Metric Field as Emergence of Hilbert Space

TL;DR

This paper introduces a novel approach where the metric field of spacetime emerges from an augmented Hilbert space constructed via quantum acceleration operators, linking quantum field theory and spacetime geometry.

Contribution

It proposes the quantum acceleration operator (QAO) and demonstrates how the metric field can be derived from an augmented Hilbert space, offering a new foundational perspective.

Findings

Quantum acceleration operators relate different vacuum states.

The metric field emerges from the augmented Hilbert space.

The approach unifies quantum field theory with spacetime geometry.

Abstract

First, we explain some ambiguities of spacetime and metric field as fundamental concepts. Then, from the Unruh effect point of view and using the Gelfand-Naimark-Segal construction, we construct an operator as a quanta of acceleration that we call quantum acceleration operator (QAO). Thereupon, we investigate the relation between the vacuum of two different frames in the Minkowski space. Also, we show that the vacuum of each accelerated frame in the Minkowski space can be obtained by applying such a QAO to the Minkowski vacuum. Furthermore, utilizing these QAOs, we augment the Hilbert space and then extract the metric field of a general frame of the Minkowski spacetime. In this approach, these concepts emerge from the Hilbert space through the constructed QAOs. Accordingly, such an augmented Hilbert space includes quantum field theory in a general frame and can be considered as a fundamental concept instead of the classical metric field and the standard Hilbert space.