[QUE/QFT-05006] QFT-PROBLEM

For a free complex Klein Gordon field find the unequal time commutator as \[ \big[\phi(x), \phi(y)\big] = i\Delta(x-y)\] and express your answer for \(\Delta(x)\) as an integral of the form \[\int dq e^{-iqx} \delta(q^2-\mu^2) \epsilon(q_0) \] You need not compute the integral.

• Argue that the function \(\Delta(x)\) is odd under change of sign of \(x\) and that it is Lorentz invariant.
• For spacelike \(x\) show that there exists a Lorentz frame such that \(x^\prime =-x\). Hence prove that the function \(\Delta(x)\) vanishes for space like \(x\).