Quantum field theory from a quantum cellular automaton in one spatial dimension and a no-go theorem in higher dimensions

TL;DR

This paper constructs a 1D quantum cellular automaton that recovers free fermion quantum field theory in the long-wavelength limit, but proves no similar construction exists in higher dimensions, highlighting fundamental limitations.

Contribution

It demonstrates a specific 1D QCA model that reproduces quantum field theory and establishes a no-go theorem for higher dimensions, revealing inherent dimensional constraints.

Findings

1D QCA recovers free fermion QFT in the continuum limit

No straightforward extension of QCA to higher dimensions is possible

Identifies fundamental barriers to higher-dimensional QCA constructions

Abstract

It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multi-particle case. We construct a one-dimensional quantum cellular automaton (QCA) model which matches the quantum walk in the single particle case, and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way, and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.