Fermion Particle Production in a Periodic Potential

In this paper we study fermion particle production in the early universe. The present work is motivated to restudy the fermion particle production from the basics and compare the results in the literature through another method developed by one of the present author. One of the authors (SB) has developed a method, known as complex trajectory WKB method, to study particle production in curved as well as flat spacetime. In the present work we have tried to compare the CWKB method with that of other works, current in the literature. In this work we have obtained the particle production amplitude starting from the basics and test our results through both analytical and numerical calculations. For fermion particle production, we first do analytical calculations with a toy example to calculate the production amplitude and verify the same doing fourth order Runge-Kutta calculation. As most problems relevant to early universe are not amenable to analytical calculations, we then take up to study the particle production in periodic potential, generally used in inflationary cosmology. We recheck two recent approaches and obtain almost identical results as that obtained by Greene and Kofman. We also verify the result through CWKB method. Boson particle production has been discussed elsewhere, we discuss it briefly in connection with CWKB. In the present work we generalize the CWKB results of boson production to fermion production. Our works will enable one to understand the various phenomena in early universe related to particle production. Using CWKB we calculate the occupation number and some other results for fermion particle production. The present work will help us clarify the variant results of fermion production current in the literature.