Measures for More Quadratic Path Integrals* Ingrid Daubechies *~

We show that the coherent state matrix elements of the quantum mechanical propagator for all quadratic Hamiltonians may be represented as the limit of path integrals with respect to appropriately modified Wiener measures as the associated diffusion constant tends to infinity. In this letter we state some results concerning path integrals with genuine mathematically welldefined measures to define matrix elements of the quantum mechanical evolution operator associated to a Hamiltonian JC. These results are a continuation, and in a certain sense also a generalization, of related work published earlier [1, 2, 3]. More specifically, we claim that for all quadratic Hamiltonian with time-dependent linear terms, the matrix elements between coherent states of the evolution operator T exp [-ifT~f(t) dt] can be written as a limit of well-defined path integrals involving Wiener measures. Our procedure follows a limiting approach introduced in [3], which is simpler and more intuitive than the projection technique used in [1], and which covers a wider class of Hamiltonians. Explicitly, our claim is that for ~c( t ) 1 2 = -~ [aP + 3Q2 + 7(PQ + QP)] + r(t)Q + s(t)P (1) we have ( p", q" l T exp l-i/or~C(t) dt } lp', q' ) = lira { exp ( 89 uT)Iv, ~c (P", q", T; p', q', o)}, /),-.~o o *Work partially supported by U.S. National Science Foundation Grant PHY-8116101. **On leave from the Dienst voor Theoretische Natuurkunde, Vrije Universiteit Brussels, Belgium and from Interuniversitair Instituut voor Kernwetenschappen, Belgium. Letters in MathematicaIPhysics 7 (1983) 229-234. 0377-9017/83/0073-0229 $00.90. Copyright 9 1983 by D. Reidel Publishing Company. (2)