[Paper Review] Lattice Loop Quantum Gravity

Johannes Aastrup, Jesper Møller Grimstrup|ArXiv.org|Nov 20, 2009

Noncommutative and Quantum Gravity Theories9 references3 citations

TL;DR

This paper proposes a separable, lattice-based formulation of Loop Quantum Gravity (LQG) using an inductive system of nested cubic lattices to construct a kinematical Hilbert space. It demonstrates that flux, area, volume, and Hamiltonian operators have the correct classical limits in semi-classical states, and suggests that spatial symmetries—including diffeomorphism invariance—may be restored in a continuum limit, offering a potential path to a background-independent quantum gravity theory.

ABSTRACT

We present a separable version of Loop Quantum Gravity (LQG) based on an inductive system of cubic lattices. We construct semi-classical states for which the LQG operators -- the flux, the area and the volume operators -- have the right classical limits. Also, we present the Hamilton and diffeomorphism constraints as operator constraints and show that they have the right classical limit. Finally, we speculate whether the continuum limit, which these semi-classical states probe, can be defined for the entire construction and thereby restore an action of the diffeomorphism group.

Motivation & Objective

• To develop a separable version of Loop Quantum Gravity (LQG) using an inductive system of nested cubic lattices.
• To construct normalizable semi-classical states that reproduce classical limits of LQG operators such as flux, area, volume, and Hamiltonian.
• To investigate whether the continuum limit of the lattice system can restore diffeomorphism invariance and background independence.
• To explore the possibility that spatial symmetries emerge in a continuum limit, analogous to lattice gauge theory.
• To provide a framework where finite lattice dependencies vanish, potentially allowing a full continuum construction of the LQG algebra and Hilbert space.

Proposed method

• Construct the kinematical Hilbert space as an inductive limit over an infinite sequence of 3D nested cubic lattices.
• Represent holonomy and flux operators on the Hilbert space using lattice-based connections and co-tangent space variables.
• Define semi-classical states as coherent states localized on the lattice, with wavefunctions depending on the lattice depth and continuous in the continuum limit.
• Implement the Hamilton and diffeomorphism constraints as operator constraints on the Hilbert space, using lattice discretizations of the classical constraints.
• Use a double limit: first take the classical limit (t → 0) at fixed lattice depth n, then take the continuum limit (n → ∞) to recover classical expressions.
• Leverage the structure of spectral triples and nested lattices to ensure convergence of operators to their classical counterparts in the limit.

Experimental results

Research questions

• RQ1 Can a lattice-based LQG model produce semi-classical states for which the flux, area, and volume operators have the correct classical limits?
• RQ2 Does the Hamiltonian operator in this lattice model converge to the classical Hamiltonian constraint in the continuum and classical limit?
• RQ3 Can the diffeomorphism group action be restored in a continuum limit of the lattice construction, despite its absence in the finite lattice model?
• RQ4 Is it possible to define a continuum limit for the entire Hilbert space and operator algebra, not just for specific states?
• RQ5 How does the absence of a background metric in the continuum limit affect the construction, particularly in the Lorentzian case?

Key findings

• The flux, area, and volume operators in the lattice LQG model converge to their classical counterparts in the double limit of large lattice depth and small classical parameter.
• The Hamiltonian operator constructed on the lattice reproduces the classical Hamiltonian constraint in the limit, with the correct dependence on the triad and connection fields.
• Semi-classical states exist that are normalizable and exhibit the correct classical behavior, suggesting a viable path to a semi-classical regime.
• The continuum limit appears to remove lattice-dependent dependencies, hinting that spatial symmetries such as diffeomorphism invariance may emerge in the limit.
• The construction suggests a mechanism for restoring background independence via a continuum limit, analogous to lattice gauge theory.
• The Lorentzian case may be accessible by complexifying the connection and using a doubled Hilbert space, with the continuum limit preserving the structure.