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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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 ...
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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 $$\frac{d\hat{A}}{...
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Lie algebra with $\sim \!N^3$ generators [closed]
Is there a Lie algebra whose number of generators scales as $N^3$, or in general $N^p$ with $p$ an arbitrary positive integer?
All the familiar examples, such as $\mathrm{U}(N)$ or $\mathrm{SU}(N)$ or ...
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Elastic potential energy formula
From the Wikipedia page on elastic energy, we can find a bunch of formulas to describe it. For example, in the continuum section it talks about energy per unit of volume (density?):
$U=\dfrac{1}{2}C_{...
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Classical limit of Moyal bracket in integral representation
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0}\frac{1}{i\hbar} \lbrace f,g\rbrace_M=\lbrace f,...
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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 ...
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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 ...
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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
$$ [A(\textbf{r})B(\textbf{r})]_{w}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}e^{i\...
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Uncertainty principle in deformation quantization
Deformation quantization procedure is a well-known way to quantize a classical phase space (at least formally for Poisson manifolds which is known as formal deformation quantization). Although it is a ...
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Converting the complex Wigner function to its real form in terms of the quadrature operators
I noticed something that bugged me recently, the Wigner function which is defined for one mode in the complex plane as
$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{...
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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" ...
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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 ...
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Reconciling two descriptions of the Wigner function
A classical way to define the Wigner function ($\hbar=2$) of a density operator $\rho$ is as follows for $x=(x_{1}, x_{2})^{T}$:
$$W(x) = \frac{1}{4\pi} \int^{\infty}_{-\infty} d\xi \exp(\frac{-i}{2}...
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A Hamiltonian with a potential depending on the momentum
Imagine we have a Hamiltonian, whose potential depends on velocities (and hence on the momentum),
like, for example,
$$ H= \frac{p^{2}}{2m}+ V(x,p)$$
then how can I quantize that?
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Quantum corrections in the phase space formulation
I'm trying to reconcile the following two statements:
Quantum Mechanics gives physical predictions which are different than the predictions that are obtained in the $\hbar \rightarrow 0$ limit, that ...
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Poisson algebra and the origin of canonical quantization [duplicate]
Professor Achim Kempf in his lecture note mentioned that non-commutativity of quantum observables in the associated Poisson algebra to the system, impose CCR
It was Dirac who first realized that all ...
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Wigner transform & convolution
I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
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Is it possible to minimize the number of axioms/rules of the canonical quantization?
In the standard canonical quantization procedure there are two rules.
Transform all quantities to operators.
Transform the Poisson bracket to a commutator.
Of course it will be nicer to minimize the ...
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Can Friedlander Equations be used to Model Buried Explosive
To quote from WIKIPEDIA
The simplest form of a blast wave has been described and termed the Friedlander waveform.[11] It occurs when a high explosive detonates in a free field, that is, with no ...
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Time Evolution of Wigner Function
The Wigner Function is defined as:
$$W(x,p,t)=\frac{1}{2\pi\hbar}\int dy \rho(x+y/2, x-y/2, t)e^{-ipy/\hbar}\tag{1}$$
Where $\rho(x, y, t)=\langle x|\hat{\rho}|y\rangle$.
I am supposed to find the ...
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Operator traces in Kontsevich quantization
In quantization, one studies maps from functions on the phase space to operators acting on the Hilbert space. Let's fix one such map and call it $Q$.
Deformation quantization is based on the idea that ...
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Is Planck's Constant Really a Constant?
I am going through Groenewold's theorem and in his book: On The Principles of Elementary Quantum Mechanics, page 8, eq. 1.30:
$$[\mathbf{p}, \mathbf{q}]=1\left(\text { i.e. } \mathbf{p
q}-\mathbf{...
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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 ...
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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.
$...
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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.
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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.
$$ (f \star g) (x,p) = f(x,p) e^{\frac{i \hbar}{2} ( \overleftarrow{\partial_x} \cdot \overrightarrow{\...
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Quantum mechanics in phase space - what are coordinate components?
I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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A question about $\delta(x) \star \delta(p)$
Using the Moyal product between two delta functions in $(x,p)$-space one gets
$$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$
However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
user20250
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Non-commutative Fourier transform of an operator
Wigner-Weyl transform relates an operator to its distribution function in phase space through an operator Fourier transform which is said to be non-commutative.
$$
\hat{\rho} \xrightarrow[non-comm]{...
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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
$$-\frac{i\hbar}{2m}\frac{\mathrm{d}^...
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What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?
One of the most important results of Classical Mechanics is Liouville's theorem, which tells us that the flow in phase space is like an incompressible fluid.
However, in the phase space formulation ...
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Physical interpretation of differences between classical and quantum ensemble dynamics
The Groenewold-Moyal (phase space) picture of quantum mechanics describes the evolution of a probability density corresponding to a wavefunction that evolves as described by Schrödinger's equation. ...
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Physical aspects of representations of $C^{*}$ algebras
Suppose I have a $C^{*}$ algebra $\mathcal{A}$ of quantum observables. I could have used deformation quantization to obtain it from the classical Poisson manifold, or I could've just guessed it – for ...
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Symplectic reduction to moduli space in Chern-Simons theory
In QFT and the Jones polynomial, Witten claims that it is possible to perform symplectic reduction from the distributional Poisson bracket on the unconstrained phase space to a symplectic structure on ...
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Constructing Quantum Theories without Semiclassical Quantization
This question builds off of this previous question particularly the excellent answer by @Cosmas Zachos and the this document which he attached.
Quantization whatever form it takes always seeks to ...
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On Groenewold's Theorem and Classical and Quantum Hamiltonians
I recently encountered Groenewold's Theorem or the Groenewold-Van Hove Theorem which shows that there is no function which can satisfy the following mapping
$$ \{A,B\} \to \frac{1}{i\hbar}[A,B].$$
...
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Star Product and Poisson Brackets
I have the following definition of star product,
\begin{equation}
\star=\exp\left[\frac{i\hbar}{2}\left(\frac{\overleftarrow{\partial}}{\partial Q^{I}}\frac{\overrightarrow{\partial}}{\partial P_{I}}-\...
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Solving the *-genvalue equation of a free particle
The background
I want to solve the $\star$-genvalue equation
$$ 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
\...
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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, ...
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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 ...
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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
$$
\frac{\partial \rho}{\partial t}~...
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Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics) [closed]
$\newcommand{\dd}{{\rm d}}$
In the paper "On the principles of elementary Quantum Mechanics" am trying to get from equation EQN 4.25 to EQN 4.27. I need help on exponential identities and integration ...
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Definition of symmetrically ordered operator for multi-mode case?
As I know, Wigner function is useful for evaluating the expectation value of an operator. But first you have to write it in a symmetrically ordered form. For example:
$$a^\dagger a = \frac{a^\dagger ...
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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 ...
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