Derivation of the Wheeler-DeWitt equation from a path integral for minisuperspace models

Journal Article · Sat Oct 15 00:00:00 EDT 1988 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.38.2468· OSTI ID:6858039

Halliwell, J J

We explore the relationship between path-integral and Dirac quantization for a simple class of reparametrization-invariant theories. The main object is to study minisuperspace models in quantum cosmology: models for quantum gravity in which one restricts attention to a finite number of degrees of freedom. Our starting point for the construction of the (Lorentzian) path integral is the very general and powerful method introduced by Batalin, Fradkin, and Vilkovisky. Particular attention is paid to the measure in the large, i.e., to the range of integration of the Lagrange multiplier. We show how to derive the Wheeler-DeWitt equation from our path-integral expression. The relationship between the choice of measure in the path integral and the operator ordering in the Wheeler-DeWitt equation is thus determined. The operator-ordering ambiguity in the Wheeler-DeWitt equation is completely fixed by demanding invariance under field redefinitions of both the three-metric and the lapse function. Our results are applied to two simple examples: the nonrelativistic point particle in parametrized form and the relativistic point particle. We also consider a simple minisuperspace example and discuss a difficulty that arises: namely, the problem of incorporating the fact that deth/sub i//sub j/>0 into the quantization procedure.

• Research Organization: Institute for Theoretical Physics, University of California, Santa Barbara, California 93106
• OSTI ID: 6858039
• Journal Information: Phys. Rev. D; (United States), Journal Name: Phys. Rev. D; (United States) Vol. 38:8; ISSN PRVDA
• Country of Publication: United States
• Language: English

Related Subjects

645400* -- High Energy Physics-- Field Theory
657003 -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
COSMOLOGICAL MODELS
DIFFERENTIAL EQUATIONS
DIRAC EQUATION
EQUATIONS
FEYNMAN PATH INTEGRAL
FIELD THEORIES
GENERAL RELATIVITY THEORY
HAMILTONIANS
INTEGRALS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
METRICS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTIZATION
QUANTUM FIELD THEORY
QUANTUM GRAVITY
QUANTUM OPERATORS
WAVE EQUATIONS