## Its about time . note

### Note

This entry is concerned with the notion of time and is a spinoff of On physical units . note.

##### Observation

A core assumption of physics that's easy to adopt is that we are able to enumerate some moments in an ordered fashion.

$t_1, t_2, t_3, ...$

Usually they are considered as point on some real axis, but I guess the only thing that really matters is that this enables us to count (other) sorts of events between such moments. If indeed the moments correspond to real numbers, $t_i, t_j$ say, then $\dfrac{1}{t_j-t_i}$ is what we want to call a frequency and as such a unit for all kinds of rates.

##### Energy as frequency

In On physical units . note we consider the Schrödinger equation

$h_\Psi\cdot\dfrac{{\mathrm d}}{{\mathrm d}t}\Psi = (-2\pi{\mathrm i})\,H\,\Psi$

and note that if $h_\Psi$ is chosen unitless, the conserved quantity of the system (what is also called energy) is a frequency. The cost associated with a process is basically inversely proportional to how quick it takes place, so that Reaching higher frequencies is more expensive. But for a more precise desciption in QM, one must look at the state spaces in the models. A Hamiltonian $H(\langle q,p\rangle)$ is like an $\langle q,p\rangle$-indexed list of prices. A Hamiltonian operator $H$ acting on state vectors in QM is like a stamp with prices.

To understand many formulas, units like e.g. mass ($kg=s/m^2$, after setting $\hbar$ unitless) can or should be multiplied by natural constants (e.g. $c^2$) so that they represent a frequency or time scale.

##### Canonical variables

If

${\mathcal H} \propto {\dot q}\frac{\partial {\mathcal L}}{\partial {\dot q}} \equiv {\dot q} \, p$

is supposed have the unit of a frequency, then

$[p]=\dfrac{1}{[q]}$.

That's also clear from

$[q,p] \propto \hbar$.

##### List of quantities that multiply to frequency/energy

(this is a temporary list for me to read formulas in terms of units)

Electrical field $[E]=\dfrac{1}{[t][x][q_x]}$ and dipole moments $[d]=[x][q_x]$.