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.
|Number of pages||42|
|Journal||Acta applicandae mathematicae|
|Publication status||Published - May-1996|
- path integral
- Markov processes