===== notes on physical theories . note ===

this is a temporary entry to link all the physics notes to, so I find them later, when I develop the theories proper.

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>https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions

>todo:
Set up mechanics (QM and classical) with a maximum number of units and all unit conversions as functions, 

E.g. $c$ with $[c]=m/s$ and possibly realized as 

$\tau_c(x) := \tfrac{1}{c}x $ 

with $x\in{\mathbb R}$

A noninvertible variant of the above would be

$\tau_c(x) := \tfrac{1}{c}||\,x\,||$ 

with $x\in{\mathcal B}$

or 

$\tau_c(\gamma) := \tfrac{1}{c}\int{\mathrm d}\gamma$

with $\gamma\in{\mathcal M}^{\mathcal I}$

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The notions of *time* and *probability* are philosophically difficult. When I ask myself the question "what's the most fundamental physical unit", I think my answer would be "quantity per time", i.e. e.g. Hertz. This is because the dynamics $F$ of a theory (Newton equations, Schrödinger equation, some rate equation,..) is determining just such a quantity. Granted, statistical physics doesn't quite fit this, but then again, or maybe because of that, I view statistical physics more to be combinatorics applied to physical models than a theory of physics itself. It produces physical results, yes, but so does combinatorics applied to anything in the world.

>Regarding indexing problems (in reference to the subsection "Classifying definitions" of [[On reading]])
>The situation

$$
\require{AMScd}

\begin{CD}          
E           
\\ 
@V{p}VV
\\           
B
\end{CD}
$$

>often presents the task to compute an inverse for the sake of $B$-indexing subsets ($\in{\mathcal P}(E)$) of $E$. 
>This is the case in physics, specifically kinetics in statistical physics, with 
$$
\begin{CD}          
X           
\\ 
@V{\Psi\,\mapsto\,\langle H\rangle_\Psi}VV
\\                
{\mathbb R}
\end{CD}
$$

>If $E_{\Psi_0}\in{\mathbb R}$ for ${\Psi_0}\in X$ is in the Image and conserved for dynamics $F$ in $X$, then the inverse Image of $\Psi_0$ are accessible points.

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