Recent results in infinite dimensional analysis and applications to Feynman integrals
The first part of this thesis proposes a general approach to infinite dimensional non-Gaussian analysis, including the important Poisson case. In particular distribution theory is developed. Different spaces of generalized functionals are constructed corresponding to the degree of singularity. Using appropriate integral transformations, generalized and test functionals are characterized in terms of holomorphy. For positive distributions a full description is given. Furthermore calculus is prepared: differential operators, Wick product and change of measure are discussed.
In the second part the Gaussian case is worked out in more detail. The White Noise measure is introduced and the connection to Brownian motion is established. On this basis, singular functionals of Brownian motion can be defined in the framework of White Noise distributions. Donsker's delta function is analyzed in detail. Furthermore operators on distribution spaces e.g. compositions with shifts and complex scaling are discussed.
The third part is due to the construction of Feynman integrals. The Feynman integrand is treated as a White Noise distribution. Its expectation is defined and yields the path integral. Starting from the free case and the harmonic oscillator, different classes of potentials are discussed to give a precise mathematical meaning to the Feynman integrand with interaction.
1. A generalization of the Khandekar Streit method is proposed. The resulting class of admissible potentials covers signed measures which can be very singular, but is restricted to the one dimensional case.
2. The Feynman integrand is also defined for potentials of the Albeverio Høegh-Krohn class, which consists of Fourier transforms of measure.
3. The third approach uses the technique of complex scaling. The so-called Doss class allows analytic potentials which obey some growth condition.
If the Feynman integrand is a well defined White Noise distribution, the calculus of differential operators can be applied. Using this, the functional form of the canonical commutation relation is derived. Finally Ehrenfest's theorem is proven.
Bielefeld University
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