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One never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables, for which the commutator is non-zero (because they take negative values, they are not probability densities). So, to avoid negative probability ...

13

As Ernesto pointed out in his comment, I've answered your first question here (which was updated on the arXiv and published very recently. As for the question about the intuition behind negative probabilities, here is my warning if you don't already have tenure: don't go there. As Feynman pointed out (and Dirac much earlier) negative probabilities are a ...

7

It is not an complete answer, but (hopefully) it may help: $$\text{tr} \left( A B \right) = \pi \int W_A(r) W_B(r) dr = \pi \int P_A(r) Q_B(r) dr,$$ where $A$ and $B$ are self-adjoint operators, $r=(p,q)$ is a point of the phase space, $W(r)$ is a Winger function, $P(r)$ - Glauber-Sudarshan P-representation and $Q(r)$ - a Husimi Q representation. The ...

6

There are two works of Feynman about negative probabilities. It is hard to add something to that, if to look for introduction to the subject. R. P. Feynman, Negative probability in Quantum implications: Essays in honor of David Bohm, edited by B. J. Hiley and F. D. Peat (Routledge and Kegan Paul, London, 1987), Chap. 13, pp 235 – 248. R. P. Feynman, ...

6

Your question has, indeed, been beaten to a pulp in the 70 years of the formulation, and, as you suggested, the necessary conditions are not all independent, so parts are redundant. For a pure state real $f(x,p)$ the sufficient condition is straightforward, eqn (6) of Ref. 1: Given its Fourier transform (the cross-spectral density) must left-right" ...

5

It's intuitively clear that this current must exist because the integral of the Wigner function is conserved by unitary evolution. This current is known as the Wigner flow, and it exists but it's not particularly pretty. For an example of the Wigner flow in use, see arXiv:1208.2970; in short, it is the current $$J=\begin{pmatrix}J_x\\J_p\end{pmatrix} ... 4 Your equation in the Liouville form is elementary for numerical integration, it is structurally just a linear advection equation with spatially varying coefficients. The transformed equation with the kernel F is not useful at all for numerical solution, don't bother with it. All we have here is a 2D advection equation (I use y instead of p):  \partial_{t} ... 4 I would say they are not entirely the same, but it depends on the context. First the definitions: the Wigner transform of an operator \hat{A} is defined as$$\tilde{W}\left[\hat{A}\right]=\int dz\left[e^{\mathbf{i}pz/\hbar}\left\langle x-z/2\right|\hat{A}\left|x+z/2\right\rangle \right]$$and this is a strange function. You see that on the left, the ... 3 Let me split the "equivalence" in two parts: Are there states with Wigner functions that are everywhere positive that show "quantum" behaviour? The answer to this question is "yes". One very famous example are Gaussian (bosonic) states - their Wigner function, by definition, is a Gaussian (which is obviously positive) for an intro see e.g. Adesso et ... 3 Citations are from here on Wikipedia. Is this really quantum mechanics? In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = |k|sinθ ≈ |k|θ in the small angle (paraxial) ... 3 In the case when B is a compact operator, a limiting process can be applied to get the operator's norm to any arbitrarily small precision. This process is a Hilbert space generalization of the linear algebra relation for a Hermitian matrix A.  ||A||_{\infty} = \lim_{n\rightarrow \infty}|tr(A^n)|^{\frac{1}{n}}. For the data given in the question the ... 3 http://arxiv.org/abs/1202.3628 This is a very recent paper demystifying the negative probability density in the Wigner Function. I hope this helps. 3 The Wigner-Weyl transform of a function f(x,p) is given by,$$\Phi[f]= \frac{1}{4\pi^2}\iiiint f(x,p) \exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, da \, db$$As you suggested, let us take the Hamiltonian of the harmonic oscillator, i.e.$$\Phi[H]=\frac{1}{8m\pi^2}\iiiint p^2\exp \left[ i \left( a(X-x)+b(P-p)\right)\right] \, dx\, dp\, ...

3

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.) Is there a simple system (hopefully a simple harmonic ...

3

I'll answer my own question and hope this information is useful to someone. I'll take $\hbar = 1$ and will deal with systems of one degree of freedom (the generalisation should be obvious). The normalisation factors are very confusing, so I'll omit most of the reasons for them to be what they are. First, let's take a look at the good old continuous Wigner ...

3

The standard name for what you are seeking, is the Wigner transform, the inverse of the Weyl transform. (As the Weyl transform maps phase-space functions to operators.) for an arbitrary operator in any ordering, the Wigner transform follows a simple 1964 formula by Kubo, eqn (111) of Ref. 1, effectively the Fourier transform of the off-diagonal matrix ...

3

The connection has been provided explicitly repeatedly, best by P Sharan (1978). In words, essentially, the time-evolution kernels of the Wigner function from each phase-space point to all other such points is computed, and then concatenated with kernels for a subsequent move, and integrated over all intermediate points. Concatenation of an infinity of such ...

3

The question might well be too broad. One may, of course, reconfigure the variables x and p into different ones, and integrate w.r.t. the "irrelevant" one , e.g. the angular variable on phase space, to produce a marginal quasi-probability distribution in the other, e.g. the angular variable squared, $(x^2+p^2)/2$ which happens to be the (rescaled variables') ...

2

I don't really have a good answer, so I'll just discuss the obvious "translation" issues involved in the Wigner transform. That is, once you have a solution ot the operator problem in Hilbert space, in principle you can map it to phase space, but phase space in itself will not expedite the solution---it will actually make it messier. The absolute value of ...

2

Indeed, the Bopp shift is a clumsy Lagrange translation operator transcription of the celebrated * product, a 4-variable integral transform, cf. eqns (12-15) in Ref. 1. There is an infinity of phase-space functions corresponding to differently ordered operators, as their ps and qs may be ordered in different ways with the intercalated * s enforcing ...

2

The answer is a resounding "yes", cf Ref. 1, provided by Groenewold in 1946, op cit, and countless imitators since. The Husimi is completely equivalent, so, injective, to the Wigner d.f., and so the answer is ipso facto "yes" here too. I do not understand your particular SU(2)-blindness attributed to the Husimi, but I trust it is just an artifact of some ...

2

The inverse of the Wigner map is the Weyl quantization map. Let $a(x,\xi)$ be a function of the phase space (i.e. the symbol, in mathematical terms). If $a$ is real-valued, then $(a)^{Weyl}$ is symmetric; Let $g(x,\xi)\in C^\infty(\mathbb{R}^{2d}; \mathbb{R}_+^*)$ such that $\partial_{(x,\xi)}^\alpha g=O(g)$ for any $\alpha\in \mathbb{N}^{2d}$ and ...

2

Negative probabilities are only possible if they are invisible. They can only be associated with joint measurements. However, we must actually forbid joint measurements. This is only possible with the additional property of measurement disturbance aka Heisenberg's uncertainty principle as properly understood. If we measure a marginal value, the very act of ...

2

As Morgan pointed out, extended probabilities, which is the technical name, means joint probability distributions may have negative probabilities, but marginal probabilities never. But this is a stretch. How can it be a joint probability distribution if we can never measure complementary observables simultaneously? Extended probabilities also means we can ...

2

Let us for simplicity work in 1D with $\hbar=1$. (The generalization to higher dimensions is straightforward.) Moreover, let us for simplicity take an operator $\hat{f}(\hat{X},\hat{P})$ without any ordering ambiguities, i.e., each monomial term in the symbol $f(x,p)$ depends only on either $x$ or $p$, but not on both. Then one possible motivation of ...

1

There is recent work that answers this question in special contexts. In particular, arXiv:1401.4174v1 establishes that for discrete systems of odd prime dimensions (i.e. we're talking about the discrete Wigner functions now, defined in arXiv:quant-ph/0401155v6 ), contextuality (a manifestly quantum property, see review by Mermin) is equivalent to negativity ...

1

You're effectively doing signal analysis. It's neither quantum mechanics nor a semi-classical approximation. Signal processing is (very) often not stochastic (random), unless thermal noise is an issue for accurate modeling of the apparatus. It's often easier to find a way to eliminate the effects of thermal noise than to calculate the effects of thermal ...

1

The Wigner function is used to describe joint probabilities between two sets of observables that do not commute. I could elaborate a bit more, but there is a lot of literature available where the authors explain this better than I can. A couple of useful articles from arXiv: Probabilistic aspects of Wigner function Negativity of the Wigner function as an ...

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