By Sergio Albeverio
Feynman direction integrals integrals, advised heuristically through Feynman within the 40s, became the foundation of a lot of latest physics, from non relativistic quantum mechanics to quantum fields, together with gauge fields, gravitation, cosmology. lately principles in response to Feynman direction integrals have additionally performed a major position in components of arithmetic like low dimensional topology and differential geometry, algebraic geometry, endless dimensional research and geometry, and quantity conception.
The 2nd variation of LNM 523 is in response to the 2 first authors' mathematical procedure of this thought provided in its 1st variation in 1976. to keep up the various advancements that have happened on account that then, a whole new bankruptcy concerning the present leading edge of study has been added. Except for this new bankruptcy, the elemental fabric and presentation of the 1st version used to be mantained, a couple of misprints were corrected. on the finish of every bankruptcy the reader also will locate notes with additional bibliographical information.
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Additional info for Mathematical Theory of Feynman Path Integrals
12) The integral on the right hand side is well defined since Im A(x,x) ~ 0 ~ 89 Je call and a(x,x) f(x)dx is continuous on D . We shall also the integral normalized with respect to A 89 of the function e f(x) , and also use the notation ~A(f) for this integral. We have now the following proposition. 1 The space of Fresnel integrable functions function-algebra in the norm ~A(f) in B -I y ~(D*) ~ l % A ( f ) I ~ Hfllo and normalized such that The condition and is a Banach is a bounded continuous linear functional on such that that llfllo = II~ll and ~(D*) A(x,By) = (x,y) for is well defined and in the range of D , A(x,y) B .
37) ~)(x) = ~ Wt(x,B ) e i~x dr(8) , where ~(x) = ~ e ix~ dr(9 ) . 37) converges strongly. 39) By the physical interpretation of the wave operators, as a function of x is the wave function at time zero of the quantum mechanical particle with asymptotic momentum t ~-co. If m ~ I W+(x,8) and ~ ~ I ~ as one finds easely, by following the previous calculation, that 0 i -~jtV(Y(T)+ 9e and we recall that a particle of momentum velocity m~ . e dy . 41). 41) in the case where such that the perturbation perturbations H = Ho+ V bation H = Ho+V is gentle.
E1 O This then proves the proposition. 4 Let E be a real separable Banach space with a bounded symmetric non-degenerate bilinear form be a splitting of such that E