The Hamiltonians of Linear Quantum Fields: II. Classically Positive Hamiltonians

TL;DR

This paper demonstrates that strictly positive classical Hamiltonians in linear bose field theories can be transformed to orthogonal form, revealing a deep link between boundedness-below properties and quantum implementability, with implications for quantum inequalities.

Contribution

It establishes a canonical transformation for positive classical Hamiltonians and explores their quantum implications, including boundedness and quantum inequalities in curved spacetime.

Findings

Positive classical Hamiltonians can be transformed to orthogonal form.

Boundedness-below of Hamiltonians relates to self-adjoint quantum implementability.

Quantum inequalities impose lower bounds on energy density in curved spacetime.

Abstract

For linear bose field theories, I show that if a classical Hamiltonian function is strictly positive, then there is a canonical transformation making the evolution orthogonal. This structure theorem is used to analyze the corresponding quantum theories. It is shown that there is an intimate connection between boundedness-below and self-adjoint implementability. Finally, it is shown that there is a broad class of "quantum inequalities:" any timelike component of the four-momentum density operator, averaged over a compact region in curved space-time, must be bounded below.