Covariant Effective Model for Spherical LQG

• The paper demonstrates that holonomy and inverse-triad corrections modify classical Hamiltonian constraints while preserving diffeomorphism invariance in spherical LQG.
• It shows that canonical transformations relate different polymerization schemes, ensuring observables and the emergent metric remain invariant under gauge changes.
• The study finds that quantum-corrected black hole solutions emerge with regular, non-singular interiors, bridging black-hole and white-hole regions.

A covariant effective model for spherical loop quantum gravity (LQG) provides a formalism where quantum corrections—stemming from the nonperturbative quantization strategies used in LQG—modify the classical constraints and dynamics of general relativity, yet maintain a consistent implementation of general covariance and gauge symmetry in symmetry-reduced (spherically symmetric) settings. Such models aim to ensure that the quantum-corrected evolution remains equivalent to diffeomorphism-invariant dynamics, with observables and the emergent metric transforming appropriately under coordinate changes, possibly after appropriate "covariantization" of both the constraints and the reconstructed geometric fields.

1. Covariance in Spherically Symmetric Loop Quantum Gravity

In spherical symmetry, the classical constraint algebra of general relativity is represented by the Dirac algebra of the Hamiltonian and diffeomorphism constraints. Quantum corrections derived from loop quantization—especially holonomy and inverse-triad corrections—modify the explicit form of these constraints and potentially deform the closure relations (structure functions) in the algebra (Bojowald et al., 2011, Bojowald, 2012, Bojowald et al., 2015, Bojowald et al., 2019, Zhang et al., 14 Jul 2024).

To preserve general covariance at the effective level, two core conditions have been identified as both necessary and sufficient (Zhang et al., 14 Jul 2024):

• The effective Hamiltonian constraint must not depend on spatial derivatives of the connection (extrinsic curvature) variables.
• The structure function in the constraint algebra (governing the bracket of two Hamiltonian constraints) must transform as an inverse spatial metric under canonical transformations and coordinate/gauge changes.

In such covariant models, the phase-space variables, constraints, and emergent metric must transform so that the quantum-corrected space-time structure remains consistent with the mechanisms of general covariance, even in the presence of deformation functions (such as $\beta$ or $\mu$) arising from LQG effects.

2. Effective Hamiltonian Constraints and Deformed Algebras

The effective Hamiltonian constraint typically receives momentum-dependent, non-quadratic (often bounded, periodic) holonomy corrections (Bojowald et al., 2011, Bojowald et al., 2015, Zhang et al., 14 Jul 2024, Belfaqih et al., 26 Nov 2024). In the spherically symmetric sector, a generic (vacuum) form is

$H_{\text{eff}} = E^2 F(s_a),$

with $F$ a scalar function constructed from phase-space scalars $s_a$ (involving the metric triads $E^1,E^2$ and extrinsic curvatures $K_1,K_2$ together with their derivatives). Quantum effects are realized by replacing $$K_2 \rightarrow \frac{\sqrt{E^1}}{\zeta} \sin \left( \zeta K_2/\sqrt{E^1} \right)$$, with $\zeta$ the fundamental quantum parameter (related to the Planck length).

The constraint algebra closes as

$$\{ H_x[N^x], H[N] \} = H[N^x \partial_x N], \qquad \{ H[N_1], H[N_2] \} = H_x[S (N_1 \partial_x N_2 - N_2 \partial_x N_1) ],$$

with the structure function $S = \mu E^1/(E^2)^2$ incorporating quantum corrections through $\mu$. The covariance conditions uniquely determine the form of $\mu$ and the allowed structure of $F(s_a)$ such that the emergent space-time metric,

$$ds^2 = -N^2 dt^2 + \frac{(E^2)^2}{\mu E^1}(dx + N^x dt)^2 + E^1 d \Omega^2,$$

transforms correctly under diffeomorphisms generated by the modified constraint algebra (Zhang et al., 14 Jul 2024, Belfaqih et al., 26 Nov 2024).

3. Canonical Transformations, Phase-Space Covariance, and Quantization Ambiguities

Different "polymerization" schemes—such as those with constant versus scale-dependent holonomy scales (the so-called $\mu_0$ and $\bar{\mu}$ schemes)—can be related by canonical transformations in the effective theory (Han et al., 2022, Belfaqih et al., 16 Jul 2024, Belfaqih et al., 26 Nov 2024). Under a transformation,

$$P_\varphi \to \frac{\lambda}{\tilde{\lambda}} P_\varphi, \quad E^x \to \frac{\tilde{\lambda}}{\lambda} E^x,$$

the holonomy parameter can shift from constant to $E^x$-dependent, or vice versa. Provided the transformation is canonical, observables and the emergent metric remain invariant. Thus, the physical effects of the holonomy scale and corresponding quantum corrections are best encoded in the covariance class of the effective theory, rather than in any specific choice of representation.

This unifies schemes previously seen as physically distinct and clarifies that the fundamental periodic parameter in the holonomy corrections sets the true scale of quantum geometry effects, while coefficient functions in the Hamiltonian can be interpreted as gauge artifacts or quantization ambiguities, subject to the covariance requirement (Belfaqih et al., 26 Nov 2024, Belfaqih et al., 16 Jul 2024).

4. Emergent Effective Space–Time and Resolution of Singularities

The covariant effective dynamics yield globally regular, quantum-corrected black hole spacetimes, often with a non-singular (wormhole-like) interior. Exact solutions are constructed in several coordinate gauges (static Schwarzschild, homogeneous, Gullstrand–Painlevé, and internal-time), all related by coordinate transformations (Alonso-Bardaji et al., 2021, Zhang et al., 14 Jul 2024, Belfaqih et al., 16 Jul 2024). At the would-be singularity, holonomy corrections induce a "bounce" at a minimal nonzero areal radius or, more generally, a quantum transition surface.

For example, an effective Schwarzschild-like metric in the "μ₀-scheme" reads

$$ds_1^2 = - f_1 dt^2 + \frac{dx^2}{f_1} + x^2 d\Omega^2, \qquad f_1 = 1 - \frac{2M}{x} + \frac{\zeta^2}{x^2}\left( 1 - \frac{2M}{x} \right)^2,$$

exhibiting a double horizon, with the curvature invariants bounded everywhere except possibly at a degenerate central point. In the μ̄-scheme, coordinate transformations and extensions further eliminate residual singularities, creating a geodesically complete wormhole structure bridging a black-hole and a white-hole region (Zhang et al., 14 Jul 2024, Belfaqih et al., 16 Jul 2024).

Quantum corrections leave the asymptotic regions unchanged, reproducing Schwarzschild or Minkowski spacetime as appropriate. The explicit identification of the effective stress-energy tensor,

$$G_{\mu\nu}^{(\text{eff})} = 8\pi T^{(\text{eff})}_{\mu\nu},$$

permits assignment of quantum-induced energy and entropy, including modified thermodynamical relations and quasilocal quantities.

5. Covariance with Matter Coupling and Extensions

For models including electromagnetic fields (electro-vacuum) or generic matter, the framework extends by constructing effective matter Hamiltonians that share the same transformation properties as the metric under gauge-generated diffeomorphisms (Yang et al., 19 Mar 2025). For the electromagnetic field, the canonical pair is modified and the matter Hamiltonian density is multiplied by a quantum correction factor $y(u)$, where $u$ encodes the gravitational quantum deformation. The total Hamiltonian constraint,

$$H_{\text{eff}} = H_{\text{gravity}}^{\text{eff}} + H_{\text{matter}}^{\text{eff}},$$

retains the covariance of the effective metric $g_{\mu\nu}^{\text{eff}}$ and vector potential $A_\mu^{\text{eff}}$, as verified by the closure of the quantum-corrected constraint algebra and preservation of gauge-consistent transformation laws.

Solutions for coupled gravity–electromagnetic systems in spherical symmetry exhibit quantum gravity effects both in the metric and the electromagnetic field, with explicit quantum corrections to, for example, the effective lapse function and the vector potential.

6. Covariance Tests and Emergent Metrics

General covariance is not guaranteed by the mere preservation of first-class constraints in the effective theory. Explicit criteria have been formulated to test general covariance: The divergence of a "quantum-corrected" Einstein tensor $\mathcal{G}^{\mu\nu}$ constructed from the emergent metric, must vanish on-shell,

$$\nabla_{(\mu)} \mathcal{G}^{\mu\nu} = 0 \big|_{H_{\text{eff}}},$$

ensuring invariance under effective diffeomorphisms (Águila et al., 6 Jan 2025). In practice, with inverse-triad or holonomy corrections, such covariance is achieved not with the classical metric reconstructed from phase-space variables, but with an emergent metric defined by appropriately mapping the modified triads or lapse. For example, inverse-triad corrections can be covariant if the areal radius is rescaled by a triad-dependent function, or the lapse is suitably renormalized; holonomy corrections require a lapse redefinition involving the holonomy function.

This stresses the necessity for thorough covariance testing of any proposed effective model, including consistent treatment of slicing dependence and operator ordering.

7. The Role of Projected Spin Networks and Lorentz Covariance

A fully covariant formulation embeds the conventional SU(2) spin network Hilbert space into a space of SL(2,ℂ) functions, via an integral transform (the Dupuis–Livine map), leading to projected spin networks (Rovelli et al., 2010). These Lorentz-covariant functions are uniquely determined by their SU(2) restriction, providing a one-to-one correspondence between canonical SU(2) and Lorentz-covariant SL(2,ℂ) states. The spinfoam dynamics in the bulk are SL(2,ℂ)-invariant, whereas the boundary theory remains effectively SU(2)-invariant—a structure clarified by Gupta–Bleuler-like projections.

This conceptual underpinning affirms that the quantum-corrected bulk dynamics are Lorentz covariant, with gauge-invariant (Dirac) observables and physical transition amplitudes encoded in the Lorentz-covariant (or generally covariant, for gravity) subspace.

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In sum, covariant effective models for spherical loop quantum gravity are characterized by quantum-corrected Hamiltonian constraints constructed to preserve a suitably deformed, or in special cases undeformed, Dirac algebra—encoding the correct geometric structure via an emergent metric that is invariant under gauge transformations. Global non-singular solutions, consistent coupling to matter, and the unification of previously ambiguous quantization schemes are now understood within this robust, covariant framework. Verification and refinement of these models depend critically on systematic covariance criteria, careful encoding of quantum corrections in both constraints and reconstructed geometric quantities, and explicit construction of invariant observables.