Dirac Equation in the Spatially Flat Friedmann Model

Abstract

The unitary transformation which diagonalizesthe field-free Dirac Hamiltonian in the spatially flatFriedmann-Robertson-Walker metric is analyzed, and apair of simultaneous first-order nonlinear differential equations is derived for the two parameters(two angles) that characterize the transformation. Theequations are solved approximately for a test particlewhose kinetic energy is small compared to its mass energy, and minimum-uncertainty wave packetsare constructcd from the solutions. It is found thatgeneral relativity limits the quantum mechanical spreadof the wave packets, but forces then to expand with the expanding space, as if they were embeddedin it. The massless Dirac equation is solved exactly forthe two-component neutrino spinor, and yieldsgeneralized nonspreading wave packets which display no quantum mechanical spread at all, but areconstrained to expand with the expanding space as theyfollow null geodesics.