A complex scaling approach to sequential Feynman integrals

Let (H,B,μ) be an abstract Wiener space. Let P be the set of all finite-dimensional orthogonal projections in H and for P∈P denote by Γ(P) the second quantization of P. It is shown that for ϕ∈⋂p>1Lp(B,μ) and z∈C+={z∈C: Re z>0}, the z−1/2-scaling σz−1/2Γ(P)ϕ of Γ(P)ϕ is well defined as an element of a distribution space over (H,B,μ). By means of this scaling, we define the sequential Feynman integral as limn→∞〈〈σzn−1/2Γ(Pn)ϕ,1〉〉 if the latter exists and has a common limit for all zn→−i, zn∈C+, Pn→I, Pn∈P. It turns out that the Fresnel integrals of Albeverio and Hoegh-Krohn coincide with this sequential Feynman integrals. The proof of a Cameron–Martin-type formula for Feynman integrals is much simplified and transparent.