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Journal of Physics Conference Series

Modeling "quantum" field theory through "classical" cellular automaton

Roman Sverdlov

Published Jun 10, 2013• 0 citations
Publisher

Abstract

The purpose of this paper is to illustrate how cellular automaton can be used in order to arrive at discrete, but "classical", model leading to the emergent "quantum" field theory.
DOI
10.1088/1742-6596/442/1/012047

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A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.
The Thirring Quantum Cellular Automaton (QCA) describes the discrete time dynamics of local fermionic modes that evolve according to one step of the Dirac cellular automaton, followed by the most general on-site number-preserving interaction, and serves as the QCA counterpart of the Thirring model in quantum field theory. In this work, we develop perturbative techniques for the QCA path sum approach, expanding both the number of interaction vertices and the mass parameter of the Thirring QCA. By classifying paths within the regimes of very light and very heavy particles, we computed the transition amplitudes in the two- and three-particle sectors to the first few orders. Our investigation into the properties of the Thirring QCA, addressing the combinatorial complexity of the problem, yielded some useful results applicable to the many-particle sector of any on-site number-preserving interactions in one spatial dimension.
One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse graining procedures, we provide a simple procedures to coarse-grain color-blind quantum cellular automata, following Goldilock rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size NNN, (ii) projecting the cells states onto its borders, connecting them with the fine dynamics, which is reminiscent of the edge-bulk correspondence in quantum field theory; (iii) describe the overall dynamics by the border states, that we call signals; (iv) construct the coarse-grained dynamics for different sizes NNN of the cells. A byproduct of this simple toy-model is a general discrete analogous of the Stokes law. Moreover, we proved that in the spacetime limit, the automaton converge to a Dirac free Hamiltonian. The QCA we introduced here, can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.