Generally covariant quantization and the Dirac field

TL;DR

This paper develops a covariant Hamiltonian framework for quantizing the Dirac field in curved spacetime, revealing unique features like non-coincidence of momentum and translation generators due to constraints.

Contribution

It introduces a covariant canonical quantization method for the Dirac field in curved spacetime and analyzes the impact of constraints on momentum and translation generators.

Findings

Field momentum differs from translation generators in curved spacetime.

Surface term modification affects momentum transfer between Dirac field and gravity.

The theory can be constraint-free but loses manifest hermiticity.

Abstract

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.