Spinor matter fields in SL(2,C) gauge theories of gravity: Lagrangian and Hamiltonian approaches

Journal Article · Sat Jun 15 00:00:00 EDT 1985 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.31.3104· OSTI ID:5547232

Antonowicz, M; Szczyrba, W

We consider the SL(2,C)-covariant Lagrangian formulation of gravitational theories with the presence of spinor matter fields. The invariance properties of such theories give rise to the conservation laws (the contracted Bianchi identities) having in the presence of matter fields a more complicated form than those known in the literature previously. A general SL(2,C) gauge theory of gravity is cast into an SL(2,C)-covariant Hamiltonian formulation. Breaking the SL(2,C) symmetry of the system to the SU(2) symmetry, by introducing a spacelike slicing of spacetime, we get an SU(2)-covariant Hamiltonian picture. The qualitative analysis of SL(2,C) gauge theories of gravity in the SU(2)-covariant formulation enables us to define the dynamical symplectic variables and the gauge variables of the theory under consideration as well as to divide the set of field equations into the dynamical equations and the constraints. In the SU(2)-covariant Hamiltonian formulation the primary constraints, which are generic for first-order matter Lagrangians (Dirac, Weyl, Fierz-Pauli), can be reduced. The effective matter symplectic variables are given by SU(2)-spinor-valued half-forms on three-dimensional slices of spacetime. The coupled Einstein-Cartan-Dirac (Weyl, Fierz-Pauli) system is analyzed from the (3+1) point of view. This analysis is complete; the field equations of the Einstein-Cartan-Dirac theory split into 18 gravitational dynamical equations, 8 dynamical Dirac equations, and 7 first-class constraints. The system has 4+8 = 12 independent degrees of freedom in the phase space.

• Research Organization: Institute of Theoretical Physics, Warsaw University, Hoa 69, 00-681 Warsaw, Poland
• OSTI ID: 5547232
• Journal Information: Phys. Rev. D; (United States), Vol. 31:12
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
GRAVITATIONAL FIELDS
GAUGE INVARIANCE
SL GROUPS
DIRAC EQUATION
FIELD EQUATIONS
HAMILTONIANS
LAGRANGIAN FUNCTION
PHASE SPACE
SPACE-TIME
SPINORS
SU-2 GROUPS
SYMMETRY BREAKING
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
INVARIANCE PRINCIPLES
LIE GROUPS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
SPACE
SU GROUPS
SYMMETRY GROUPS
WAVE EQUATIONS
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