Geometric flavours of quantum field theory on a Cauchy hypersurface. II: Methods of quantization and evolution (Q6596134)

In this two-part series of papers (for the first part see [\textit{J. L. Alonso} et al., J. Geom. Phys. 203, Article ID 105264, 25 p. (2024; Zbl 07904733)]) the authors propose a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved space-times 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 (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 the first paper of the series the necessary results of Gaussian analysis in infinite-dimensional spaces of distributions are collected and summarized.\N\NIn the second part the tools of Gaussian analysis in infinite-dimensional spaces collected in the first part of the series are used to describe the procedures of geometric quantization in the space of Cauchy data. These tools are also applied to describe the geometrization of the space of pure states of quantum field theory as a Kähler manifold. A modification of the Schrödinger equation via a quantum connection is discussed. The authors use the Bochner-Minlos theorem to define the measure of the prequantum Hilbert space and the Malliavin derivative as the central tool to define a connection on such space.\N\NThey discuss the (anti)holomorphic, Schrödinger and field-momentum pictures of the quantum field theory resulting according to the choice of polarization by using the methods of Gaussian analysis such as the Wiener-Ito decomposition theorem that provides the articles interpretation of the theory through the Segal isomorphism with the bosonic Fock space. It is shown how the Segal-Bargmann transform preserves the quantization between the Schrödinger and holomorphic (field-momentum and antiholomorphic) prescriptions.