Pictures and equations of motion in Lagrangian quantum field theory

TL;DR

This paper explores various pictures of motion in Lagrangian quantum field theory, introduces a new momentum picture, and discusses their covariant formulations and equations of motion.

Contribution

It introduces a novel momentum picture in quantum field theory and derives equations of motion for different covariant pictures, expanding the conceptual framework.

Findings

Derived equations of motion for arbitrary Lagrangians.

Established covariant formulations of different pictures.

Proposed and analyzed the momentum picture as a 4D analogue of Schrödinger picture.

Abstract

The Heisenberg, interaction, and Schr\"odinger pictures of motion are considered in Lagrangian (canonical) quantum field theory. The equations of motion (for state vectors and field operators) are derived for arbitrary Lagrangians which are polynomial or convergent power series in field operators and their first derivatives. The general links between different time-dependent pictures of motion are derived. It is pointed that all of them admit covariant formulation, similar to the one of interaction picture. A new picture, called the momentum picture, is proposed. It is a 4-dimensional analogue of the Schr\"odinger picture of quantum mechanics as in it the state vectors are spacetime-dependent, while the field operators are constant relative to the spacetime. The equations of motion in momentum picture are derived and partially discussed. In particular, the ones for the field operators turn to be of algebraic type. The general idea of covariant pictures of motion is presented. The equations of motion in these pictures are derived.