A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry

TL;DR

This paper compares Laplacian and Dirac operators in Riemannian and antisymplectic geometries, highlighting their similarities and differences, and explores implications for quantum Hamiltonians in curved spaces.

Contribution

It introduces an antisymplectic Dirac operator and gamma matrices, establishing parallels with Riemannian spin geometry and analyzing curvature terms in quantum Hamiltonians.

Findings

Antisymplectic Dirac operator and gamma matrices introduced

Similarities between Riemannian and antisymplectic Laplacians analyzed

Connections to quantum Hamiltonian curvature terms discussed

Abstract

We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth-order term proportional to the Levi-Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order \Delta operator in antisymplectic geometry, which in general has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.