The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator

TL;DR

This paper explores the complex eigenvalue structure of the 1+1 Dirac oscillator, revealing a Riemann surface that connects particle and antiparticle states, and explains the absence of negative energy states at n=0.

Contribution

It analytically characterizes the eigenvalue structure of the Dirac oscillator by extending the frequency into the complex plane, uncovering a Riemann surface linking particle and antiparticle states.

Findings

Identifies a twofold Riemann surface for eigenvalues.

Shows transition between positive and negative energy states via complex frequency.

Provides explanation for the absence of negative energy state at n=0.

Abstract

We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator, by analytically continuing its frequency on the complex plane. A twofold Riemann surface is found, connecting the two states of a pair of particle and antiparticle. One can, at least in principle, accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane, without changing the Hamiltonian after transition. This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.