§ 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.
Let an experiment be a tuple of the phase space , action space ,
and an action of the actions onto the phase space
. We will write
to denote the new state of the system
. So the experiment is the data
§ Coordinate systems.
The existence of the action allows us to
write the evolution of the system recursively:
However, to understand the final state , 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 means introducing coordinates into phase space of which are in triangular form under the action of the inputs of .
§ Conservation laws as triangular form
We identify certain interesting invariants of a system by two criteria:
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 .
For example, consider a system with an energy function . If we
perform an action on the system , then we can predict the action
given just and --- here,
is the action of the system on .
- The parameter determines some obviously important aspects of the system. That is, there is a deterministic function which maps to "measure" some internal state of the system.
- If the values of such a parameter is known at time (denoted ) and it is also known what inputs are presented to the system from time to time (denoted ), then the new value of is a deterministic function of and .
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 of the states of the system such that under action , depends only on and , and not on .
§ 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 . Then we define
is a symmetry of iff:
We say that the theory is transitive (in the action sense) if for
all , there exists
such that .
Facts of the symmetries of a system:
- is a permutation of .
- commutes with the action of each : .
To show that this gives rise to a triangulation, we first construct
a semigroup of the actions of the experiment:
Now, let , the full symmetry group of . One can apparently
express the symmetry group in terms of:
- We know that the symmetries of a theory form a group.
- If is transitive, then each symmetry is a regular permutation --- If there exists an such that (a fixed point), then this implies that for all .
- Let the action split into disjoint orbits from whom we choose representatives . Then, if is transitive, there is exactly one action that sends a particular to a particular . So, on fixing one component of an action, we fix all components .