Wick Rotation and Fermion Doubling in Noncommutative Geometry

TL;DR

This paper addresses the Euclidean nature and fermion doubling in noncommutative geometry models of the Standard Model, proposing a Wick rotation method to eliminate extra degrees of freedom and recover a Lorentzian theory.

Contribution

It provides a precise prescription for Wick rotation in noncommutative geometry, resolving fermion doubling issues and connecting Euclidean and Lorentzian formulations.

Findings

Wick rotation can eliminate extra fermionic degrees of freedom.

The approach correctly reproduces the Lorentzian Fock space.

Differences between Euclidean and Lorentzian cases are clarified.

Abstract

In this paper, we discuss two features of the noncommmutative geometry and spectral action approach to the Standard Model: the fact that the model is inherently Euclidean, and that it requires a quadrupling of the fermionic degrees of freedom. We show how the two issues are intimately related. We give a precise prescription for the Wick rotation from the Euclidean theory to the Lorentzian one, eliminating the extra degrees of freedom. This requires not only projecting out mirror fermions, as has been done so far, and which leads to the correct Pfaffian, but also the elimination of the remaining extra degrees of freedom. The remaining doubling has to be removed in order to recover the correct Fock space of the physical (Lorentzian) theory. In order to get a Spin(1,3) invariant Lorentzian theory from a Spin(4) invariant Euclidean theory such an elimination must be performed after the Wick rotation. Differences between the Euclidean and Lorentzian case are described in detail, in a pedagogical way.