Quantum cellular automata and free quantum field theory

TL;DR

This paper derives quantum field theory from quantum cellular automata principles, showing how discrete automata can recover relativistic field dynamics and exploring extensions to interacting theories.

Contribution

It provides a principled derivation of quantum field theory from automata on Cayley graphs, connecting discrete quantum information models to continuous relativistic fields.

Findings

Automata on Cayley graphs can produce Weyl, Dirac, and Maxwell equations in the relativistic limit.

The discrete automaton structure is set at the Planck scale, unifying quantum information and field theory.

Extension towards non-linear automata for interacting quantum fields is discussed.

Abstract

In a series of recent papers it has been shown how free quantum field theory can be derived without using mechanical primitives (including space-time, special relativity, quantization rules, etc.), but only considering the easiest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the simple principles of unitarity, homogeneity, locality, and isotropy. This has opened the route to extending the axiomatic information-theoretic derivation of the quantum theory of abstract systems to include quantum field theory. The inherent discrete nature of the informational axiomatization leads to an extension of quantum field theory to a quantum cellular automata theory, where the usual field theory is recovered in a regime where the discrete structure of the automata cannot be probed. A simple heuristic argument sets the scale of discreteness to the Planck scale, and the customary physical regime where discreteness is not visible is the relativistic one of small wavevectors. In this paper we provide a thorough derivation from principles that in the most general case the graph of the quantum cellular automaton is the Cayley graph of a finitely presented group, and showing how for the case corresponding to Euclidean emergent space (where the group resorts to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics in the relativistic limit. We conclude with some perspectives towards the more general scenario of non-linear automata for interacting quantum field theory.