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25

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

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" ...


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

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

Probabilities have a physical meaning only in a context where measureemnt is possible. Between states of a pointer basis in a measurement context, the matrix elements of a density matrix have the standard probabilistic meaning. In any other basis, they are just mathematical expressions intermediate to other calculations of interest. (I wouldn't give a penny ...


6

I) 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 ...


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

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 ...


4

Your statement on a pseudo-probability interpretation is slightly off, in that you most certainly did not write the Wigner quasi-probability distribution in your edit. Taking your two bases to be $x$ and $p$ for specificity, what you wrote, $ρ(x,p)$, up to a phase: $e^{(ixp/ħ)}$, is the "standard ordering prescription" for quasi-distribution functions, ...


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 ...


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} ...


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 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 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') ...


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

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 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

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

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 classic review you quote is absolutely right. It means position and momentum at the same time! Well, if you write down something that replicates QM, it must somehow comport with the uncertainty principle. The expression (bilinear in wave functions, not the present squared version!) you wrote down is actually quite close to the standard ordering ...


2

The probabilistic interpretation of the Wigner function is already flawed at the level of the Kolmogorov axioms, even for unremittingly positive values. That is to say, two points in phase space within a distance less than $\hbar$ are not mutually exclusive sample space contingencies. In physics parlance, these two points are not distinguishable in any ...


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

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 ...



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