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An Operator Derivation of the Path Decomposition Expansion

The path decomposition expansion is a path integral technique for decomposing sums over paths in configuration space into sums over paths in different spatial regions. It leads to a decomposition of the configuration space propagator across arbitrary surfaces in configuration space. It may be used, for example, in calculations of the distribution of first crossing times. The original proof relied heavily on the position representation and in particular on the properties of path integrals. In this paper, an elementary proof of the path decomposition expansion is given using projection operators. This leads to a version of the path decomposition expansion more general than the configuration space form previously given. The path decomposition expansion in momentum space is given as an example.