# Covariant Canonical Quantization

## TL;DR

This paper introduces a covariant canonical quantization framework that extends the Hilbert space, unifies aspects of canonical and functional integral methods, and rederives key quantum field theory quantities.

## Contribution

It presents a novel covariant quantization formalism with a covariant number operator and symmetric vacua, bridging canonical and path integral approaches.

## Key findings

- Reconstruction of the LSZ formula via projection limit
- Introduction of a covariant number operator and symmetric vacua
- Vacuum energy divergences arise when splitting spacetime

## Abstract

A formulation of Covariant Canonical Quantization is discussed, which works on an extended Hilbert space and reduces to conventional canonical quantization when constraining to the solution of the field equation a priori. From the formal point of view it may be seen as a formalism between the canonical operator and the functional integral approach. A covariant number operator and two symmetric vacua are constructed. By that means, certain well-known quantities like the LSZ formula are rederived via a projection limit. The time-ordering operator can be replaced by taking into account the mirrored vacuum as well. Then the quantum field theoretical divergences like the vacuum energy arise a posteriori when a spacetime split is performed. The role of the vacuum energy in different contexts is then discussed in general.