I have a problem with proof of causality in Peskin & Schroeder, An Introduction to QFT, page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for y=0 as follows:
[ϕ(x,t),ϕ(0,0)]=∫d3p(2π)312√p2+m2(e−ip.x−it√p2+m2−eip.x+it√p2+m2)
The book goes on about how the integrand being Lorentz invariant makes this integral zero for the x out of the light cone. But I (not being a special relativity expert) want to see it more rigorously:
after changing variables p→−p in the first term, the equation simplifies to:
[ϕ(x,t),ϕ(0,0)]=∫d3p(2π)3−2i2√p2+m2eip.xsin(t√p2+m2)
using spherical coordinates:
[ϕ(x,t),ϕ(0,0)]=∫dpdϕdθp2sinθ(2π)3−i√p2+m2eipxcosθsin(t√p2+m2)[ϕ(x,t),ϕ(0,0)]=∫∞0dpp(2π)2−2ix√p2+m2sin(px)sin(t√p2+m2)
again after another change of variables u=√p2+m2,
[ϕ(x,t),ϕ(0,0)]=−2ix∫∞mdu(2π)2sin(x√u2−m2)sin(tu)
I cannot see how this integral should be zero for x>t !!! Can somebody please explain this to me?
This post imported from StackExchange Physics at 2014-03-22 17:27 (UCT), posted by SE-user Blackie