Complex Ashtekar variables and reality conditions for Holst's action

Abstract

From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our result perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.

24 pages, 2 pictures

References in corpus (7)

• LQG vertex with finite Immirzi parameter
• Flipped spinfoam vertex and loop gravity
• Stepping out of Homogeneity in Loop Quantum Cosmology
• Lorentz covariance of loop quantum gravity
• Lifting SU(2) Spin Networks to Projected Spin Networks
• Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
• Surface terms, Asymptotics and Thermodynamics of the Holst Action

Cited by in corpus (27)

• The Spin Foam Approach to Quantum Gravity
• A Short Review of Loop Quantum Gravity
• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Integral quantizations with two basic examples
• New boundary variables for classical and quantum gravity on a null surface
• New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory
• New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis
• The twistorial structure of loop-gravity transition amplitudes
• Imaginary action, spinfoam asymptotics and the 'transplanckian' regime of loop quantum gravity
• Geometric temperature and entropy of quantum isolated horizons
• On the Relations between Gravity and BF Theories
• On the Implementation of the Canonical Quantum Simplicity Constraint
• Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity
• First order gravity on the light front
• Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector
• New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling
• Twisted geometries, twistors and conformal transformations
• Generating functional for gravitational null initial data
• Discrete gravity as a topological field theory with light-like curvature defects
• Hamiltonian spinfoam gravity
• The gravitational Hamiltonian, first order action, Poincaré charges and surface terms
• Area Law from Loop Quantum Gravity
• Corner symmetry and quantum geometry
• On the on-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries
• Loop quantum gravity with optimal control path integral, and application to black hole tunneling
• Revisiting EPRL: All Finite-Dimensional Solutions by Naimark's Fundamental Theorem
• Consistent and non-consistent deformations of gravitational theories