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.


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 $x\in{\mathcal B}$


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

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

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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