A comparative study of Laplacians and Schroedinger- Lichnerowicz-Weitzenboeck identities in Riemannian and antisymplectic geometry

Journal Article · Wed Jul 15 00:00:00 EDT 2009 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3152575· OSTI ID:21294216

Batalin, Igor A ^[1]; Bering, Klaus ^[1]

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1. Niels Bohr Institute, Niels Bohr International Academy, Blegdamsvej 17, DK 2100 Copenhagen (Denmark)

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 torsion-free 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.

• OSTI ID: 21294216
• Journal Information: Journal of Mathematical Physics, Vol. 50, Issue 7; Other Information: DOI: 10.1063/1.3152575; (c) 2009 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
DIRAC EQUATION
DIRAC OPERATORS
GEOMETRY
HAMILTONIANS
LAPLACIAN
MATRICES
SCALARS
SCHROEDINGER EQUATION
SPIN
TORSION