# 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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### What does the superposition of fields mean in the context of the convolution of two Glauber-Sudarshan $P$-representations?

In his 1963 paper, in which he introduces his formulation of the Glauber-Sudarshan $P$-representation (https://doi.org/10.1103/PhysRev.131.2766), Glauber refers to the convolution of the $P$-...
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### Critical Points of a Wigner function

I am interested in calculating the critical points of a Wigner function $$W(x,p)=\frac{1}{\pi}\int_{-\infty}^\infty\left\langle x+y\middle|\rho\middle|x-y\right\rangle e^{-2ipy}\mathrm{d}y$$ ...
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### Is there a non-negative normalized Wigner function that doesn't correspond to a physical state?

This is related to Is the Wigner function non-negative only for convex mixtures of Gaussian states? and Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator ...
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### Is the Wigner function non-negative only for convex mixtures of Gaussian states?

Hudson's theorem, the result usually cited in this context, tells us that for a pure state, the Wigner is non-negative iff the state is Gaussian, but doesn't in general say anything about mixed states....
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### Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator function?

Given the characteristic function defined as: $$\chi(\beta)=\text{tr}[\rho D(\beta)],$$ with $D(\alpha)=e^{\alpha a^\dagger-\bar\alpha a}$ the displacement operator. Is it possible that for some $\rho$...
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### How is Hudson's theorem for the Wigner function proved?

Hudson's theorem tells us that a pure state has non-negative Wigner function iff it's Gaussian. This was originally proven in [Hudson 1974], and then generalised to multidimensional systems in [Soto ...
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### Generating function for Weyl-ordered spin correlation functions

Consider the standard 2-dimensional phase space: $[\hat{x},\hat{p}] = \mathrm{i}$. Upon taking the Wigner transform of the density matrix $\hat{\rho}$, W(x,p) = \int\mathrm{d}y\, \...
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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 ...
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### 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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