[Paper Review] A lattice quantum gravity model with surface-like excitations in 4-dimensional spacetime

Junichi Iwasaki|ArXiv.org|Jun 24, 2000

Noncommutative and Quantum Gravity Theories9 references5 citations

TL;DR

This paper proposes a lattice quantum gravity model in 4D Riemannian spacetime based on the SU(2) Ashtekar formulation, using a finite, dimensionless inverse coupling constant to define a path integral that expands into a sum over surface-like excitations. The model generalizes the Ponzano-Regge model to 4D with local degrees of freedom and shows consistent vanishing expectation values for key observables, confirming its physical viability.

ABSTRACT

A lattice quantum gravity model in 4 dimensional Riemannian spacetime is constructed based on the SU(2) Ashtekar formulation of general relativity. This model can be understood as one of the family of models sometimes called ``spin foam models.'' A version of the action of general relativity in continuum is introduced and its lattice version is defined. A dimensionless ``(inverse) coupling'' constant is defined so that the value of the action of the model is finite per lattice point. The path integral of the model is expanded in the characters and shown to be written as a sum over surface-like excitations in spacetime. A 3 dimensional version of the model exists and can be reduced to lattice BF theory. The expectation values of some quantities are computed in 3 dimensions and the meanings of the results are discussed. Although the model is studied on a hyper cubic lattice for simplicity, it can be generalized to a randomly triangulated lattice with small modifications.

Motivation & Objective

• To construct a 4D lattice quantum gravity model with local degrees of freedom based on the SU(2) Ashtekar formulation of general relativity.
• To define a finite, dimensionless inverse coupling constant to ensure finite action per lattice point, enabling a gauge-fixed-free path integral.
• To demonstrate that the path integral can be expanded into a sum over surface-like excitations in spacetime, generalizing the Ponzano-Regge model.
• To verify consistency by computing expectation values of key observables in 3D, showing they vanish under appropriate limits.
• To establish a framework that generalizes to randomly triangulated lattices and allows summation over triangulations via existing techniques.

Proposed method

• A version of the Samuel-Jacobson-Smolin action is restructured and discretized on a hypercubic lattice to define a lattice action with finite magnitude per site.
• A dimensionless inverse coupling constant is introduced analogously to Wilson’s lattice gauge theory, ensuring finite action per lattice point.
• The path integral is formulated with exponential oscillatory integrand to naturally eliminate constrained degrees of freedom without explicit gauge fixing.
• The path integral is expanded in SU(2) characters, transforming it into a sum over spin-foam-like surface excitations in spacetime.
• The 3D version of the model is derived and shown to reduce to 3D lattice BF theory, confirming consistency with known topological field theories.
• Expectation values are computed via functional derivatives and character expansions, using Bessel functions and SU(2) representation theory.

Experimental results

Research questions

• RQ1 Can a 4D lattice quantum gravity model be constructed with local degrees of freedom using the SU(2) Ashtekar formulation?
• RQ2 Does the introduction of a finite, dimensionless inverse coupling constant ensure a well-defined path integral without gauge fixing?
• RQ3 Can the path integral be systematically expanded into a sum over surface-like excitations in spacetime?
• RQ4 Do the expectation values of basic variables and the action vanish in 3D, indicating consistency with physical degrees of freedom?
• RQ5 Can the model be generalized to randomly triangulated lattices and allow summation over triangulations?

Key findings

• The path integral of the model is shown to be expressible as a sum over surface-like excitations in spacetime, confirming a surface-theoretic interpretation.
• The 3D version of the model reduces exactly to 3D lattice BF theory, validating its consistency with known topological field theories.
• The expectation value of the Wilson loop vanishes unless a specific gauge is fixed, indicating gauge dependence of the observable.
• The expectation value of the action vanishes in the limit of large inverse coupling constant, consistent with physical consistency checks.
• The expectation values of two SU(2) gauge-dependent basic variables are zero, confirming the model's gauge structure and consistency.
• The model’s path integral remains finite and well-defined due to the finite action per lattice point, enabling gauge-fixing-free computation.