Stress-tensor operator for fermion field interacting with classical gravitational field

Abstract

The interaction between a quantized spin- 1 2 fermion field and a classical gravitational field is studied. The S-matrix elements are expressed in terms of a functional integral of the vacuum expectation value of the fermion stress tensor (and the first quantized fermion Green's functions). The basic problem for such a system is to find an expression for the stress tensor operator that avoids paradoxes analogous to those examined by Schwinger in current commutators. This is solved to first order in the gravitational potentials by “spreading” the points in the usual stress tensor expression and introducing a modified version of the Mandelstam-De Witt path-dependent functions. The two-sided fermion closed loop (with gravitational vertices) is calculated and is explicitly seen to be conserved, invariant under gravitational gauge transformations, and Lorentz covariant.