On momentum operator in quantum field theory

TL;DR

This paper explores the relationship between two definitions of the momentum operator in quantum field theory, analyzing their differences and implications for the connection between quantum field theory and quantum mechanics.

Contribution

It provides a detailed comparison of the canonical energy-momentum tensor and translation operator definitions of momentum in quantum field theory, highlighting their distinct roles.

Findings

The two momentum operators are similar but generally different.

Each momentum operator has a unique role within the theory.

Speculations on the link between quantum field theory and quantum mechanics are discussed.

Abstract

The interrelations between the two definitions of momentum operator, via the canonical energy-momentum tensorial operator and as translation operator (on the operator space), are studied in quantum field theory. These definitions give rise to similar but, generally, different momentum operators, each of them having its own place in the theory. Some speculations on the relations between quantum field theory and quantum mechanics are presented.