Covariance in self dual inhomogeneous models of effective quantum geometry: Spherical symmetry and Gowdy systems

Abstract

When applying the techniques of Loop Quantum Gravity (LQG) to symmetry-reduced gravitational systems, one first regularizes the scalar constraint using holonomy corrections, prior to quantization. In inhomogeneous system, where a residual spatial diffeomorphism symmetry survives, such modification of the gauge generator generating time reparametrization can potentially lead to deformations or anomalies in the modified algebra of first class constraints. When working with self-dual variables, it has already been shown that, for spherically symmetric geometry coupled to a scalar field, the holonomy-modified constraints do not generate any modifications to general covariance, as one faces in the real variables formula- tion, and can thus accommodate local degrees of freedom in such inhomogeneous models. In this paper, we extend this result to Gowdy cosmologies in the self-dual Ashtekar formulation. Furthermore, we show that the introduction of a $\bar{\mu }$-scheme in midisuperspace models, as is required in the `improved dynamics' of LQG, is possible in the self-dual formalism while being out of reach in the current effective models suing real-valued Ashtekar-Barbero variables. Our results indicate the advantages of using the self-dual variables to obtain a covariant loop regularization prior to quantization in inhomogeneous symmetry reduced polymer models, implementing additionally the crucial $\bar{\mu }$-scheme, and thus a consistent semiclassical limit.

24 pages, references added

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• (b,v)-type variables for black to white hole transitions in effective loop quantum gravity
• Perspectives on the dynamics in a loop quantum gravity effective description of black hole interiors
• Quantum evolution of black hole initial data sets: Foundations
• Bouncing compact objects. Part II: Effective theory of a pulsating Planck star
• Anomaly-free deformations of spherical general relativity coupled to matter
• Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology
• Asymptotically de Sitter Universe inside a Schwarzschild black hole
• Non-covariance of "covariant polymerization" in models of loop quantum gravity
• Periodic orbits and gravitational waveforms in quantum-corrected black hole spacetimes
• Holonomy and inverse-triad corrections in spherical models coupled to matter
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• Effective Field Theory of Loop Quantum Cosmology
• Classical polymerization of the Schwarzschild metric
• Loop effective model for Schwarzschild black hole interior: a modified $\bar{\mu }$ dynamics
• Deformed covariance in spherically symmetric vacuum models of loop quantum gravity: Consistency in Euclidean and self-dual gravity
• Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
• Loop Quantum Corrected Einstein Yang-Mills Black Holes
• Quasiclassical solutions for static quantum black holes
• Emergent modified gravity: Polarized Gowdy model on a torus
• Black-hole models in loop quantum gravity