The Geometric Method
This chapter deals with a construction of heat kernels from the geometric point of view. Each operator will be associated with a geometry. Investigating the geodesic flow in this geometry, one can describe the heat kernels for a large family of operators. The idea behind this method is that the heat flow propagates along the geodesics of the associated geometry. The “density” of the heat flow is described by a volume function that satisfies a transport equation which is an analog of the continuity equation from fluid dynamics. This corresponds to the density of paths given by the van Vleck determinant in the path integral approach. This method works for elliptic operators with or without potentials or linear terms. The method can be modified to work even in the case of sub-elliptic operators, as the reader will become familiar with in Chaps. 9 and 10. This method was initially applied for the Heisenberg Laplacian; see, for instance .
KeywordsTransport Equation Heat Kernel Classical Action Hamiltonian Function Jacobi Equation
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