Remarks on QFT in the Coordinate Space

TL;DR

This paper transforms Poincaré group generators for a free massive scalar field from momentum space to coordinate space, using the Newton-Wigner-Pryce operator, to better understand the structure of spatial coordinates in QFT and explore non-commutative extensions.

Contribution

It provides a detailed transformation of Poincaré generators into coordinate space based on the Newton-Wigner-Pryce operator, offering insights into the spatial structure in quantum field theory.

Findings

Explicit form of generators in coordinate space

Insights into the commutative nature of spatial coordinates in QFT

Foundation for studying non-commutative space in QFT

Abstract

Generators of the Poincar\'e group, for a free massive scalar field, are usually expressed in the momentum space. In this work we perform a transformation of these generators into the coordinate space. This (spatial)-position space is spanned by eigenvectors of the Newton-Wigner-Pryce operator. The motivation is a deeper understanding of the commutative spatial coordinate space in QFT, in order to investigate the non-commutative version thereof.