[Paper Review] Discrete space-time

Rodolfo Gambini, Jorge Pullin|ArXiv.org|May 4, 2005

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper proposes the consistent discretization approach to quantum gravity, where discrete space-time is constructed by discretizing the action rather than the equations of motion, eliminating constraints and enabling unitary, computable dynamics. The method avoids singularities, naturally implements relational time, and connects with loop quantum gravity's kinematics while offering a path to numerical quantum gravity and potential resolution of the black hole information paradox.

ABSTRACT

We review recent efforts to construct gravitational theories on discrete space-times, usually referred to as the ``consistent discretization'' approach. The resulting theories are free of constraints at the canonical level and therefore allow to tackle many problems that cannot be currently addressed in continuum quantum gravity. In particular the theories imply a natural method for resolving the big bang (and other types) of singularities and predict a fundamental mechanism for decoherence of quantum states that might be relevant to the black hole information paradox. At a classical level, the theories may provide an attractive new path for the exploration of issues in numerical relativity. Finally, the theories can make direct contact with several kinematical results of continuum loop quantum gravity. We review in broad terms several of these results and present in detail as an illustration the classical treatment with this technique of the simple yet conceptually challenging model of two oscillators with constant energy sum.

Motivation & Objective

• To develop a consistent, constraint-free formulation of gravitational theories on discrete space-time for use in quantum gravity.
• To resolve longstanding issues in canonical quantum gravity, such as the problem of time and unitarity loss, via discrete dynamics.
• To provide a numerically stable and convergent approach to classical and quantum gravity that preserves constraints and avoids singularities.
• To establish a bridge between discrete theories and the kinematical framework of loop quantum gravity.
• To explore the implications of discrete time evolution for black hole information and cosmological singularities.

Proposed method

• Discretize the gravitational action in time while keeping space continuous, using a variational principle to derive equations of motion.
• Approximate time-derivative terms via holonomies along finite time-like plaquettes, preserving gauge invariance and unitarity.
• Introduce Lagrange multipliers to enforce unitarity of parallel transport matrices, ensuring SU(2) gauge invariance.
• Treat the diffeomorphism constraint not as a constraint but as a conserved quantity, which can be imposed after evolution.
• Use relational time via a physical clock variable to define dynamics, bypassing the problem of time in canonical quantum gravity.
• Construct a unitary evolution operator in the quantum theory, enabling direct numerical computation of quantum dynamics.

Experimental results

Research questions

• RQ1 Can a consistent discretization of the gravitational action eliminate constraints in canonical quantum gravity while preserving key physical symmetries?
• RQ2 How does the consistent discretization approach resolve cosmological and black hole singularities in discrete space-time?
• RQ3 Can the discrete theory reproduce the kinematical structure of loop quantum gravity, particularly the Hilbert space and gauge invariance?
• RQ4 Does the discrete evolution scheme preserve the constraints of general relativity in the continuum limit?
• RQ5 Can the relational time framework in discrete gravity provide a mechanism for decoherence relevant to the black hole information paradox?

Key findings

• The consistent discretization approach yields a theory without first-class constraints, simplifying quantization and enabling direct unitary evolution.
• Singularities are naturally avoided because they typically do not lie on the discrete computational grid, implying zero probability for their occurrence in quantum theory.
• The diffeomorphism constraint is not imposed as a constraint but as a conserved quantity, allowing consistent evolution while preserving the kinematical structure of loop quantum gravity.
• The method produces numerically stable and convergent classical evolutions that respect constraints, unlike standard free-evolution schemes in numerical relativity.
• The discrete theory allows for a relational time framework that breaks unitarity, offering a potential mechanism for decoherence relevant to the black hole information paradox.
• By discretizing only time and using holonomy-based actions, the approach reproduces the physical state space of loop quantum gravity in 2+1 dimensions, confirming consistency with continuum loop quantization.