
Adrien Amour's user avatar
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Uncertainty principle in deformation quantization

Deformation quantization procedure is a well-known way to quantize a classical phase space (at least formally for Poisson manifolds which is known as formal deformation quantization). Although it is a ...
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Converting the complex Wigner function to its real form in terms of the quadrature operators

I noticed something that bugged me recently, the Wigner function which is defined for one mode in the complex plane as $$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{...
user135520's user avatar
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Commutators as contour integrals in 2D CFT, and classical limits

In a 2D CFT, the commutator of two operators $$A_i=\oint a_i(z)dz$$ can be given by $$[A_1,A_2]=\oint_0dw\oint_wdza_1(z)a_2(w)$$ where the $z$ integral is taken over a contour around $w$ and the $w$ ...
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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 ...
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How can Planck's constant take different values?

I have seen books and papers mentioning "In the semiclassical limit, $\hbar$ tends to zero", "the scaled Planck's constant goes as $1/N$ where $N$ is the Hilbert space dimension" ...
Raman's user avatar
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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 ...
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How to obtain a star product of a pair of functionals in matter sector?

In lecture notes D-branes, Tachyons and String field theory by Washington Taylor and Barton Zwiebach in the topic "Witten's cubic bosonic SFT" there they define a star product and when star ...
Arshid's user avatar
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Reconciling the expression for the Wigner function involving $\langle x+\xi/2|\rho|x-\xi/2\rangle$ with the one using the characteristic function

A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$: $$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
user135520's user avatar
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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?
Jose Javier Garcia's user avatar
3 votes
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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 ...
Prof. Legolasov's user avatar
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Poisson algebra and the origin of canonical quantization [duplicate]

Professor Achim Kempf in his lecture note mentioned that non-commutativity of quantum observables in the associated Poisson algebra to the system, impose CCR It was Dirac who first realized that all ...
John Patrikov's user avatar
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2 answers
578 views

Wigner transform & convolution

I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
surrutiaquir's user avatar
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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 ...
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Can Friedlander Equations be used to Model Buried Explosive

To quote from WIKIPEDIA The simplest form of a blast wave has been described and termed the Friedlander waveform.[11] It occurs when a high explosive detonates in a free field, that is, with no ...
majordoctor's user avatar
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Time Evolution of Wigner Function

The Wigner Function is defined as: $$W(x,p,t)=\frac{1}{2\pi\hbar}\int dy \rho(x+y/2, x-y/2, t)e^{-ipy/\hbar}\tag{1}$$ Where $\rho(x, y, t)=\langle x|\hat{\rho}|y\rangle$. I am supposed to find the ...
eeqesri's user avatar
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6 votes
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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 ...
Prof. Legolasov's user avatar
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3 answers
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Is Planck's Constant Really a Constant?

I am going through Groenewold's theorem and in his book: On The Principles of Elementary Quantum Mechanics, page 8, eq. 1.30: $$[\mathbf{p}, \mathbf{q}]=1\left(\text { i.e. } \mathbf{p q}-\mathbf{...
Quantally's user avatar
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Geometric Intuition for the Moyal Product

I've recently been reading into deformation quantization as another formulation of quantum mechanics. I have focused on understanding the Moyal product in particular, as it contains the seeds for the ...
Noah M's user avatar
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Vanishing Poisson bracket with non-vanishing Moyal bracket

Let $M=\mathbb{R}^{2n}$ be the phase space with standard Poisson bracket on smooth functions on $M$. Fix a classical hamiltonian $h$ (function on $M$) and function $f$ generating symmetry of $h$ i.e. $...
Alex's user avatar
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7 votes
2 answers
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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.
Dr. user44690's user avatar
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2 answers
201 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{\...
user avatar
1 vote
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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 ...
fred's user avatar
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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^{...
user avatar
4 votes
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443 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]{...
Mark Robinson's user avatar
3 votes
3 answers
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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}^...
MH Yip's user avatar
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20 votes
3 answers
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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 ...
jak's user avatar
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3 votes
2 answers
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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. ...
aghostinthefigures's user avatar
2 votes
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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 ...
Prof. Legolasov's user avatar
6 votes
2 answers
438 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 ...
Prof. Legolasov's user avatar
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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 ...
Jake Xuereb's user avatar
17 votes
1 answer
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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 $$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$ ...
Jake Xuereb's user avatar
1 vote
1 answer
185 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}}-\...
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