6.2.5 A Caveat on Marginal Distributions

If $X$ and $Y$ are random variables and $f(x,y)$ is their joint probability distribution, the marginal distribution $g(x)$ of $X$ is given by

$$g(x) = \int f(x,y) dy .$$ (6.54)

It can be interpreted as a probability density of the single variable x. The random variables $X$ and $Y$ are independent if and only if $f$ can be factorized by the marginal distributions $g(x), h(y)$ as

$$f(x,y) = g(x) h(y) .$$ (6.55)

The Wigner distribution $f$ has the properties

$$\frac{1}{2\pi\hbar} \int dp f(q,p) = \langle q \vert \rho \vert q \rangle$$ (6.56)

$$\frac{1}{2\pi\hbar} \int dq f(q,p) = \langle p \vert \rho \vert p \rangle .$$ (6.57)

For an observable $A$ with corresponding symbol $a$ we have for its expectation value

$$\begin{gather*}\begin{split}\bar{A} = &{\mathrm{tr} (}\rho A) = {\mathrm{tr} ... ...ar a(q,p) = \frac{1}{2\pi\hbar}\int dp dq f(q,p) a(q,p) \end{split}\end{gather*}$$ (6.58)

where $\star$ denotes the Moyal star product.

Classically the definition of the local (marginal) expectation values of observables is not ambiguous due to the commutativity of all observables. However, quantum mechanically to each way of calculating the expectation value in 6.59 corresponds a definition for the local expectation value $a(q)$ and in general these definitions give different results

$$\langle q \vert\rho A\vert q \rangle \neq \langle q\vert A \rho \... ...}\int dp f(q,p) \star a(q,p) \neq \frac{1}{2\pi\hbar}\int dp f(q,p) a(q,p) .$$ (6.59)

In general the operator $A \rho$ is not selfadjoint, so the definitions using Dirac brackets above are different. We can rephrase this by using the explicit expression for the density operator for a pure state $\rho = \vert\psi \rangle \langle \psi \vert$. For a single wave function $\psi$ we get two definitions for the local expectation $a(q)$

$$\langle q \vert \psi \rangle \langle \psi \vert A \vert q \rangle \neq \langle q \vert A \vert \psi \rangle \langle \psi \vert q \rangle .$$ (6.60)

A good way to define the local expectation is to symmetrize the definitions

$$a(q) = \frac{1}{2} \langle q \vert (\rho A + A \rho) \vert q \rangle .$$ (6.61)

This definition has the property that $\rho A + A \rho$ is again a selfadjoint operator, hence the local expectation is real. This is the way in which the current $j(x)$ is conventionally defined.

A special case which is not treated in this way is the definition of local expectation for an energy-like operator $E = P^2$ where $P$ is selfadjoint. Then we can define

$$P^2(q) = \langle q \vert P \rho P \vert q \rangle ,$$ (6.62)

which is a positive quantity. We use this form to define a positive phase space distribution function in Section 6.2.6.

However, in the Wigner formalism one usually defines energy as the second $p$-moment in the form

$$E(q) = \frac{1}{2\pi\hbar}\int dp \frac{p^2}{2m} f(q,p) .$$ (6.63)

With this definition for the local expectation value the local energy $E(q)$ can become negative, as was observed in Wigner function simulations.

To calculate the Wigner transformation of $P \rho P$ we use a suitable definition for the star product. We get

$$\tilde{E}(q) = \frac{1}{2\pi\hbar}\int dp \frac{1}{2m}(p^2 + ({\frac{\hbar}{2}})^2 \partial_q^2) f(q,p)$$ (6.64)

which is seen to differ from the second order moment definition 6.64.

The question of marginalization is also discussed in [Wlo99].