Effects of non-locality in the star-product of two fields

Question

My question regards an argument appearing on page 19 of the review: Quantum Field Theory on Non-commutative Spaces - Szabo. The Fourier integral kernel representation of the star-product of two fields is given previously as,

\begin{align} \operatorname{Tr}\left(\hat{\mathcal{W}}[f] \hat{\mathcal{W}}[g] \hat{\Delta}(x)\right)=\frac{1}{\pi^{D}|\operatorname{det} \theta|} \iint d^{D} y d^{D} z f(y) g(z) \mathrm{e}^{-2 i\left(\theta^{-1}\right)_{i j}(x-y)^{i}(x-z)^{j}} \tag{2.17} \end{align}

In this expression, there are fields $f$ and $g$, and $\hat{\mathcal{W}}[f]$ is the Weyl operator corresponding to the field $f$. The object $\theta$ defines the non-commutativity of the coordinates. Of expression (2.17), the author writes,

The oscillations in the phase of the integration kernel there suppress parts of the integration region. Precisely, if the fields $f$ and $g$ are supported over a small region of size $\delta \ll \sqrt{\|\theta\|}$, then $f \star g$ is non-vanishing over a much larger region of size $\|\theta\| / \delta$.

I don't understand this argument. I see that (2.17) is an oscillatory integral, and so oscillations in the phase may lead to suppression of different parts of the integration region. I don't understand the second part. Why consider a region of size $\delta \ll \sqrt{\|\theta\|}$? How can one see that $f \star g$ is non-vanishing over a region larger than $\|\theta\| / \delta$?

Answer 1

This is a basic feature of the star product, easiest to see in bland QM in phase space. (Just take D=2 dimensions x and p, and $\theta_{ij} =\hbar \epsilon_{ij}$; beyond the stringer tart-up, the mathematical structure is identical to QM's.) You might, or might not profit from the Real Mc Coy: Zurek, W. H. (2001), Sub-Planck structure in phase space and its relevance for quantum decoherence, Nature 412(6848), 712-717.

The integral representation of the $\star$-product (2.17) you are discussing (Baker 1958; Srinivasan & Wolf 1975) are eqns (15) & (163) of our revised tutorial booklet.

Basically, in words, the $\star$-product entangles translations of f and g, so the smaller the support area of the functions, the larger the gradients involved, hence the bigger the translations (non locality). This is all mere verbiage, of course, unless it summarizes basic paradigm calculations you could perform yourself, like your author's (3.8) —a constant, infinitely broad! (Exercise 0.4 in our booklet.)

An easy sketch for this (prove it!) is our (rapidity composition) Corollary (82), namely $$ \exp( -a(x^2+p^2)/\hbar ) \star \exp( -b(x^2+p^2)/\hbar )\\ \approx \exp\left ( -{(a+b)(x^2+p^2)\over \hbar( 1+ab)}\right ). $$ Take both widths of the 2d Gaussians $\star$-composed to be small: $\hbar/a=\hbar/b=\delta^2$, so their star product has a width $\sqrt{(\delta^2 + \hbar^2/\delta^2)/2}$, effectively a huge width ℏ/δ√2. As δ goes to zero to get to the Dirac δ-function limit, the width of the respective star product goes to infinity!

To flesh out the initial verbiage paragraph then, the "Bopp shift" representation of the star product is $$f(x,p)\star g(x,p)=f\left(x+\frac{i\hbar}{2}\partial_p,~p-\frac{i\hbar}{2}\partial_x\right) g(x,p), $$ where the gradients only act on g, not f. But if g is confined to a small ambit of linear dimension δ, the gradients are effectively huge, so the shifts of the arguments of f are huge, ℏ/δ, and the non-locality augments dramatically. You can dress this up with symplectic Fourier transform verbiage, but it is a simple, primitive phenomenon. Dreaming of miracle cancellations is path-integral-think overkill, all of which was slyly bypassed by phase-space quantization.