[Paper Review] Space-time foam in 2D and the sum over topologies

R. Loll, W. Westra|ArXiv.org|Aug 31, 2003

Noncommutative and Quantum Gravity Theories17 references3 citations

TL;DR

This paper formulates a well-defined 2D Lorentzian quantum gravity model that includes a sum over topologies by restricting topology-changing geometries to those preserving causality. Using Lorentzian random triangulations and causality constraints, the path integral converges and admits a double-scaling limit yielding a non-perturbative theory of space-time foam with a finite, renormalized partition function and a phase transition at maximal gravitational coupling.

ABSTRACT

It is well-known that the sum over topologies in quantum gravity is ill-defined, due to a super-exponential growth of the number of geometries as a function of the space-time volume, leading to a badly divergent gravitational path integral. Not even in dimension 2, where a non-perturbative quantum gravity theory can be constructed explicitly from a (regularized) path integral, has this problem found a satisfactory solution. -- In the present work, we extend a previous 2d Lorentzian path integral, regulated in terms of Lorentzian random triangulations, to include space-times with an arbitrary number of handles. We show that after the imposition of physically motivated causality constraints, the combined sum over geometries and topologies is well-defined and possesses a continuum limit which yields a concrete model of space-time foam in two dimensions.

Motivation & Objective

• To resolve the ill-defined nature of the sum over topologies in quantum gravity, which suffers from super-exponential divergence due to factorial growth of geometries.
• To construct a non-perturbative 2D quantum gravity theory that includes topology change while maintaining causality and finiteness.
• To demonstrate that causality constraints eliminate divergent configurations and lead to a well-behaved path integral with a continuum limit.
• To establish a double-scaling limit that yields a finite, renormalized partition function and a concrete model of space-time foam in 2D.
• To explore the physical implications of topology change, particularly the scattering of light rays due to baby universes (holes) in the spacetime.

Proposed method

• Use Lorentzian random triangulations to discretize 1+1 dimensional spacetimes, ensuring a globally defined proper time and causal structure.
• Impose causality constraints that exclude topology-changing geometries violating global causality, allowing only isolated, short-lived holes (one time-step duration).
• Construct a state sum over geometries and topologies (genus g) by summing over all triangulated Lorentzian spacetimes with arbitrary genus and causal structure.
• Perform an exact summation over all such configurations using generating functions and recurrence relations derived from the one-step propagator.
• Apply a double-scaling limit by tuning both the cosmological constant Λ and gravitational coupling G to their critical values, while rescaling the lattice spacing a.
• Renormalize the coupling constants via logarithmic subtraction to obtain a finite, renormalized partition function Z^R(Λ, G) in the continuum limit.

Experimental results

Research questions

• RQ1 Can a sum over topologies in 2D quantum gravity be made mathematically well-defined despite the factorial divergence of geometries?
• RQ2 How do causality constraints affect the inclusion of topology-changing spacetimes in a Lorentzian path integral?
• RQ3 Does the inclusion of arbitrary genus (holes) lead to a finite, non-perturbative partition function in a 2D Lorentzian quantum gravity model?
• RQ4 What is the physical interpretation of the resulting theory in terms of observable effects like light ray scattering?
• RQ5 Can a double-scaling limit be defined that yields a finite, unambiguous continuum theory with a non-trivial dependence on the gravitational coupling?

Key findings

• The sum over topologies is rendered finite by causality constraints that exclude configurations with large-scale causality violations, eliminating the factorial divergence in entropy.
• The partition function admits a double-scaling limit that yields a finite, renormalized partition function Z^R(Λ, G) = (1/a²) * (1/4Λ) * (1 - √(1 - 4e^(-2/G))) in the continuum.
• The expectation value of the spacetime volume is ⟨V⟩ = 1/Λ, consistent with the cosmological constant setting the overall scale.
• The average number of holes per time interval is ⟨g⟩ = -½(1 - 1/√(1 - 4e^(-2/G))), which increases with G and diverges at G = 2/log 4, signaling a phase transition.
• The model exhibits a non-local observable effect: the fraction of a light beam scattered by holes is proportional to ⟨g⟩ × (T/L), linking topology to measurable physics.
• For small G, the theory reduces to standard 2D Lorentzian quantum gravity without holes; for larger G, topology change becomes significant, realizing a concrete model of space-time foam.