James Camacho comments on Arrows of time and space

[James Camacho]

I think it’s because Schroedinger’s equation is

$$\psi \left(t\right)⟩=e^{iHt}\psi \left(0\right)⟩$$

which, breaking $H$ into its energy eigenstates gives

$$|\psi \left(t\right)⟩=\sum p_n{e}^{i{E}_{n}t}|{\psi }_{E_n}⟩.$$

In Minkowski space, the time $t$ is literally imaginary. What if we rotated it ${90}^{\circ }$ in the complex plane so that it’s just like the other dimensions? This is called the Wick rotation, $it=-\beta$, and we recover the Boltzmann (Gibbs) distribution

$$|\psi \left(t\right)⟩=\sum a_n{e}^{-\beta E_n}|{\psi }_{E_n}⟩$$

or equivalently

$$Pr\left[\psi \right(t)={\psi }_{E_n}]\propto e^{-\beta E_n}.$$

$\beta$ takes on the role of inverse-temperature. Now, what is the entropy-maximizing distribution? Let

$$p_n=Pr\left[\psi \right(t)={\psi }_{E_n}].$$

We want

$$\begin{array}{cc}max\text{entropy}& =-\sum p_n\ln p_n,\\ \text{s.t.}\sum p_n{E}_n& =\text{constant}.\end{array}$$

Lagrange multipliers give

$$\ln p_n=\beta E_n+\text{constant}$$

or the same Boltzmann (Gibbs) distribution we saw earlier. So basically, if we switch to a coordinate axis where time is identical to the other distributions, we find that evolution through time is equivalent to a maximum-entopy distribution in this other coordinate axis.

Now, this is almost circular reasoning, because why is Schroedinger’s equation the way it is? Basically, if you have some observer $t$ (careful here! we’re reusing variable names with different meanings!), and you have some object $\psi$, and you say that $\psi$ looks the same to the observer as the observer changes, that’s written mathematically as

$$\frac{\partial \psi }{\partial t}=A\psi$$

where $A$ is some matrix/​linear transformation. So

$$\psi \left(t\right)=e^{At}\psi \left(0\right)=e^{\stackrel{\text{scaling}}{\stackrel{}{Zt}}+\stackrel{rotation}{\stackrel{}{iHt}}}\psi \left(0\right).$$

After a long enough time, we’re really only looking at rotation. So, why is our “observer” the same as the time dimension? Well, it’s not. But imagine it’s moves mostly in that dimension, maybe around $299,792,458$ times faster, with a little mixing in of the other three. Then the mixing in of the other three will make it so the entropy-maximizing distribution we see in 4 space dimensions should also be entropy-maximizing if we Wick rotate any one of those dimensions.