The Green's function for a spinless relativistic particle subjected to the action of an electromagnetic plane wave, with local gauge, is determined according to the stochastic quantum mechanics of G. Parisi and Wu. The evaluation was done in two steps: first the classical action is extracted and next the fluctuation factor is calculated. The treatment has been carried out in the phase and configuration spaces.
The problem of relativistic particles moving in the background of a weak gravitational plane wave is solved by the use of Parisi–Wu stochastic quantization method. After having solved the corresponding Langevin equations, we have calculated exactly the correlation product of two fields known as a Green's function at the thermal-equilibrium limit for the fictitious time. By addition of an infinitesimal imaginary part of the mass, the existence of the limit of the correlation function at the equilibrium is assured. Analytical and exact expressions of the wave functions are obtained both for Klein–Gordon (KG) and Dirac particles.
Using the Parisi and Wu stochastic quantization method, the Green's function for a Klein-Gordon particle subjected to the action of two plane electromagnetic waves propagating along particular directions is determined exactly thanks to a perturbative treatment. The calculation is performed in the phase space.
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