# Klein-Gordon field commutator integral identity

+ 3 like - 0 dislike
304 views

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
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
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
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
@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
@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
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
@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, C&eacute;cile 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).

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.