Manifestly covariant canonical quantization I: the free scalar field

TL;DR

This paper develops a covariant canonical quantization framework for the free scalar field by reformulating classical physics as a constrained Hamiltonian system in history phase space, incorporating observer trajectories as quantum fields.

Contribution

It introduces a manifestly covariant quantization method using history phase space and observer trajectories, extending traditional approaches with cohomological and BRST-inspired techniques.

Findings

Recovered standard quantum field results in classical observer approximation.

Applied formalism to harmonic oscillator and free scalar field.

Established a covariant quantization framework consistent with classical constraints.

Abstract

Classical physics is reformulated as a constrained Hamiltonian system in the history phase space. Dynamics, i.e. the Euler-Lagrange equations, play the role of first-class constraints. This allows us to apply standard methods from the theory of constrained Hamiltonian systems, e.g. Dirac brackets and cohomological methods. In analogy with BRST quantization, we quantize in the history phase space first and impose dynamics afterwards. To obtain a truly covariant formulation, all fields must be expanded in a Taylor series around the observer's trajectory, which acquires the status of a quantized physical field. The formalism is applied to the harmonic oscillator and to the free scalar field. Standard results are recovered, but only in the approximation that the observer's trajectory is treated as a classical curve.