The noncommutative geometry of Zitterbewegung

TL;DR

This paper uses noncommutative geometry to model Dirac fermions in curved spacetime, revealing how Zitterbewegung arises from the causal structure and remains unaffected by electromagnetic fields, with implications for quantum simulations.

Contribution

It introduces a novel noncommutative geometric model of Dirac fermions that captures Zitterbewegung and explores its invariance and potential experimental tests.

Findings

Zitterbewegung frequency corresponds to maximum internal change speed.

Internal state remains unchanged under external electromagnetic fields.

Proposes quantum simulation experiments to test the model.

Abstract

Based on the mathematics of noncommutative geometry, we model a 'classical' Dirac fermion propagating in a curved spacetime. We demonstrate that the inherent causal structure of the model encodes the possibility of Zitterbewegung - the 'trembling motion' of the fermion. We recover the well-known frequency of Zitterbewegung as the highest possible speed of change in the fermion's 'internal space'. Furthermore, we show that the latter does not change in the presence of an external electromagnetic field and derive its explicit analogue when the mass parameter is promoted to a Higgs-like field. We discuss a table-top experiment in the domain of quantum simulation to test the predictions of the model and outline the consequences of our model for quantum gauge theories.