Logico-algebraic approach to spacetime quantization

TL;DR

This paper develops a logico-algebraic framework for spacetime quantization, removing the need for a fundamental spacetime manifold and making spacetime relative to measurement outcomes.

Contribution

It introduces a novel algebraic approach that combines differential and non-commutative structures to describe quantum spacetime without assuming an underlying manifold.

Findings

Spacetime points emerge from eigen-subalgebras of the basic algebra.

The formalism makes spacetime relative to measurement, not fundamental.

Provides a new perspective on quantum gravity and spacetime structure.

Abstract

In the last decades the logico-algebraic approach to quantum mechanics turned to be a successful tool to render the quantum mechanical formalism on a steady operationalistic background. The algebraic approach to general relativity first proposed by Geroch is used to build a smearing procedure for events in the quantized theory of spacetime. It stems from the notion of basic algebra which possesses both the differential structure (as functional algebras) and non-commutativity (as algebras of observables in quantum theories). The main essence of this formalism is that it deprives the notion of spacetime as event support of its fundamental status making it relative to the measurement performed. Before the observation is executed, it is pointless to speak of events as points of spacetime: there is no underlying manifold. Only when an eigen-subalgebra of the basic algebra is outlined as eigenstate of the observable, it gives rise to its functional representation on the appropriate set thought of as spacetime.