A DEWITT EXPANSION OF THE HEAT KERNEL FOR MANIFOLDS WITH A BOUNDARY

The DeWitt (DW) expansion of the heat kernel G-DELTA (chi, chi'; tau), for a second order elliptic differential operator DELTA is extended to manifolds with a boundary. The boundary conditions are satisfied by including an additional term to the usual DW contribution appropriate for manifolds without a boundary, which provides an asymptotic expansion as tau --> 0 for x almost-equal-to x'. The extra piece is based on geodesic paths linking x and x', which undergo reflection on the boundary, just as the DW expression is based on the direct geodesic from chi to x'. The procedure is straightforward and manifestly covariant throughout, requiring no expansion about flat space, and allows for both the Dirichlet and generalized Neumann boundary conditions. The boundary contributions to Tr(e-tau-DELTA), which are an expansion in tau-1/2, are found in agreement with previous indirect methods, although the corresponding contributions to G-DELTA involve non-polynomial functions of tau. Conservation equations for vector and tensor fields obtained from the heat kernel, which include terms restricted to the boundary, are also verified. The results are applied to determine the leading singular behaviour of the Green function for DELTA at x = x' in the neighbourhood of the boundary.