Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution

$$[F\star G](x,p) = \int \!dy\,dk\, F(y,k)G(x-y,p-k)e^{i (xk-yp)}$$

or more compactly

$$[F\star G](\alpha) = \int \!d\beta\, F(\beta)G(\alpha-\beta)e^{i \alpha \wedge \beta}$$

where $\alpha = (x,p)$ and $\wedge$ is the symplectic form. Unlike the symplectic Fourier transform,

$$\hat{F}(\xi) = \int \!d\alpha\, F(\alpha)e^{i\alpha\wedge\xi}$$

which can be completely understood as normal Fourier transform followed by a change of variables $(\xi_x,\xi_p) \to (-\xi_p,\xi_x)$, the symplectic convolution appears to have nontrivial properties. However, I have not been able to find almost any references that discuss it. The most clear reference to it I could find is in "Toeplitz and Hankel operators and Dixmier traces on the unit ball of $\mathbb{C}^n$" by Englis et al., but they seem to use it without much discussion.

Does the symplectic convolution go by another name? What introductory references discuss its basic properties?

  • 2
    In the harmonic analysis literature this is known as "twisted convolution", and can be viewed as a component of convolution on the Heisenberg group. The last few chapters of Stein's "Harmonic analysis" has some discussion of this operator, if I recall correctly. One may also wish to check Folland's "Harmonic analysis on phase space". – Terry Tao Feb 3 '16 at 3:11
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    Using your notation, $F\star \hat{G}$ is none other than the Wigner-Weyl-Moyal formula for the $\star$-product of $F$ and $G$ in "phase space quantum mechanics". Sometimes the name Groenewold gets appended as well. It appears in many references of varying sophistication. Probably, googling around using these keywords you can find one that appeals to you. – Igor Khavkine Feb 3 '16 at 3:46
  • Thanks very much, both of you. Exactly what I needed to know, and very helpful. Happy to accept an official answer if either of you want to submit one. – Jess Riedel Feb 3 '16 at 19:35
up vote 5 down vote accepted

This operation (generalised slightly by replacing $e^{i (xk-yp)}$ by $e^{i\lambda (xk-yp)}$ for a parameter $\lambda$) is known as "twisted convolution" in the harmonic analysis literature, see e.g. Chapter XII.3.3 of Stein's "Harmonic analysis". (Side note: the citation tool does not seem to cover books, which is very strange.) As noted in that text, it arises naturally in studying convolution on the Heisenberg group (with group law $(x,p,t) (y,k,s) = (x+y,p+k,t+s+ xk-yp)$), after reducing to an isotypic component of the action of the centre (i.e. to functions of the form $f(x,p,t) = F(x,p) e^{i \lambda t}$ for some fixed $\lambda$). It also shows up in the composition law for pseudodifferential operators under Weyl quantisation: if $Op(a) Op(b) = Op(c)$, then the Fourier transform of $c$ is the twisted convolution of the Fourier transforms of $a$ and $b$ for a suitable choice of parameter $\lambda$. Equivalently, as noted above by Igor, $c$ is the Moyal product of $a$ and $b$.

I'm partial to Folland's "Harmonic analysis in phase space" for a treatment of all of these topics (e.g. twisted convolution is introduced on page 25).

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