The Hamiltonians of Linear Quantum Fields: I. Existence Theory

TL;DR

This paper investigates the conditions under which classical Hamiltonian vector fields in linear quantum field theories have self-adjoint quantum operators, revealing geometric constraints in curved spacetime.

Contribution

It characterizes classical Hamiltonian vector fields with self-adjoint quantum operators and establishes geometric conditions for their existence in curved spacetime.

Findings

Self-adjoint Hamiltonians exist only if the second fundamental form vanishes.

Provides a characterization of Hamiltonian vector fields with quantum self-adjointness.

Links geometric properties of spacetime to quantum evolution operators.

Abstract

For linear scalar field theories, I characterize those classical Hamiltonian vector fields which have self-adjoint operators as their quantum counterparts. As an application, it is shown that for a scalar field in curved space-time (in a Hadamard representation), a self-adjoint Hamiltonian for evolution along the unit timelike normal to a Cauchy surface exists only if the second fundamental form of the surface vanishes identically.