Construct and study analytically suitable quantum-mechanical targets

Construct and study smooth, descent-compatible geometric target categories for quantum-mechanical functorial field theories, including suitable versions of the category of Hilbert spaces that accommodate invertible time evolution and the required dualizability properties.

Background

The paper identifies analytic and categorical obstacles to using Hilbert spaces as targets for one-dimensional geometric functorial field theories. For a positive unbounded Hamiltonian, forward Euclidean evolution is bounded while inverse evolution is generally unbounded; rigged Hilbert spaces are proposed as one possible remedy. The paper also discusses free rigid constructions for addressing dualizability, but emphasizes that a satisfactory target must additionally possess appropriate smoothness and descent properties. The construction and subsequent study of such targets are left unresolved.