Formulation Of Canonical Gravity Research Articles

Gauge symmetries in Ashtekar's formulation of general relativity

It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekar's formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einstein's equations and are thus intimately related to the so-called “problem of time” in an eventual quantum theory of gravity.

The Ashtekar Formulation for CanonicalN= 2 Supergravity

We extend the Ashtekar formulation of canonical gravity to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt a chiral formulation, left- and right-handed supersym­ metries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersym­ metries are realized by constrained parameters.

Nonminimally coupled scalar field and Ashtekar variables

The non-minimal coupling of a scalar field is considered in the framework of Ashtekar's new variables formulation of gravity. A first order action functional for this system is derived in which the field variables are a tetrad field, and an SL(2,C) connection, together with the scalar field. The tetrad field and the SL(2,C) connection are related to the Ashtekar variables for the vacuum case by a conformal transformation. A canonical analysis shows that for this coupling the equations of Ashtekar's formulation of canonical gravity are non-polynomial in the scalar field. (to be published in Phys. Rev. D)

Self-dual connections, torsion and Ashtekar’s variables

Two identities are derived in terms of an arbitrary self-dual connection and the associated spin connection which show that torsion contributes to the simple complex-valued structure of the Hamiltonian and vector constraints in Ashtekar’s formulation of canonical gravity. A comparison is established between the Hamiltonian formulations of the Einstein–Cartan–Sciama–Kibble (ECSK) theory and the one in terms of Ashtekar’s variables, both in the matter-free case, in order to emphasize that in the latter the torsion tensor restricted to three-dimensional spacelike hypersurface does not vanish prior to the imposition of the reality conditions.

Symmetry properties of Ashtekar’s formulation of canonical gravity

We provide a simple derivation of Ashtekar’s formulation of canonical gravity and study the symmetry properties of the new variables. TheSO(3) group structure of the Hamiltonian formulation is found to be related to the local Lorentz symmetry of the Einstein-Cartan-Sciama-Kibble-Trautman theory. In view of these symmetry properties we suggest a definition for the connection operator in the quantum theory.

THE ASHTEKAR FORMULATION FOR CANONICAL N=2 SUPERGRAVITY

Ashtekar’s formulation of canonical gravity is extended to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt the chiral formulation, left- and right-handed supersymmetries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersymmetries are realized by constrained parameters. By means of the canonical formalism, simple polynomial constraints are derived and we can show that the Poisson brackets of supersymmetry currents are actually closed. This guarantees the consistency of our supersymmetry transformations, which is unclear at the level of the algebra on the fields.

The GN→∞limit of quantum gravity

Dirac quantisation of the GN to infinity limit of general relativity is carried out. This is done in the framework of Ashtekar's new formulation of canonical gravity (1986). Using the new variables, a complete set of commuting operators are constructed for this limit. These operators are used to construct a large set of quantum states that are annihilated by the constraints (in this limit). It is shown that these states are also annihilated by the new form of the full Hamiltonian constraint.