Quantum Field Theory in Categorical Quantum Mechanics

TL;DR

This paper develops a categorical quantum mechanics framework for quantum field theory using non-standard analysis, defining operators and propagators in a diagrammatic, relativistically invariant manner.

Contribution

It introduces a novel categorical formulation of quantum field theory with explicit operator definitions and propagator derivations within a non-standard analysis setting.

Findings

Defined quantum fields as states in a Hilbert space object

Constructed ladder and field operators as endomorphisms

Derived propagator using diagrammatic commutation relations

Abstract

We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as states and we show that the usual ladder and field operators can be defined as suitable endomorphisms. We deal with relativistic normalisation and we obtain the Lorentz invariant Heisenberg picture operators. By moving to a coherent perspective -- where the classical time and momentum parameters are replaced by wavefunctions over the parameter spaces -- we show that ladder operators and field operators can be obtained by applying the same morphism to plane waves and delta functions respectively. Finally, we formulate the commutation relations diagrammatically and we use them to derive the propagator.