§ Energy as triangulaizing state space

This comes from The wild book by John Rhodes, which I anticipate I'll be posting more of in the coming weeks.

§ Experiments

Let an experiment be a tuple of the phase space XX, action space AA, and an action of the actions onto the phase space :A×XX\curvearrowright: A \times X \rightarrow X. We will write x=axx' = a \curvearrowright x to denote the new state of the system xx. So the experiment EE is the data E(X,A,:A×XX)E \equiv (X, A, \curvearrowright : A \times X \rightarrow X).

§ Coordinate systems.

The existence of the action \curvearrowright allows us to write the evolution of the system recursively: xt+1=axtx_{t+1} = a \rightarrow x_t. However, to understand the final state xt+1x_{t+1}, we need to essentially "run the recursion", which does not permit us to understand the experiment . What we really need is the ability to "unroll" the loop. To quote:
Informally, understanding an experiment EE means introducing coordinates into phase space of EE which are in triangular form under the action of the inputs of EE.

§ Conservation laws as triangular form

We identify certain interesting invariants of a system by two criteria:
  1. The parameter Q(t)Q(t) determines some obviously important aspects of the system. That is, there is a deterministic function M(Q(t))M(Q(t)) which maps Q(t)Q(t) to "measure" some internal state of the system.
  2. If the values of such a parameter QQ is known at time t0t_0 (denoted Q(t0)Q(t_0)) and it is also known what inputs are presented to the system from time tt to time t+ϵt + \epsilon(denoted I[t0,t0+ϵ]I[t_0, t_0 + \epsilon]), then the new value of QQ is a deterministic function of Q(t0)Q(t_0) and I[t0,t0+ϵ]I[t_0, t_0+ \epsilon].
Such parameters allow us to understand a system, since they are deterministic parameters of the evolution of the system, while also provding a way to measure some internal state of the system using MM. For example, consider a system xx with an energy function e(x)e(x). If we perform an action aa on the system xx, then we can predict the action e(x=ax)e(x' = a \curvearrowright x) given just e(x)e(x) and aa --- here, (x=ax)(x' = a \curvearrowright x) is the action of the system aa on xx.
In general, conservation principles give a first coordinate of a triangularization. In the main a large part of physics can be viewed as discovering and introducing functions ee of the states qq of the system such that under action aa, e(aq)e(a \curvearrowright q) depends only on e(q)e(q) and aa, and not on qq.

§ Theory: semidirect and wreath products

§ Symmetries as triangular form

We first heuristically indicate the construction involved in going from the group of symmetries to the triangularization, and then precisely write it out in all pedantic detail.
Let an experiment be E(X,A,)E \equiv (X, A, \curvearrowright). Then we define Π\Pi is a symmetry of EE iff:
  1. Π:XX\Pi: X \rightarrow X is a permutation of XX.
  2. Π\Pi commutes with the action of each aa: Π(ax)=aΠ(x) \Pi(a \curvearrowright x) = a \curvearrowright \Pi(x) .
We say that the theory EE is transitive (in the action sense) if for all x1,x2X,x1x2x_1, x_2 \in X, x_1 \neq x_2, there exists a1,a2,ana_1, a_2, \dots a_n such that x2=an(a1x1) x_2 = a_n \curvearrowright \dots (a_1 \curvearrowright x_1) . Facts of the symmetries of a system:
  1. We know that the symmetries of a theory EE form a group.
  2. If EE is transitive, then each symmetry Π\Pi is a regular permutation --- If there exists an xx such that Π(xf)=xf\Pi(x_f) = x_f (a fixed point), then this implies that Π(x)=x\Pi(x) = x for all xx.
  3. Let the action split XX into disjoint orbits O1,O2,OkO_1, O_2, \dots O_k from whom we choose representatives x1O1,x2O2,xkOkx_1 \in O_1, x_2 \in O_2, \dots x_k \in O_k. Then, if EE is transitive, there is exactly one action that sends a particular xix_i to a particular xjx_j. So, on fixing one component of an action, we fix all components .
To show that this gives rise to a triangulation, we first construct a semigroup of the actions of the experiment: S(E){a1an:n1 and aiA}S(E) \equiv \{ a_1 \dots a_n : n \geq 1 \text{~and~} a_i \in A \}. Now, let G=Sym(E)G = Sym(E), the full symmetry group of EE. One can apparently express the symmetry group in terms of:
(X,S)(G,G)({O1,O2,Ok},T)(X, S) \leq (G, G) \wr (\{ O_1, O_2, \dots O_k\}, T)