Covariant path integrals on hyperbolic surfaces

Journal Article · Sat Nov 01 04:00:00 EST 1997 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.532158· OSTI ID:544814

Schaefer, J ^[1]

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1. Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651 (United States)

DeWitt{close_quote}s covariant formulation of path integration [B. De Witt, {open_quotes}Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,{close_quotes} Rev. Mod. Phys. {bold 29}, 377{endash}397 (1957)] has two practical advantages over the traditional methods of {open_quotes}lattice approximations;{close_quotes} there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded self-adjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature {minus}1. The Pauli{endash}DeWitt curvature correction term arises, as in DeWitt{close_quote}s work. Introducing a Fuchsian group {Gamma} of the first kind, and a continuous, bounded, {Gamma}-automorphic potential V, we obtain a Feynman{endash}Kac formula for the automorphic Schr{umlt o}dinger equation on the Riemann surface {Gamma}{backslash}H. We analyze the Wick rotation and prove the strong convergence of the so-called Feynman maps [K. D. Elworthy, {ital Path Integration on Manifolds, Mathematical Aspects of Superspace}, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47{endash}90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, {open_quotes}The path integral on the Poincare upper half plane and for Liouville quantum mechanics,{close_quotes} Phys. Lett. A {bold 123}, 319{endash}328 (1987). {copyright} {ital 1997 American Institute of Physics.}

• OSTI ID: 544814
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 11 Vol. 38; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
FEYNMAN PATH INTEGRAL
QUANTUM MECHANICS
RIEMANN SPACE
SCHROEDINGER EQUATION
SURFACES
SYMMETRY