Dimensional Reduction without Extra Continuous Dimensions

TL;DR

This paper introduces a new method for dimensional reduction in classical field theory using extended algebras inspired by noncommutative geometry, revealing natural origins for axion, dilaton, and Higgs fields.

Contribution

It presents a novel formalism for dimensional reduction employing extended differential forms and generalized connections inspired by noncommutative geometry.

Findings

Derived natural geometrical origins for axion, dilaton, and Higgs fields.

Applied formalism to gauge and gravitational theories.

Provided a new perspective on space-time with discrete extra dimensions.

Abstract

We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives and generalized connections associated with the "geometry" of space-times with discrete extra dimensions. We apply our formalism to theories of gauge- and gravitational fields and find natural geometrical origins for an axion- and a dilaton field, as well as a Higgs field.