Klein-Gordon Feynman Green's Function

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

(+m2)ϕ(x,t)=0,\big(\Box + m^2\big) \phi(x,t) = 0,

where \Box indicates the d'Alembert wave operator:

=2t2+2x2+2y2+2z2.\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=(E,k)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: abs()\text{abs}(\cdot) denotes the absolute value function.

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