Klein-Gordon Feynman Green's Function

A massive scalar field \(\phi (x,t)\) of mass \(m\) in quantum field theory satisfies the Klein-Gordon equation

\[\big(\Box + m^2\big) \phi(x,t) = 0,\]

where \(\Box\) indicates the d'Alembert wave operator:

\[\Box = -\dfrac{\partial^2}{\partial t^2} +\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} +\dfrac{\partial^2}{\partial z^2}. \]

Find the Green's function of the Klein-Gordon operator in momentum space by Fourier transform. Note that \(k = \big(E,\vec{k}\big)\) is a vector with four components: the energy of a particle, and its three components of spatial momentum.

Notation: \(\text{abs}(\cdot) \) denotes the absolute value function.


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