[Paper Review] A covariant polymerized scalar field in loop quantum gravity

Florencia Benítez, Rodolfo Gambini+1 more|arXiv (Cornell University)|Feb 18, 2021

Noncommutative and Quantum Gravity Theories1 references4 citations

TL;DR

This paper introduces a covariant polymerization scheme for scalar fields in loop quantum gravity by formulating it as a non-bijective canonical transformation, preserving the theory's covariance and constraint algebra. The method ensures quantum corrections are non-trivial and compatible with loop quantum gravity's Hilbert space, with effective dynamics in spherically symmetric gravity showing agreement with standard polymerized models, including singularity avoidance via a bounce at maximal extrinsic curvature.

ABSTRACT

We propose a new polymerization scheme for scalar fields coupled to gravity. It has the advantage of being a (non-bijective) canonical transformation of the fields and therefore ensures the covariance of the theory. We study it in detail in spherically symmetric situations and compare to other approaches.

Motivation & Objective

• To address the lack of covariance in standard polymerization schemes used in loop quantum gravity, which are slicing-dependent and break manifest covariance.
• To develop a polymerization procedure that is a canonical transformation—preserving the constraint algebra and spacetime covariance—while still introducing non-trivial quantum corrections.
• To apply the new scheme to spherically symmetric gravity coupled to a scalar field, testing its consistency with known results in effective dynamics and singularity resolution.
• To explore whether this approach can be generalized to the full theory, offering a more covariant framework for semiclassical loop quantum gravity.

Proposed method

• The polymerization is defined as a non-bijective canonical transformation on phase space variables, mapping classical variables to their sine-transformed forms, e.g., $ x \to \sin(kx)/k $, with $ k $ as the polymerization parameter.
• The transformation preserves the Poisson bracket algebra and constraint structure, ensuring the theory remains generally covariant despite the non-invertibility.
• The method is applied to spherically symmetric gravity with a scalar field, using a redefined Hamiltonian and diffeomorphism constraint with rescaled lapse and shift to achieve an Abelian constraint algebra.
• Effective dynamics are derived from the polymerized Poisson brackets, showing agreement with standard models in the case of gravitational collapse.
• The approach maintains compatibility with the Ashtekar-Lewandowski measure and allows for a well-defined Hilbert space representation.
• The polymerization parameter $ k $ is treated as constant in this work, though the possibility of a dynamical $ k $ depending on phase space variables is noted for future work.

Experimental results

Research questions

• RQ1 Can a polymerization scheme for scalar fields in loop quantum gravity be made covariant by formulating it as a canonical transformation?
• RQ2 Does the proposed non-bijective canonical transformation preserve the constraint algebra and spacetime covariance while still introducing non-trivial quantum corrections?
• RQ3 How do the effective dynamics of this covariant polymerization compare to standard polymerized models in spherically symmetric gravitational collapse?
• RQ4 Is singularity resolution still achieved in this new framework, and does it rely on the same physical mechanisms as in previous models?
• RQ5 Can this approach be extended to the full theory of loop quantum gravity, or is it limited to symmetric reductions?

Key findings

• The proposed polymerization is a non-bijective canonical transformation that preserves the constraint algebra and ensures general covariance, resolving a key limitation of prior approaches.
• Effective dynamics in spherically symmetric gravity with a scalar field show no significant qualitative differences from standard polymerized models, including in the Choptuik collapse scenario.
• The theory exhibits a bounce at maximal extrinsic curvature, preventing the formation of a classical singularity, consistent with previous loop quantum gravity results.
• The metric components transform correctly under diffeomorphisms, as confirmed by the tensorial behavior derived from the polymerized constraint algebra.
• The method allows for a consistent Hilbert space representation using the Ashtekar-Lewandowski measure, supporting its viability in quantum theory.
• The framework is compatible with the same physical mechanisms for singularity resolution as in $ \bar{\mu} $-scheme models, such as the complexification of Dirac observables and removal of spin network nodes at the bounce.