Posted Tuesday November 7 2023.

A “spatio-temporal” dynamical system consists of

- a space
*X*of points such as*n*-dimensional euclidean space, - a space
*Y*of states, - a monoid
*T*representing the time variable, - an evolutionary operator (a monoid action)
*Φ*:*T*×*Y*^{X}→*Y*^{X}acting on maps*X*→*Y*

In physics these are called “field theories”. For example, a classical scalar field theory is heat diffusion in 3 dimensions, in which case *X* = *R*^{3}, *Y* = *R*, *T* = [0, ∞), and we can actually write down the evolutionary operator *Φ*^{t} using the heat kernel in 3 dimensions. On the other hand, we could model Conway’s Game of Life by taking *X* = *Z*^{2}, *Y* = 0, 1, and *T* = *N*.

These two example systems are pretty different, but they do share one thing in common: they have spatial symmetry, in the sense that the evolutionary operator *Φ*^{t} is equivariant with respect to the action of some group *G* acting on *X*. For a group element *g* ∈ *G* and a point *x* ∈ *X* we will denote the action of *g* on *x* by *g* * *x*. Notably, any group action of *G* on *X* extends to a group action on *Y*^{X}. So if *g* ∈ *G* and *u* ∈ *U* we can write *g* * *u* for the action of *g* to *u*.

Definition 1: Let *G* be a group, let *X* be a set, and let the mapping (*g*, *x*) ↦ *g* * *x* be a group action of *G* on *X*. The group action is called *transitive* if for any *x*_{1}, *x*_{2} ∈ *X*, there exists *g* ∈ *G* such that *g* * *x*_{1} = *x*_{2}.

By “vacuum state” we just mean a constant state - that is a function *u* : *X* → *Y* such that there exists some *y* ∈ *Y* such that *u*(*x*) = *y* for all *y*. Some people just write *u* ≡ *y*. Anyway, because of the transitive group action, if *u* is constant, then so is *Φ*(*t*, *u*) for all *t* ∈ *T*. Let’s prove it!

Lemma 1: Suppose the group action (*g*, *x*) ↦ *g* * *x* is transitive. Then *u* ∈ *Y*^{X} is constant iff. *u* = *g* * *u*, ∀*g* ∈ *G*.

Proof: For the forward direction, suppose *u* is constant, *u* ≡ *y*, and let *g* ∈ *G*. Then *u*(*x*) = (*g* * *u*)(*x*) = *y* for all *x* ∈ *X*, so *g* * *u* = *u*. For the other direction, suppose *g* * *u* = *u* for all *g* ∈ *G*. We want to show *u* is constant, so let *x*_{1}, *x*_{2} ∈ *X*. Since *G* is transitive, we know there exists *g* ∈ *G* such that *x*_{2} = *g* * *x*_{1}. Thus *u*(*x*_{2}) = *u*(*g* * *x*_{1}) = (*g* * *u*)(*x*_{1}) = *u*(*x*_{1}). Thus, *u* is constant. QED.

The next lemma shows that if the group action is transitive, and the dynamical system is equivariant with respect to the group action, then constant states will remain constant.

Lemma 2: Suppose the action (*g*, *x*) ↦ *g* * *x* is transitive, and suppose *Φ*, the action of *T* on *Y*^{X}, is *G*-equivariant, i.e. *Φ*(*t*, *g* * *u*) = *g* * *Φ*(*t*, *u*) for all *u* ∈ *Y*^{X}, *g* ∈ *G*, and *t* ∈ *T*. Then if *u* ∈ *Y*^{X} is constant, so is *Φ*(*t*, *u*) for any *t* ∈ *T*.

Proof: Suppose *u* is constant, i.e. *u* ≡ *y* for some *y* ∈ *Y*. Now let *t* ∈ *T*. We wish to show *Φ*(*t*, *u*) is constant. By Lemma 1, it suffices to show *g* * *Φ*(*t*, *u*) = *Φ*(*t*, *u*) for all *g* ∈ *G*. So let *g* ∈ *G*. By Lemma 1, since *u* is constant we know *g* * *u* = *u*. By assumption we have *g* * *Φ*(*t*, *u*) = *Φ*(*t*, *g* * *u*). Replacing *g* * *u* = *u*, we have *g* * *Φ*(*t*, *u*) = *Φ*(*t*, *u*), the desired result. QED.

Corollary: If the action $(g,x) g x $ is transitive and the action of *Φ*^{t} on *Y*^{X} is *G*-equivariant, then there is a unique monoid action of *ϕ* : *T* × *Y* → *Y* satisfying

*Φ* ∘ (1_{T} × *i*) = *i* ∘ *ϕ*