Klein-Gordon field commutator integral identity

Question

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

Comments

You posted the same question on math.stackexchange. You should only post it to one of the sites. As I have explained there, there is something seriously wrong with your first formula as the term in the bracket simply vanishes. Furthermore, the integral over $k$ should be in 2d.

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

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As Fabian comments, the first integral is nonsensical, and hence I'm voting to close as unclear what you're asking. Also, "a Klein-Gordon field with frequency $\omega_0$" also non-sensical, since you can excite arbitrary energies in a free quantum field.

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

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Apologies. There was an extra negative sign on the second exponential term. The integral is taken over 3-space.

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

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@spitespike: if the integral is taken over $\mathbb{R}^3$ the result quoted is wrong...

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

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@Fabian Totally possible. I don't trust it which is why I want to verify. Why wrong?

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

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I don't think this is incorrect. See Theory of Quantized Fields tinyurl.com/ng3xysv circa page 150.

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

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@Spitespike, that Bogoliubov&Shirkov reference is precisely what I was going to send you to for the derivation. You could also look at "Fourier transforms of Lorentz invariant functions" for background, Alexander Wurm, Nurit Krausz, Cécile DeWitt-Morette, and Marcus Berg, J. Math. Phys. 44, 352 (2003); doi: 10.1063/1.1522817 (which, however, derives only the real part instead of the imaginary part you're asking about).