# Algebraic Classical and Quantum Field Theory on Causal Sets

## TL;DR

This paper develops a framework for constructing classical and quantum field theories on causal sets using perturbative algebraic quantum field theory, introducing new discretised wave operators and a wide class of interacting models.

## Contribution

It introduces a novel discretised wave operator based on a 'preferred past' and constructs a broad class of interacting quantum field models on causal sets for the first time.

## Key findings

- Defined the SJ state as a quantum state on free theories.
- Constructed classical free and interacting models on causal sets.
- Developed a deformation quantization approach using pAQFT.

## Abstract

The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a `preferred past', which we also introduce, and show how they may be used to construct classical free and interacting field theory models on a fixed causal set; additionally, we describe how the sensitivity of observables to changes in the background causal set may be encapsulated in a relative Cauchy evolution. These structures are used as the basis of a deformation quantization, using the methods of pAQFT. The SJ state is defined and discussed as a particular quantum state on the free quantum theory. Finally, using the framework of pAQFT, we construct interacting models for arbitrary interactions that are smooth functions of the field configurations. This is the first construction of such a wide class of models achieved in QFT on causal sets.