Operator-Valued Tensors on Manifolds: A Framework for Field Quantization

TL;DR

This paper develops a geometric framework using operator-valued tensors on manifolds, extending classical concepts like metrics and curvature to facilitate field quantization in quantum physics.

Contribution

It introduces operator-valued tensors and extends semi-Riemannian geometry, providing a novel mathematical foundation for field quantization.

Findings

Defined operator-valued tensors on manifolds

Extended semi-Riemannian metrics to operator-valued metrics

Reformulated key geometric concepts like curvature and connections

Abstract

In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commutative C-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered.