Wigner transformation in curved space-time and the curvature correction of the Vlasov equation for semiclassical gravitating systems

Journal Article · Tue Oct 15 04:00:00 EDT 1985 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.32.1871· OSTI ID:5213412

Winter, J

A covariant generalization of the Wigner transformation of quantum equations is proposed for gravitating many-particle systems, which modifies the Einstein-Liouville equations for the coupled gravity-matter problem by inclusion of quantum effects of the matter moving in its self-consistent classical gravitational field, in order to extend their realm of validity to higher particle densities. The corrections of the Vlasov equation (Liouville equation in one-particle phase space) are exhibited as combined effects of quantum mechanics and the curvature of space-time arranged in a semiclassical expansion in powers of h/sup 2/, the first-order term of which is explicitly calculated. It is linear in the Riemann tensor and in its gradient; the Riemann tensor occurs in a similar position as the tensor of the Yang-Mills field strength in a corresponding Vlasov equation for systems with local gauge invariance in the purely classical limit. The performance of the Wigner transformation is based on expressing the equation of motion for the two-point function of the Klein-Gordon field, in particular the Beltrami operator, in terms of a midpoint and a distance vector covariantly defined for the two points. This implies the calculation of deviations of the geodesic between these points, the standard concept of which has to be refined to include infinitesimal variations of the second order. A differential equation for the second-order deviation is established.

• Research Organization: Sektion Physik der Universitat Muenchen, Theresienstrasse 37 D-8000 Muenchen 2, West Germany
• OSTI ID: 5213412
• Journal Information: Phys. Rev. D; (United States), Journal Name: Phys. Rev. D; (United States) Vol. 32:8; ISSN PRVDA
• Country of Publication: United States
• Language: English

Related Subjects

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657003 -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BOLTZMANN-VLASOV EQUATION
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
GAUGE INVARIANCE
GRAVITATIONAL FIELDS
INVARIANCE PRINCIPLES
KLEIN-GORDON EQUATION
MANY-BODY PROBLEM
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PHASE SPACE
QUANTUM MECHANICS
RIEMANN SPACE
SEMICLASSICAL APPROXIMATION
SPACE
SPACE-TIME
WAVE EQUATIONS
WIGNER THEORY
YANG-MILLS THEORY