Consider a Klein-Gordon field $\phi$ on points $x,y$ of $\mathbb R^4$ Minkowski-spacetime. Here I'm writing $x=(x^0, \stackrel \rightarrow x)$ so that $\stackrel \rightarrow x$ gives the spatial components and $x^0$ is time. The field commutator for local points is defined by
$$
[\phi(x),\phi(y)]=c \int_{\mathbb R^3} d^3\stackrel \rightarrow k \frac{1}{(2\pi)^3 2 \omega}\left ( e^{-i\omega (x^0-y^0) + i \stackrel \rightarrow k\cdot (\stackrel \rightarrow x-\stackrel \rightarrow y)} - e^{+i\omega (x^0-y^0) + i \stackrel \rightarrow k\cdot (\stackrel \rightarrow x-\stackrel \rightarrow y)} \right )
$$
where $\omega=\sqrt{|\stackrel \rightarrow k|^2 +\omega_0^2}$ and $\omega_0$ is a constant (rest mass). I want to verify that this integeral is equal to
$$
[\phi(x),\phi(y)]=c \mbox{sign}(x^0-y^0) \left ( i \omega_0 \theta (\tau^2) \frac{J_1(\omega_0 \tau)}{4\pi \tau} - \frac i {2\pi} \delta (\tau^2) \right )
$$
where $\tau=\sqrt{-(x-y)^2}$ is proper time, $\theta$ is the Heavyside function, and $J_1$ is the Bessel function. This identity looks intractable to me. How can you verify it? I have tried seeing if the original integral appropriately modified will satisfy the Bessel's defining differential equation. I have also tried viewing the original integral as a Fourier transform, but I'm stumped. Any suggestions? Thanks.

This post imported from StackExchange Physics at 2015-01-20 23:21 (UTC), posted by SE-user spitespike