But where did it come from? ##\theta(x^0 - y^0)##, right? The restricted Lorentz group (identity-connected part) preserves time orientation, so it's ok to use ##\theta(x^0 - y^0)## in that circumstance. The rest just involves expressing it in momentum space.realanswers said:
Feynman Green's function, also known as the Feynman propagator, is a fundamental solution to the equations of motion in quantum field theory. It describes the probability amplitude for a particle to propagate from one point to another in spacetime and is essential for calculating scattering amplitudes in Feynman diagrams.
The 4-momentum integral is important because it allows the Feynman Green's function to be expressed in momentum space, which simplifies calculations in quantum field theory. By transforming the problem into momentum space, one can take advantage of the symmetries and conservation laws that are more naturally expressed in terms of momentum.
The Feynman Green's function \( G_F(x - y) \) in position space can be expressed as a 4-momentum integral in momentum space by using the Fourier transform. The expression is given by:\[ G_F(x - y) = \int \frac{d^4 p}{(2\pi)^4} \frac{e^{-ip \cdot (x - y)}}{p^2 - m^2 + i\epsilon} \]where \( p \) is the 4-momentum, \( m \) is the mass of the particle, and \( \epsilon \) is an infinitesimally small positive number ensuring the correct causal structure.
The infinitesimal parameter \( \epsilon \) ensures the correct causal structure of the propagator by specifying the boundary conditions. It shifts the poles of the propagator slightly off the real axis in the complex plane, which dictates the time-ordering of the fields and ensures that the propagator decays appropriately for large time separations.
Yes, the Feynman Green's function can be computed numerically using the 4-momentum integral, although it can be challenging due to the oscillatory nature of the integrand and the need to handle the infinitesimal parameter \( \epsilon \). Numerical techniques such as contour integration or regularization methods are often employed to obtain accurate results.