An $xp$ model on $AdS_2$ spacetime

TL;DR

This paper formulates an $xp$ model on AdS$_2$ spacetime, revealing a spectrum akin to a massive Dirac fermion and connecting it to conformal quantum mechanics through isometry generators.

Contribution

It introduces an $xp$ model on AdS$_2$, showing its spectrum matches that of a massive Dirac fermion and establishing a link to conformal quantum mechanics.

Findings

Spectrum of the Hamiltonian has positive and negative eigenvalues with harmonic oscillator form.

$xp$ models are equivalent to massive Dirac fermions on certain spacetimes.

Constructed generators of SO(2,1) isometry group and discussed conformal quantum mechanics relation.

Abstract

In this paper we formulate the $xp$ model on the AdS$_2$ spacetime. We find that the spectrum of the Hamiltonian has positive and negative eigenvalues, whose absolute values are given by a harmonic oscillator spectrum, which in turn coincides with that of a massive Dirac fermion in AdS$_2$. We extend this result to generic $xp$ models which are shown to be equivalent to a massive Dirac fermion on spacetimes whose metric depend of the $xp$ Hamiltonian. Finally, we construct the generators of the isometry group SO(2,1) of the AdS$_2$ spacetime, and discuss the relation with conformal quantum mechanics.