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The problem of a relativistic spinning particle in interaction with an electromagnetic plane wave field is treated via path integrals. The dynamics of the spin of the particle is described using the supersymmetric action proposed by Fradkin and Gitman. The problem has been solved by using two identities, one bosonic and the other fermionic, which are related directly to the classical equations of motion. The exact expression of the relative Green's function is given and the result agrees with those of the literature. Further, the suitably normalized wave functions are also extracted.

We determine explicitly the exact transcendental bound states energies equation for a one-dimensional harmonic oscillator perturbed by a single and a double point interactions via Green’s function techniques using both momentum and position space representations. The even and odd solutions of the problem are discussed. The corresponding limiting cases are recovered. For the harmonic oscillator with a point interaction in more than one dimension, divergent series appear. We use to remove this divergence an exponential regulator and we obtain a transcendental equation for the energy bound states. The results obtained here are consistent with other investigations using different methods.

The Green's functions for charged particles of spin zero and 1/2, which are subjected to the action of the field of an electromagnetic plane wave, are calculated in the path integral formalism. It is also shown that in the case of spin 0, the semi-classical Green's function obtained via a canonical transformation, is accurate. These Green's functions are obtained under a compact form. The waves in the case of spin 0 and the wave functions in the case of spin 1/2 are then deduced.