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Path integrals on finite sets. (English) Zbl 0853.58026

The author uses the formalism of the Feynman path integral to “solve” \[ -{1 \over i} {\partial \over \partial t} \phi_t = H_0 \phi_t \] where \(H_0\) is a self-adjoint operator on \(L^2(M) = \mathbb{C}^M\), \(M\) being a finite set. The paths are functions of \(\mathbb{R}\) with values in \(M\). The path integral here is a family of measures parametrized by \(t'\) and \(t\) with values in the space of operators on \(L^2(M)\). The relationship of these measures to the measures of a Markov process is studied and it is shown that these measures are concentrated on a certain set of step measures. Some applications to certain hypothetical systems are also considered.

MSC:

58D30 Applications of manifolds of mappings to the sciences
81S40 Path integrals in quantum mechanics
46G10 Vector-valued measures and integration
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