# Questions tagged [wigner-transform]

The Wigner transform is the bridge between Hilbert space operators to phase-space quantities (c-numbers). Use for issues relating to the Weyl correspondence (the inverse of the Wigner transform), the Wigner function (the Wigner-transform of the density matrix) and, in general, Quantum Mechanics in phase space issues, such as the *-product, the Wigner transform of the operator multiplication operation. May also use for distributions such as the Husimi.

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### How does the Stratonovich-Weyl operator kernel, used to find the Wigner function, work?

Recently during my studies, I came across an alternative construction of the Wigner function. This construction starts from the notion of the Stratonovich-Weyl operator kernel. I saw this construction ...
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### Quantum mechanics representations $x$, $p$, and not $(x,p)$? [duplicate]

In Quantum Mechanics, generally speaking we work with space representation $\Psi(x)$ and/or the momentum representation $\Psi(p)$ of wavefunctions. Are there representations with mixing of $(x,p)$, i....
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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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### Construct the Density Matrix of a Gaussian State from its First and Second Moments / Wigner Function

Borrowing some description for the setup from a question I posted earlier here; Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the ...
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### What is the Wigner representation of $\left(\hat{x}^2+\hat{p}^2\right)^n$?

I would like to calculate the Wigner representation of the operators $\left(\hat{x}^2+\hat{p}^2\right)^3$ and $\left(\hat{x}^2+\hat{p}^2\right)^4$. I know at least two ways to do it, but both rely on ...
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### Calculating the Wigner transform of operators

Recently I started to study the formulation of quantum mechanics in the phase space. So I was introduced to the concept of Wigner function and Weyl transform. I learned that if F is an operator, then ...
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### Understanding derivation of Wigner function for the Harmonic oscillator

In the document https://www.hep.anl.gov/czachos/aaa.pdf, they derive the Wigner functions, $f_n$ for the harmonic oscillator. However, I have some problems understanding some of the steps. On page 37 ...
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### Effects of non-locality in the star-product of two fields

My question regards an argument appearing on page 19 of the review: Quantum Field Theory on Non-commutative Spaces - Szabo. The Fourier integral kernel representation of the star-product of two fields ... 260 views

### How is it possible to find the Wigner function for spin coherent states?

I studied Wigner function distribution for Glauber coherent state and I know that by using this function we can find the probability distribution for particle's position, but How can me find and ...
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### Wigner functional for fermionic fields (QFT in phase space)

I'm curently studying the Wigner functional formulation of Quantum Field Theory, which is derived from the Schrödinger picture: the operators which act on the states of the Fock space are functions of ...
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### Formulations of the Wigner function in Quantum Field Theory (QFT in Phase Space)

I'm studying the phase space formulation of quantum field theory for my final degree project, and I have found two very different ways to construct the Wigner funtion. In the first method, a phase ...
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### Multiplicative inverse of Weyl symbol and invertibility of operator

If the Weyl symbol $A_W$ of an operator $\hat{A}$ has a multiplicative inverse at every point of the phase-space, can I conclude that $\hat{A}$ is invertible?
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### Wigner transform of a thermal density matrix (Schrödinger Hamiltonian)

I am interested in computing the Wigner transform in $R^{2n}$ of $$e^{-\beta H}$$ where $$H = \sum_{k=1}^n P_k^2 + V(x_1, \ldots,x_n)$$ I am assuming that $V$ is a polynomial bounded below and ...
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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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### General quantum operator

Is it true that any operator can be expressed as (e.g. in one dimension) $$\hat{A}=\sum_{n=0, \, m=0}^{\infty}c_{n,m}\hat{x}^n\hat{p}^m \, ?$$ It seems true because any classical observable is a ...
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### Wigner-Weyl transform for a function of coordinates only

I am reading this paper by Tatarskii, which serves as an introduction to the Wigner representation of quantum mechanics. There is a step in the paper involving the Weyl transform that does not seem ...
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http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/Wigner_function.pdf From the Appendix in the above PDF (page 945), below equation (A3) the following expressions are given: $$u = ... 3 votes 0 answers 128 views ### Is the Wigner function a signed measure? I have read in Wikipedia that quasiprobability distributions in phase space quantum mechanics may fail to be \sigma-additive, but I don't know in which sense this is true. If I have a Wigner ... 0 votes 0 answers 86 views ### How does the Wigner function differ from quantum distribution functions? I understand that Wigner functions are quantum-mechanical phase-space distribution functions (quasi-distribution to be more specific). For spin-1/2 particles the Wigner function is a 4\times 4 ... 0 votes 1 answer 243 views ### Prove that f_\psi(x,p) is the Wigner Function of a pure state iff H\star f_\psi= E f_\psi Given a pure state |\psi\rangle with position wavefunction x\mapsto\psi(x), define its Wigner function as$$f_\psi(x,p) = \frac{1}{2\pi} \int dy e^{-iyp} \psi(x+y/2)\psi^*(x-y/2) \equiv \frac{1}{2\...
This is something I got curious about. Let's say I have the Wigner function for an $n$ particle system: $$W \equiv W(x_1,\dots,x_n,;p_1,\dots,p_n)$$ Now, let's say this system obeys has spin. As far ...
I want to generalize the Wigner overlap formula, $Tr( F G ) = 2 \pi \int_{-\infty}^{\infty} dq \int_{-\infty}^{\infty} dq W_F(q,p) W_G(q,p)$, where $W_F(q,p)$ and $W_G(q,p)$ are the Wigner functions ...