Scalar field on non-integer dimensional spaces

TL;DR

This paper explores deformed spectral triples on tori with non-integer dimensions, analyzing the differential algebra, scalar field actions, and one-loop quantum effects, revealing new structures without loss of covariance.

Contribution

It introduces non-integer dimensional spectral triples with finite quantum corrections and studies their differential algebra and scalar field dynamics.

Findings

No junk forms for non-zero deformation parameter

Finite one-loop quantum contributions

Presence of non-trivial extra structure in scalar field action

Abstract

Deformations of the canonical spectral triples over the n-dimensional torus are considered. These deformations have a discrete dimension spectrum consisting of non-integer values less than n. The differential algebra corresponding to these spectral triples is studied. No junk forms appear for non-vanishing deformation parameter. The action of a scalar field in these spaces is considered, leading to non-trivial extra structure compared to the integer dimensional cases, which does not involve a loss of covariance. One-loop contributions are computed leading to finite results for non-vanishing deformation.