External Fields as Intrinsic Geometry

TL;DR

This paper explores how external fields can be integrated into the intrinsic geometry of space-time, including noncommutative geometries, and discusses their relation to bosonic field theories and gauge theories.

Contribution

It introduces a framework for absorbing external fields into noncommutative geometry, extending previous work on re-expressing bosonic theories as gauge theories in noncommutative spaces.

Findings

External fields can be incorporated into noncommutative geometry.

Some bosonic theories can be reformulated as abelian gauge theories in noncommutative spaces.

Noncommutative structure contains extra modes governed by a single abelian action.

Abstract

There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one. We shall also recall some previous incidences of the same phenomena involving bosonic field theories. It is known that some such theories on the commutative geometry of space-time can be re-expressed as abelian-gauge theory in an appropriate noncommutative geometry. The noncommutative structure can be considered as containing extra modes all of whose dynamics are given by the one abelian action.