By Gert Roepstorff

This booklet has been written two times. After having written and released it in German in 1990, i began allover back and rewrote the complete tale for an English talking viewers. in the course of the first around I acquired encouraging phrases and significant feedback from scholars and associates alike that have helped to maintain me the second one time round. within the preface the writer frequently states that his or her ebook resulted from a direction that she or he gave at a few collage. i can't declare that the current ebook is any exception to the guideline. yet I increased and remodelled the unique fabric which circulated as a manuscript in order that the broadcast model might persist with a extra stringent and coherent architectural plan. In doing so i've got targeting the conceptual difficulties inherent within the course indispensable formalism instead of on yes hugely really good options utilized in functions. however, i've got additionally incorporated these tools which are of primary curiosity and feature taken care of particular difficulties in most cases to demonstrate them.

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Pn the random variable (1. J, kPjPkG(Sj, Sk). 10) All finite-dimensional distributions of the process X. 10) with respect to P (PI, ... ,Pn). Remark. Assume Sj f. Sk if j f. k. The condition E(y2) ;::: 0 (= 0 only if Pi = 0, i = 1, ... , n) implies that the numbers gjk = G(Sj, Sk) may be regarded as elements of some positive n x n matrix. Since this is the case for all time sequences Sl, ... ,Sn (all n), it is clear that the operator D has to be positive. If the positivity condition is violated, there is no Gaussian process attached to D.

Yet, a precise answer cannot be given; the matter is being debated among scientists and philosophers. Whenever we have a solution of the Schrodinger equation before us, we may look upon it as an analytic continuation, in some sense or another, of a solution of the diffusion equation. We thus connect the evolution of a quantum system of Schrodinger particles with the classical, but indeterministic, motion of a single Brownian particle living in the same configuration space. The role played by the interaction potential, though crucial, has not yet been clarified.

Obviously, any simple function is of this type; hence L eLand any f E L may be approximated by simple functions f n' For any such function f, the path integral reduces to an ordinary finite-dimensional integral. 12) we may state: Suppose the function f : fl ---+ lR is such that it depends only on finitely many coordinates of the path w. If these are the positions Xi = W(Si) E lRd at the times Si, i = 1, ... 19) provided the integral on the right-hand side exists. 44 2 The Feynman-Kac Formula Since the natural number n assumes arbitrarily large (finite but unbounded) values, one may justly speak of the path integral as an extension, to infinite dimensions, of the traditional notion of a finite-dimensional integral.