A Hamiltonian-like formalism that treats one spatial coordinate -- rather than time -- differently

TL;DR

This paper proposes a novel Hamiltonian-like formalism in quantum field theory that treats one spatial coordinate differently from the others, aiming to make Lorentz invariance more explicit.

Contribution

It introduces a new formalism that treats one spatial coordinate differently, bridging the gap between Hamiltonian and Lagrangian approaches regarding Lorentz invariance.

Findings

Formalism explicitly shows Lorentz invariance

Treats one spatial coordinate differently from others

Provides a new perspective on Hamiltonian formalism

Abstract

The Hamiltonian and Lagrangian formalisms of Qunatum Field Theory (QFT) are equivalent. But while Lorentz invariance can be clearly seen in the Lagrangian formalism, it is not so explicit in the Hamiltonian one. This is because time is treated a little differently from the spatial coordinates in the Hamiltonian formalism. In this paper, I explore whether it is possible to devise another formalism that is just like the Hamiltonian one (with operators and state vectors) but which treats time on an equal footing with two of the spatial coordinates, while the third one is treated differently, the way time is in the usual Hamiltonian formalism.