Heat Equation

Abstract

AbstractThis Chapter introduces the most important technical tool of this book—the heat kernel expansion. The Chapter starts with main definitions and continues with examples of the heat trace for operators with known spectrum. The universal form of the heat trace asymptotics is stated. The rest of the Chapter is devoted to calculations of the coefficients in the asymptotic expansion (the celebrated heat kernel coefficients). First, the DeWitt method based on the recurrence relations is reviewed. Though this method dominated in physics for a long time, the Gilkey approach, based on “functorial” relations is more flexible and is the main tool in this book. It is shown, that in this approach the heat kernel coefficients are calculated in a simple and transparent way on manifolds with and without boundaries. After suitable modifications the same approach allows one to compute the heat kernel expansion in the presence of codimension one defects (branes) and on manifolds with conical singularities. The calculations are presented in full detail, and particular examples of fields of various spins are considered.KeywordsAsymptotic ExpansionHeat EquationHeat KernelSpectral ProblemConical SingularityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.