Diracs equation for particles.

Relativistic particles according to their behavior obey with the Dirac equation in curved spaces, especially in expanding universe is of great importance in astrophysics and cosmology. A very important but at the same time puzzling problems of the quantum field theory in curved spacetime is the problem to find an analytic solution of Dirac equation on given backgrounds. Barut and Duru investigated the exact solutions of the Dirac equation for three typical models of expanding universe. The Dirac equation in the line element becomes

where dot refers to differentiation with respect to conformal time η and flat-Dirac matrices

are defined as follows

Substituing

the Dirac equation becomes

and we recieve a pair of equations:

After inserting

we obtain

1.Birrel, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge Univ. Press, Cambridge

(1982)

2. Linde, A.D.: Particle Physics and Inflationary Cosmology. Harwood Academic, Reading