Path integrals on finite sets

EGF Thomas

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2 Citations (Scopus)

Abstract

We construct an analogue of the Feynman path integral for the case of

-1/i partial derivative/partial derivative t phi t = H-o phi t

in which H-o is a self-adjoint operator in the space L(2)(M) = C-M, where M is a finite set, the paths being functions of R with values in M. The path integral is a family of measures F-t',F-t with values in the operators on L(2)(M), or equivalently, a family of complex measures corresponding to matrix coefficients.

It is shown that these measures on path space are in some sense dominated by the measure of a Markov process. This implies that F-t',F-t is concentrated on the set of step functions S[t,t'].

This allows one to make sense of, and prove, the analogue of Feynman's formula for the propagator of the Hamiltonian H = H-o + V, where V is a potential, namely the formula:

e(-i(t'-t)H) = integral(s[t,t']) e(-i integral?t?(t')) (v(x)s))ds) F-t',F-t(dx)

and the corresponding formulas for the matrix coefficients, in which the integral extends over the paths beginning and ending in the appropriate points. We show that the measures F-t',F-t are completely determined by these equations and by a certain multiplicative property.

The path integral corresponding to a 'two-particle system without interaction' is the direct product of the corresponding path integrals. The propagator for a 'two-particle system with interaction' can be obtained by repeated integration.

Finally, we show that the above integral formula can be generalized to the case where the potential is time dependent.

Original languageEnglish
Pages (from-to)191-232
Number of pages42
JournalActa applicandae mathematicae
Volume43
Issue number2
Publication statusPublished - May-1996

Keywords

  • path integral
  • Feynman
  • Markov processes
  • DEFINITION

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