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 $X$, action space $A$, and an action of the actions onto the phase space $\curvearrowright: A \times X \rightarrow X$. We will write $x' = a \curvearrowright x$ to denote the new state of the system $x$. So the experiment $E$ is the data $E \equiv (X, A, \curvearrowright : A \times X \rightarrow X)$.

§ Coordinate systems.

The existence of the action $\curvearrowright$ allows us to write the evolution of the system recursively:

$x_{t+1} = a \rightarrow x_t$.

However, to understand the final state $x_{t+1}$, 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 $E$ means introducing coordinates into phase space of $E$ which are in triangular form under the action of the inputs of $E$.

§ Conservation laws as triangular form

We identify certain interesting invariants of a system by two criteria:

1. The parameter $Q(t)$ determines some obviously important aspects of the system. That is, there is a deterministic function $M(Q(t))$ which maps $Q(t)$ to "measure" some internal state of the system.
2. If the values of such a parameter $Q$ is known at time $t_0$ (denoted $Q(t_0)$) and it is also known what inputs are presented to the system from time $t$ to time $t + \epsilon$(denoted $I[t_0, t_0 + \epsilon]$), then the new value of $Q$ is a deterministic function of $Q(t_0)$ and $I[t_0, t_0+ \epsilon]$.

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 $M$.

For example, consider a system $x$ with an energy function $e(x)$. If we perform an action $a$ on the system $x$, then we can predict the action $e(x' = a \curvearrowright x)$ given just $e(x)$ and $a$ --- here, $(x' = a \curvearrowright x)$ is the action of the system $a$ on $x$.

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 $e$ of the states $q$ of the system such that under action $a$, $e(a \curvearrowright q)$ depends only on $e(q)$ and $a$, and not on $q$.

§ Theory: semidirect and wreath products

• For semidirect products, I refer you to the cutest way to write semidirect products Line of investigation to build physical intuition for semidirect 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 $E \equiv (X, A, \curvearrowright)$. Then we define $\Pi$ is a symmetry of $E$ iff:

1. $\Pi: X \rightarrow X$ is a permutation of $X$.
2. $\Pi$ commutes with the action of each $a$: $\Pi(a \curvearrowright x) = a \curvearrowright \Pi(x)$.

We say that the theory $E$ is transitive (in the action sense) if for all $x_1, x_2 \in X, x_1 \neq x_2$, there exists $a_1, a_2, \dots a_n$ such that $x_2 = a_n \curvearrowright \dots (a_1 \curvearrowright x_1)$.

Facts of the symmetries of a system:

1. We know that the symmetries of a theory $E$ form a group.
2. If $E$ is transitive, then each symmetry $\Pi$ is a regular permutation --- If there exists an $x$ such that $\Pi(x_f) = x_f$ (a fixed point), then this implies that $\Pi(x) = x$ for all $x$.
3. Let the action split $X$ into disjoint orbits $O_1, O_2, \dots O_k$ from whom we choose representatives $x_1 \in O_1, x_2 \in O_2, \dots x_k \in O_k$. Then, if $E$ is transitive, there is exactly one action that sends a particular $x_i$ to a particular $x_j$. 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: $S(E) \equiv \{ a_1 \dots a_n : n \geq 1 \text{~and~} a_i \in A \}$.

Now, let $G = Sym(E)$, the full symmetry group of $E$. One can apparently express the symmetry group in terms of: