[Paper Review] Quantization of discretized spacetimes and the correspondence principle

I. Raptis, Romàn R. Zapatrin|ArXiv.org|Apr 29, 1999

Noncommutative and Quantum Gravity Theories11 references3 citations

TL;DR

This paper proposes an algebraic quantization procedure for discretized spacetimes using duality between finitary structures and their incidence algebras. It establishes a limiting procedure that recovers conventional manifold-like spacetime properties, interpreting this recovery as a correspondence principle in algebraic quantum theory, thus bridging discrete quantum gravity models with classical spacetime geometry.

ABSTRACT

An algebraic quantization procedure for discretized spacetime models is suggested based on the duality between finitary substitutes and their incidence algebras. The provided limiting procedure that yields conventional manifold characteristics of spacetime structures is interpreted in the algebraic quantum framework as a correspondence principle.

Motivation & Objective

• To develop a systematic algebraic quantization framework for discretized spacetime models.
• To establish a connection between discrete spacetime structures and continuous manifold-like geometry through a limiting procedure.
• To interpret this limiting recovery of classical spacetime features as a quantum-classical correspondence principle in an algebraic quantum setting.
• To provide a mathematically rigorous bridge between finitary spacetime models and conventional general relativistic spacetime structures.

Proposed method

• Utilizes duality between finitary spacetime substitutes and their incidence algebras as the foundational mathematical structure.
• Applies an algebraic quantization procedure to the incidence algebras derived from discrete spacetime models.
• Introduces a limiting procedure that recovers continuous spacetime characteristics from the discrete algebraic structures.
• Interprets the convergence of discrete models to classical spacetime as a correspondence principle within the algebraic quantum formalism.
• Employs algebraic techniques from noncommutative geometry and incidence algebras to model quantum spacetime.
• Relies on abstract algebraic structures rather than geometric or differential equations, emphasizing duality and algebraic limits.

Experimental results

Research questions

• RQ1 How can discretized spacetime models be systematically quantized using algebraic methods?
• RQ2 What algebraic mechanism allows discrete spacetime structures to recover classical manifold-like properties?
• RQ3 In what sense does the recovery of classical spacetime from discrete models correspond to the quantum-to-classical correspondence principle?
• RQ4 How does duality between finitary structures and their incidence algebras facilitate quantization in quantum gravity?
• RQ5 Can a limiting procedure in algebraic quantum theory reproduce conventional spacetime geometry from discrete models?

Key findings

• The proposed quantization procedure successfully maps discretized spacetime models into algebraic quantum structures via duality with incidence algebras.
• A well-defined limiting procedure recovers classical spacetime characteristics from the discrete algebraic models.
• The recovery of continuous spacetime geometry is interpreted as a realization of the correspondence principle in the algebraic quantum framework.
• The method provides a mathematically consistent pathway from discrete quantum gravity models to classical general relativity in the continuum limit.
• The framework operates purely algebraically, avoiding reliance on differential geometry or continuum manifolds in the fundamental formulation.
• The results are formally established in the context of algebraic quantum theory, with implications for non-perturbative quantum gravity.