Real Ashtekar variables for Lorentzian signature space-times

1994/10/11 by J. Fernando Barbero G., J. Fernando Barbero · 60 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Computer science #Constraint (computer-aided design) #Cosmology and Gravitation Theories #Formalism (music) #General relativity #Geometry #Hamiltonian (control theory) #Hamiltonian constraint #Loop quantum gravity #Mathematical optimization #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Phase space #Physics #Quantum #Quantum gravity #Quantum mechanics #Theoretical physics #Wheeler–DeWitt equation #gr-qc

paper · pdf · doi:10.1103/physrevd.51.5507

published as Phys.Rev.D51:5507-5510,1995 · 10 pages, LATEX, Preprint CGPG-94/10-3

arxiv created 1994/10/11 · openalex publication_date 1995/05/15 · arxiv updated 2017/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

I suggest in this paper 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 a 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.

Citations

• Knot Theory and Quantum Gravity
• Loop space representation of quantum general relativity
• New Variables for Classical and Quantum Gravity
• New Hamiltonian formulation of general relativity

Cited by

• Knot theory and a physical state of quantum gravity
• Polymer quantization of the free scalar field and its classical limit
• On the resolution of the big bang singularity in isotropic loop quantum cosmology
• Hilbert space structure of covariant loop quantum gravity
• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
• Loop quantum gravity: an outside view
• Reality Conditions and Ashtekar Variables: a Different Perspective
• The Phoenix Project: master constraint programme for loop quantum gravity
• New variables for classical and quantum gravity in all dimensions: I. Hamiltonian analysis
• Generic degeneracy and entropy in loop quantum gravity
• BF gravity
• Towards Loop Quantum Supergravity (LQSG)
• Irreducibility of the Weyl Algebra in Loop Quantum Gravity
• Twistorial structure of loop-gravity transition amplitudes
• Effective constraints of loop quantum gravity
• Factor ordering and large-volume dynamics in quantum cosmology
• Tensorial Methods and Renormalization in Group Field Theories
• Dirac and Lagrangian reductions in the canonical approach to the first-order form of the Einstein–Hilbert action
• Shortcomings of the big bounce derivation in loop quantum cosmology
• Loop Quantum Gravity: An Inside View
• Inducing Barbero-Immirzi connections along SU(2) reductions of bundles on spacetime
• New scalar constraint operator for loop quantum gravity
• New length operator for loop quantum gravity
• Discussion on possible effects of the Barbero-Immirzi parameter at the TeV-scale particle physics
• A Discrete Evolution Formulation of the Spin Foam Models of BF theory and gravity
• Gravity quantized: Loop quantum gravity with a scalar field
• SO (4, C )-covariant Ashtekar-Barbero gravity and the Immirzi parameter
• Generalized Ashtekar variables for Palatini f(R) models
• Spontaneously broken Lorentz symmetry for Hamiltonian gravity
• Spontaneous Breaking of Lorentz Symmetry for Canonical Gravity
• Concise symplectic formulation for tetrad gravity
• Gravity as a constrained BF theory: Noether charges and Immirzi parameter
• Simplicity constraints: A 3D toy model for loop quantum gravity
• Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity
• Quantum gravity and spin-1/2 particle effective dynamics
• Degenerate limits of Scalar-Tensor gravity
• New variables for classical and quantum gravity in all dimensions: V. Isolated horizon boundary degrees of freedom
• Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics
• Black-hole entropy from complex Ashtekar variables
• Real and complex connections for canonical gravity
• Immirzi ambiguity in the kinematics of quantum general relativity
• Loop Quantum Cosmology
• Einstein-Cartan wormhole solutions
• Proof of Bekenstein–Mukhanov ansatz in loop quantum gravity
• First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian
• Spin foam models for quantum gravity
• Revisiting the solution of the second-class constraints of the Holst action
• Deformed general relativity and effective actions from loop quantum gravity
• Kinematic quantum states for the Teleparallel Equivalent of General Relativity
• The Reduced Phase Space of Palatini–Cartan–Holst Theory
• Space of quantum states built over metrics of fixed signature
• On the Discrete Version of the Schwarzschild Problem
• The length operator in Loop Quantum Gravity
• Commuting simplicity and closure constraints for 4D spin-foam models
• Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity
• Einstein-Cartan gravity with Holst term and fermions
• Lemaitre-Tolman-Bondi collapse from the perspective of loop quantum gravity
• Torsion at Different Scales: From Materials to the Universe
• Fusion basis for lattice gauge theory and loop quantum gravity
• Loop quantum gravity [wikipedia]