By S. Albeverio, P. Combe, R. Hscogh-Krohn, G. Rideau, M. Sirugue-Collin, M. Sirugue, R. Stora

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Extra info for Feynman Path Integrals: Proceedings of the International Colloquium Held in Marseille, May 1978

Example text

And defining then~for f Po2 - A m(P)2 Y(P) , p2+l P = (Po' ~)' One v e r i f i e s where Po R(B)c D Y l ' Y2 E D, Ac (yi,Y2) = P f dP° ~l (p) (Po 2 - m(~)2)-Z ~2 (p) ( p 2 + l ) 2 dp RS i. ^YZ (p) ~ (PJ - m(p)2) + f ~s+l with P the principal value d i s t r i b u t i o n measurable function, we see t h a t ( [ Z ] ) ( H , Hence the normalized Entegral defined and we have for all c In p a r t i c u l a r , IA C e wi th i n 2 j ,k g (y) d y . 2) and c(~) an a r b i t r a r y non negative D, B, A ) is a Fresnel fourtuple.

19] [14] , [2 0 , []~ U n f o r t u n a t e l y , f o r a p p l i c a t i o n s to quantum mechanical problems, the functions to be i n t e g r a t e d being of the form f(y) = exp(W(y)), not as powerful as might be expected from the f i n i t e In f a c t our space F(H) for W is d i f f i c u l t to be extended preserving the discussion of the asymptotic expansion in powers of is replaced by -~ Iyl 2 Tr(A~

Moreover ~AR ~ (T,~) converges pointwise to AR (~' ~) outside the cone [~I = ITI. 13) with d e n s i t y v i n F(R) and with i n i t i a l condition ( i . e. f AR E F (s'c~s)) s [#2_ (vy)2_m2~]d7 dr -~ R e - i ~o ~ v(y~(T,7)+~n(~,7))dTdT -tA ~(o,x)= o ko (y ( - t , o ) + @n ( - t , o ) ) dy. 13) and is a bounded operator on F(S'(]RS)), uniformly bounded in ~ and in fact satisfying t IAI Ilvll II U ~ ' ~ mll o ~ e o II~II o , where II H is the norm on the Fresnel space of Fourier transforms of measures O (given by the total variation of the correspondent measure).

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