A mathematical base for Fibre bundle formulation of Lagrangian Quantum Field Theory

TL;DR

This paper develops a differential-geometric framework for Lagrangian quantum field theory using fibre bundles, replacing Hilbert spaces with Hilbert bundles and operators with morphisms to provide a geometric foundation.

Contribution

It introduces a fibre bundle formulation of quantum field theory, replacing Hilbert spaces with Hilbert bundles and operators with morphisms, offering a new geometric perspective.

Findings

Replaces Hilbert space with Hilbert bundle in quantum field theory

Maps quantum operators to bundle morphisms

Provides a differential-geometric foundation for the theory

Abstract

The paper contains a differential-geometric foundations for an attempt to formulate Lagrangian (canonical) quantum field theory on fibre bundles. In it the standard Hilbert space of quantum field theory is replace with a Hilbert bundle; the former playing a role of a (typical) fibre of the letter one. Suitable sections of that bundle replace the ordinary state vectors and the operators on the system's Hilbert space are transformed into morphisms of the same bundle. In particular, the field operators are mapped into corresponding field morphisms.