Zitterbewegung and symmetry switching in Klein's four-group

TL;DR

This paper extends the concept of Zitterbewegung to systems where symmetry changes induce trembling motion, demonstrated through a quantum chaos model with symmetry switching in the Klein four-group.

Contribution

It shows that symmetry switching in a quantum chaos model can cause Zitterbewegung-like trembling motion, linking symmetry, energy spectrum, and quantum fluctuations.

Findings

Trembling motion occurs due to symmetry subgroup switching.

Trembling is enhanced in highly excited states.

Link established between trembling and uncertainty relations.

Abstract

Zitterbewegung is the exotic phenomenon associated either with the relativistic electron-positron rapid oscillation or to the electron-hole transitions in the narrow gap semiconductors. In the present work, we enlarge concept of Zitterbewegung and show that the trembling motion may occur due to the dramatic changes in the symmetry of the system. In particular, we exploit a paradigmatic model of quantum chaos, quantum mathematical pendulum (universal Hamiltonian). The symmetry group of this system is the Klein's four-group that possess three invariant subgroups. The energy spectrum of the system parametrically depends on the height of the potential barrier, and contains degenerate and non-degenerate areas, corresponding to the different symmetry subgroups. Change in the height of the potential barrier switches the symmetry subgroup and leads to the trembling motion. We analyzed mean square fluctuations of the velocity operator and observed that trembling enhances for the highly excited states. We observed the link between the phenomena of trembling motion and uncertainty relations of noncommutative operators of the system.