Real Ashtekar Variables for Lorentzian Signature Space-times

Abstract

I suggest in this letter a new strategy to attack the problem of the reality conditions in the Ashtekar approach to classical and quantum general relativity. By writing a modified Hamiltonian constraint in the usual $SO(3)$ Yang-Mills phase space I show that it is possible to describe space-times with Lorentzian signature without the introduction of complex variables. All the features of the Ashtekar formalism related to the geometrical nature of the new variables are retained; in particular, it is still possible, in principle, to use the loop variables approach in the passage to the quantum theory. The key issue in the new formulation is how to deal with the more complicated Hamiltonian constraint that must be used in order to avoid the introduction of complex fields.

10 pages, LATEX, Preprint CGPG-94/10-3

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• Abelian Ashtekar formulation from the ADM action
• Spectrum of area in the Faddeev formulation of gravity
• Geometrodynamics and Lorentz symmetry
• Cosmology in Loop Quantum Gravity: symmetry reduction preserving gauge degrees of freedom
• Hamiltonian gravity in tetrad-connection variables
• Big-Bounce in Quantum f(R)-Cosmology: Polymer Dynamics with Internal Time
• Degenerate Limits of Scalar-Tensor Gravity
• Independent 4-tetrahedra connection representation of Regge calculus
• Holographic bound in covariant loop quantum gravity
• General structure of the solutions of the Hamiltonian constraints of gravity
• First order discrete Faddeev gravity at the strongly varying fields
• Spontaneous breaking of Lorentz symmetry for canonical gravity
• Connection dynamics of a gauge theory of gravity coupled with matter
• Attributing sense to some integrals in Regge calculus
• Spin Foams for the SO(4,C) BF theory and the SO(4,C) General Relativity
• Global Barbero-Immirzi Connections
• On the Hamiltonian formalism of the Tetrad-Gravity with fermions
• Simplicial Palatini action
• Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity
• Complete loop quantization of a dimension 1+2 Lorentzian gravity theory
• Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem
• Asymptotically flat boundary conditions for the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Boundary conditions for Ashtekar-Barbero variables in the context of asymptotically flat spacetimes which lead to supertranslations at spatial infinity
• Inflationary assessment of $F(\mathcal{R},\tilde{\mathcal{R}})$ Einstein-Cartan models
• Covariant quantization of the Einstein-Hilbert theory in first-order form
• Barbero-Immirzi parameter in Regge calculus
• Motion in Quantum Gravity
• On the Faddeev gravity on the piecewise flat manifold
• Gravitational quantum states as finite representations of the Lorentz group
• Denseness of Ashtekar-Lewandowski states and a generalized cut-off in loop quantum gravity
• The Space of Connections as the Arena for (Quantum) Gravity
• Generators of Local Lorentz Transformation in ADM-Vielbein Formalism of Gravitational Relativity
• A version of the connection representation of Regge action
• Discrete Faddeev action for the tetrad fields strongly varying along different coordinates
• Generalized BF state in quantum gravity
• Proof of Bekenstein-Mukhanov ansatz in loop quantum gravity