Semiclassical structure of trace formulas

Journal Article · Sat Dec 01 04:00:00 EST 1990 · Journal of Mathematical Physics (New York); (USA)

DOI:https://doi.org/10.1063/1.528949· OSTI ID:6146707

Littlejohn, R G ^[1]

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1. Theoretical Physics Institute, School of Physics and Astronomy, 116 Church Street S.E., Minneapolis, MN (USA)

Trace formulas provide the only general relations known connecting quantum mechanics with classical mechanics in the case that the classical motion is chaotic. In particular, they connect quantal objects such as the density of states with classical periodic orbits. In this paper, several trace formulas, including those of Gutzwiller, Balian and Bloch, Tabor, and Berry, are examined from a geometrical standpoint. New forms of the amplitude determinant in asymptotic theory are developed as tools for this examination. The meaning of caustics in these formulas is revealed in terms of intersections of Lagrangian manifolds in phase space. The periodic orbits themselves appear as caustics of an unstable kind, lying on the intersection of two Lagrangian manifolds in the appropriate phase space. New insight is obtained into the Weyl correspondence and the Wigner function, especially their caustic structures.

• DOE Contract Number: AC03-76SF00098
• OSTI ID: 6146707
• Journal Information: Journal of Mathematical Physics (New York); (USA), Journal Name: Journal of Mathematical Physics (New York); (USA) Vol. 31:12; ISSN JMAPA; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
BOUND STATE
CLASSICAL MECHANICS
DIFFERENTIAL EQUATIONS
ENERGY-LEVEL DENSITY
EQUATIONS
EQUATIONS OF MOTION
FIELD THEORIES
FUNCTIONS
LAGRANGIAN FUNCTION
MATHEMATICAL SPACE
MECHANICS
ORBITS
PARTIAL DIFFERENTIAL EQUATIONS
PHASE SPACE
QUANTUM MECHANICS
SEMICLASSICAL APPROXIMATION
SPACE
UNIFIED-FIELD THEORIES
WEYL UNIFIED THEORY
WIGNER COEFFICIENTS