# Towards Quantum Field Theory in Categorical Quantum Mechanics

## TL;DR

This paper develops a categorical framework for infinite-dimensional quantum systems and quantum fields using non-standard analysis, enabling algebraic manipulation of unbounded operators and modeling of various quantum field scenarios.

## Contribution

It introduces a novel dagger compact category for infinite-dimensional quantum systems and quantum fields within Categorical Quantum Mechanics, expanding the framework's applicability.

## Key findings

- Constructed a dagger compact category *Hilb for unbounded operators
- Modeled quantum systems with periodic boundary conditions and on lattices
- Extended the framework to include non-separable Hilbert spaces and quantum fields

## Abstract

In this work, we use tools from non-standard analysis to introduce infinite-dimensional quantum systems and quantum fields within the framework of Categorical Quantum Mechanics. We define a dagger compact category *Hilb suitable for the algebraic manipulation of unbounded operators, Dirac deltas and plane-waves. We cover in detail the construction of quantum systems for particles in boxes with periodic boundary conditions, particles on cubic lattices, and particles in real space. Not quite satisfied with this, we show how certain non-separable Hilbert spaces can also be modelled in our non-standard framework, and we explicitly treat the cases of quantum fields on cubic lattices and quantum fields in real space.