Conserved quantities in discrete dynamics: what can be recovered from Noether’s theorem, how, and why?

Abstract

The connections between symmetries and conserved quantities of a dynamical system brought to light by Noether’s theorem depend in an essential way on the symplectic nature of the underlying kinematics. In the discrete dynamics realm, a rather suggestive analogy for this structure is offered by second-order cellular automata. We ask to what extent the latter systems may enjoy properties analogous to those conferred, for continuous systems, by Noether’s theorem. For definiteness, as a second-order cellular automaton we use the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that—and why—energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant. We then begin to explore the issue of whether, in these systems, it may hold as well that translation invariance entails momentum conservation.

Appendix: Continuity and all that

“Continuity” has two quite distinct meanings in mathematics. A continuum is a set, such as the real line or the complex plane, that is uncountable but “only mildly so,” namely, one whose cardinality equals that of the set ${2^{\mathbb{Z}}}$ (where ${{\mathbb{Z}}}$ denotes the integers—the countable set par excellence). In common parlance, a continuous variable (or parameter) means “one taking values in a continuum.”

On the other hand, a continuous function is one for which “the counterimage of every open set is an open set.” This kind of continuity is the foundation stone of general topology, and is moreover a categorical aspect of cellular automata, as explained below.

Without further qualifications, a dynamics is discrete if it is the iteration of a transformation—intuitively, a game where an onlooker can ask “How many moves have you made yet?” and get an answer “t moves,” with $t=0,1,2,3,\ldots$. If instead the time parameter t runs along a one-dimensional continuum—and thus is a “continuous variable” in the above sense—one may be allowed, at least informally, to speak of a “continuous” dynamics. But when the dynamics is thought of as a function f _t that maps a state q ₀ to a new state q _t (where t is the parameter of a one-parameter group—which may be continuous or discrete) this function need not, in general, be continuous in the topological sense. To avoid confusion, to just make the point that t is meant to be a continuous (rather than discrete) parameter, one should preferably speak of a continuum dynamics.

A cellular automaton is a discrete dynamics, in the sense that time ranges over a discrete set—the integers—rather than a continuum—the reals. But the discreteness of cellular automaton in not limited to the parameter t. The local state (the state of a site) ranges over a finite alphabet, and thus is a discrete variable (with, without loss of generality, the discrete topology). The local map of a cellular automaton makes use of information brought in to a site from a finite number of locations, and is thus a finite table; however, these finite neighborhoods overlap, and thus generate by iteration an infinite, though discrete, mesh. By raising the finite state-alphabet to the countable cardinality of this mesh, the global state set (the set of all possible states for the entire mesh) is a continuum; incidentally, this continuum is compact with respect to its natural topology—which is by construction the product topology of the single-site topologies.

Finally, and most importantly, a characterizing property of cellular automata is that, in spite of their pervasive discreteness, their dynamics—a mapping of that continuum into itself—is a continuous function on a metric space.