Fermion picture for cellular automata - Wetterich, Christof (Arxiv, 2022)
Many cellular automata admit an interpretation in terms of fermionic particles. Reversible automata on space-lattices with a local updating rule can be described by a partition function or Grassmann functional integral for interacting fermions moving in this space. We discuss large classes of automata that are equivalent to discretized fermionic quantum field theories with various types of interactions. Two-dimensional models include relativistic Thirring or Gross-Neveu type models with abelian or non-abelian continuous global symmetries, a particle in a potential, local gauge symmetries, and spinor gravity with local Lorentz symmetry as well as diffeomorphism invariance in the (naive) continuum limit. Probabilistic cellular automata are characterized by a probability distribution over initial bit-configurations. They can be described by the quantum formalism with wave functions, density matrix and non-commuting operators associated to observables. This formalism is useful for a discussion of concepts as vacuum states, spontaneous symmetry breaking, coarse graining and the continuum limit for probabilistic cellular automata. The formalism of fermionic quantum field theories opens a new view on the statistical behavior of automata with a very large number of cells. In the other direction, some specific many body systems or discrete quantum field theories for fermions can be understood by solving or simulating the associated probabilistic cellular automaton. This opens a new access to non-equilibrium quantum field theory.
Gesehen am 26.10.2022