Number operators for Riemannian manifolds

TL;DR

This paper explores the spectrum of number operators derived from the Dirac operator on Riemannian manifolds, linking spectral properties to harmonic functions and providing quantum analogs of classical geometric theorems.

Contribution

It establishes conditions under which the number operator has integer spectrum, connecting measure density, harmonic functions, and geometric splitting theorems.

Findings

Spectrum of the number operator contains nonnegative integers.

Existence of harmonic distance functions characterizes measure density conditions.

Provides examples of nonflat, nonproduct manifolds with these properties.

Abstract

The Dirac operator d+delta on the Hodge complex of a Riemannian manifold is regarded as an annihilation operator A. On a weighted space L_mu^2 Omega, [A,A*] acts as multiplication by a positive constant on excited states if and only if the logarithm of the measure density of mu satisfies a pair of equations. The equations are equivalent to the existence of a harmonic distance function on M. Under these conditions N=A*A has spectrum containing the nonnegative integers. Nonflat, nonproduct examples are given. The results are summarized as a quantum version of the Cheeger--Gromoll splitting theorem.