Wigner function = Fourier transform + Coordinate rotation

[Follow-up post: In what sense is the Wigner function a quasiprobability distribution?]

I’ve never liked how people introduce the Wigner function (aka the Wigner quasi-probability distribution). Usually, they just write down a definition like

(1)   \begin{align*} W(x,p) = \frac{1}{\pi \hbar} \int \mathrm{d}y \rho(x+y, x-y) e^{-2 i p y/\hbar} \end{align*}

and say that it’s the “closest phase-space representation” of a quantum state. One immediately wonders: What’s with the weird factor of 2, and what the heck is y? Usually, the only justification given for the probability interpretation is that integrating over one of the variables recovers the probability distribution for the other (if it were measured):

(2)   \begin{align*} \int \! \mathrm{d}p \, W(x,p) = \vert \rho(x,x) \vert^2 , \\ \int \! \mathrm{d}x \, W(x,p) = \vert \hat{\rho}(p,p) \vert^2 , \end{align*}

where \hat{\rho}(p,p') is just the density matrix in the momentum basis. But of course, that doesn’t really tell us why we should think of W(x,p), as having anything to do with the (rough) value of x conditional on a (rough) value of p.

Well now I have a much better idea of what the Wigner function actually is and how to interpret it. We start by writing it down in sane variables (and suppress \hbar):

(3)   \begin{align*} W(\bar{x},\bar{p}) = \frac{1}{2 \pi} \int \! \mathrm{d}\Delta x \,\rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) e^{-i \bar{p} \Delta x}. \end{align*}

So the first step in the interpretation is to consider the function

(4)   \begin{align*} M(\bar{x},\Delta x) \equiv  \rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) , \end{align*}

which appears in the integrand. This is just the (position-space) density matrix in rotated coordinates \bar{x} \equiv (x+x')/2 and \Delta x = x-x'. There is a strong sense in which the off-diagonal terms of the density matrix represent the quantum coherence of the state between different positions, so \Delta x indexes how far this coherence extends; large values of \Delta x indicate large spatial coherence. On the other hand, \bar{x} indexes how far down the diagonal of the density matrix we move; it’s the average position of the two points between which the off-diagonal terms of the density matrix measures coherence. (See the figure below.)


The function M(\bar{x},\Delta x) is just the position-space density matrix \rho(x,x') rotated in new coordinates: (\bar{x},\Delta x) = ((x+x')/2,x-x'). \bar{x} moves down the diagonal, and \Delta x reaches off-diagonal.

So then we see that the Wigner function W(\bar{x},\bar{p}) is just the Fourier transform of M(\bar{x},\Delta x) with respect to the coherence variable \Delta x.

(5)   \begin{align*} W(\bar{x},\bar{p}) = \int \! \mathrm{d}\Delta x \, M(\bar{x},\Delta x) e^{-i \bar{p} \Delta x}. \end{align*}

We can express this more compactly as

(6)   \begin{align*} W = \mathcal{F}_{\Delta x \to p}[R_{x,x' \to \bar{x},\Delta x}[\rho]]. \end{align*}

where \mathcal{F}_{\Delta x \to p} denotes the Fourier transform from \Delta x to \bar{p} and R_{x,x' \to \bar{x},\Delta x} denotes the relevant 45^\circ coordinate rotation from (x,x') to (\bar{x},\Delta x). This interpretation has several of advantages.

First, all the numerical constants in the formula for the Wigner function can be immediately traced back to the Fourier transform. (Different Fourier transform conventions will of course have different constants.)

Second, transforming the density matrix into a Wigner function is a manifestly reversible operation, since both the Fourier transform and the rotation are reversible.Remember that, aside from being a complex-valued function on R^2, the density matrix \rho must also be positive. I haven’t bothered to figure out how that requirement translate into restrictions on the Wigner function beyond the simple fact that it’s a real-valued function on R^2. The unit-trace requirement on \rho is equivalent to the requirement that integrating W(\bar{x},\bar{p}) over its domain equals unity.a   So the Wigner function is just another lens with which we can look at the state of a quantum system. We could just as well always work with Wigner functions when analyzing the dynamics of a quantum system, although the Hamiltonian evolution turns out to be a little uglier to express.

Third, we can also see how the Wigner function is related to the momentum-space representation of the density matrix by “filling in” all the possible applications of Fourier transforms and coordinate rotations. (See the next figure.) This lets us immediately and transparently derive that

(7)   \begin{align*} W(\bar{x},\bar{p}) &= \int \! \mathrm{d}\Delta p \, \hat{M}(\bar{p},\Delta p) e^{+i \bar{x} \Delta p} \\ &= \int \! \mathrm{d}\Delta p \, \hat{\rho} \left(\bar{p}+\frac{\Delta p}{2},\bar{p}-\frac{\Delta p}{2} \right) e^{+i \bar{x} \Delta p}. \end{align*}

where \hat{M}(\bar{p},\Delta p) is the relevant coordinate rotation of \hat{\rho},(p,p').


Diagram of how the position- and momentum-space representations of the density matrix are related to the Wigner function by Fourier transforms and coordinate rotations. Auxillary functions \sigma, \hat{\sigma}, M, \hat{M}, and \hat{W} have been introduced to fill in possibilities. Each function can be transformed into any other.

The (doubly) Fourier transformed Wigner functionI’m going to stop trying to bother with normalization.b  

(8)   \begin{align*} \hat{W}(\Delta p,\Delta x) \propto \int \int \! \mathrm{d}\bar{x} \, \mathrm{d}\bar{p} \, W(\bar{x},\bar{p}) e^{+i \bar{x} \Delta p} e^{-i \bar{p} \Delta x} \\ \end{align*}

(aka the Moyal characteristic function, aka the Wigner characteristic function, especially when the arguments are swapped) also has a nice relationship to the Husimi Q function Q(\vec{\alpha}) = \langle \vec{\alpha} \vert \rho \vert \vec{\alpha} \rangle. Here, \vec{\alpha} = (\bar{x},\bar{p}) is just a condensed notation for phase space coordinates and \vert \vec{\alpha} \rangle is the (unsqueezed) coherent state centered at (\bar{x},\bar{p}). It’s well known that the Husimi Q function is just the Wigner function convolved with a Gaussian:

(9)   \begin{align*} Q(\vec{\alpha}) \propto \int \! \mathrm{d}\vec{\beta} \, W(\beta) e^{-\vert \alpha - \beta \vert^2/2} \\ \end{align*}

where \mathrm{d}\vec{\beta} = \mathrm{d}\bar{x} \, \mathrm{d}\bar{p} and \vert \vec{\alpha} - \vec{\beta} \vert^2 = \bar{x}^2+\bar{p}^2. (Up to normalization, the Wigner representation of the coherent state at \vec{\alpha}_0 is a Gaussian: e^{-\vert \vec{\alpha}_0 - (\bar{x},\bar{p}) \vert^2/2}.)

But of course, by the convolution theorem, a convolution in real space is just multiplication in Fourier space:

(10)   \begin{align*} \hat{Q}(\vec{\nu}) \propto e^{-\vert \vec{\nu} \vert^2/2} \hat{W}(\vec{\nu})  \\ \end{align*}

where

(11)   \begin{align*} \hat{Q}(\vec{\mu}) &\propto \int \! \mathrm{d}\vec{\alpha} Q(\vec{\alpha}) e^{-i \vec{\mu} \cdot \vec{\alpha}},  \\ \hat{W}(\vec{\mu}) &\propto \int \! \mathrm{d}\vec{\alpha} W(\vec{\alpha}) e^{-i \vec{\mu} \cdot \vec{\alpha}}. \end{align*}

In fact, the Fourier transformed Wigner function \hat{W} has more meaning than you might think. Structure of the Wigner function inside areas smaller than \hbar are known to be associated with the type of long-scale coherence that is typically destroyed by environmental decoherence.Furthermore, this structure is responsible for superpositions being useful in quantum enhanced measurements.c   But the Fourier transformed Wigner function inverts this; the sub-Planckian structure now is on large scales, while the supra-Planckian structures (which gives the classical phase space probability distribution) is compressed to small scales. When calculating the Husimi Q function, we see that the Gaussian just suppresses the large-scale structure (i.e. the coherence) while leaving the small-scale structure untouched. (Of course, we could probably have read this off directly from the convolution.)

[Edited 2014-6-10]

Footnotes

(↵ returns to text)

  1. Remember that, aside from being a complex-valued function on R^2, the density matrix \rho must also be positive. I haven’t bothered to figure out how that requirement translate into restrictions on the Wigner function beyond the simple fact that it’s a real-valued function on R^2. The unit-trace requirement on \rho is equivalent to the requirement that integrating W(\bar{x},\bar{p}) over its domain equals unity.
  2. I’m going to stop trying to bother with normalization.
  3. Furthermore, this structure is responsible for superpositions being useful in quantum enhanced measurements.
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One Comment

  1. Oh.. man! That is one of the *BEST* explanations I have seen so passionately explained on the internet on that too a serious subject! Thanks a bunch! Keep writing. 🙂

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