ABSTRACT
The short-time asymptotic expansion of the heat kernel of a linear differential operator has many applications in physics and pure mathematics. For second-order operators whose leading term is a Laplacian or Laplace–Beltrami operator, algorithms for calculating the terms in the expansion are well known. The algorithm most popular in the physics literature is that of DeWitt. Attempts in the physics literature to develop methods for treating such operators have been inconclusive and sometimes contradictory. It is clear that the problem can be solved in principle by the calculus of pseudodifferential operators, but the calculational complexity is such that computerization is necessary. The increasing availability of symbolic computation software on relatively accessible computers will greatly facilitate applications. The chapter describes the pseudodifferential approach and its present status.
