Geometric flavors of quantum field theory on a Cauchy hypersurface. I: Gaussian analysis and other mathematical aspects (Q6596133)

In this two-part series of papers (for the second part see [\textit{J. L. Alonso} et al., J. Geom. Phys. 203, Article ID 105265, 41 p. (2024; Zbl 07904734)], see also [\textit{J. L. Alonso} et al., Classical Quantum Gravity 41, No. 10, Article ID 105004, 48 p. (2024; Zbl 1544.83041)]), the authors propose a mathematically comprehensive framework for the Hamiltonian pictures of quantum field theory in curved spacetimes in order to study the kinematics and the dynamics from the point of differential geometry in infinite dimensions. The authors' construction is based upon the results of Gaussian analysis and Hida-Malliavin calculus developed to study stochastic variational systems and applied with huge success to model financial systems (see, for example, [\textit{Y. Hu}, Analysis on Gaussian spaces. Hackensack, NJ: World Scientific (2017; Zbl 1386.60005)]). They use the symplectic characterization of quantum mechanics proposed in the paper by \textit{T. W. B. Kibble} [Commun. Math. Phys. 65, 189--201 (1979; Zbl 0412.58006)]. In that paper a Poisson structure on the space of pure states of the Schrödinger-Hamiltonian description of quantum field theory is described.\N\NIn this first paper of the series the necessary results of Gaussian analysis in infinite-dimensional spaces of distributions are collected and summarized. These spaces serve as the basis for the description of the Schrödinger and holomorphic pictures, over arbitrary Cauchy hypersurfaces, using tools of Hida-Malliavin calculus in the second paper of the series.