New video. Published a video about the Born rule/entropy equivalence. While doing that, I forced myself to be as geometrical as possible, so I was able to make the derivations only use euclidean geometry and basic trigonometry! As a bonus, I also found a visual way to see the cosine half-angle formula.
Preparing for the workshop. Most of my time was preparing for the upcoming workshop on non-additive measure in physics. The idea is to have a discussion on how to connect ideas from different fields, more than just presenting results at each other. It will be hybrid, as some invited participants couldn’t come to Michigan. I am very curious to see how (if?) it works out!
Nothing else to report… except maybe that I went to Weird Al’s concert on Friday, and that I have hundreds of e-mail of backlog. I’ll be really glad what things settle down a bit.
I will have to go through the derivation a few more times. But I think it is always important to write down my initial thoughts and later see what I got wrong after going through it more times.
But it makes sense that you are using the Bloch sphere, and the entropy of mixing is a trigonometric proof of mixing between two pure states on the Bloch sphere. I was wondering where the geometry in the other videos came from.
My guess is you get the metric by using ye olde Pythagorean trick when finding the distance between two points on the sphere? As long as both observers can disagree on something like the pure or mixed state but agree on the metric, it should work itself out. But GR also assumes an arrow of time.
In terms of a physical mechanism. A pure state |psi> is a linear superposition of observable states the particle can exist in simultaneously. Never dispute this or Bohr writes nastygrams.
If the arrow of time is an increase in entropy due to mixing states during a measurement, then it is impossible to define causality. The particle would be in all states simultaneously because, without causality, the system is nonlocal. Without a background of entropy, it is impossible to tell the direction two billiard balls came from before they collided and after they reflected. Similarly, a particle traveling faster than the speed of light would arrive before it left; in relativity, this would be spacelike on the light cone, or that literal time arrow of the light cone is zero, but this is called nonlocal in quantum mechanics.
The time-independent Schrödinger equation does spread out the wavefunction over time. The wavefunction diffuses, a phenomenon that can be observed, for example, with repeated and delayed measurements in the quantum Zeno effect. The Schrodjnger equation has the same form as the partial differential heat equation.
https://en.wikipedia.org/wiki/Heat_equation
In a steady state, we get Poisson’s or Laplace's equation, which can be used to describe a lot of physics….
https://en.wikipedia.org/wiki/Poisson%27s_equation
Why does heat diffuse? Entropy. Why does the Schrödinger equation diffuse? Entropy.
The uncertainty principle contains the squared wavefunctions with the observable being measured. The geometric argument for the Born rule -> uncertainty principle. I should note that, with the uncertainty principle, the probability density is real because the wavefunctions are squared.
What is interesting is when the wavefunction diffuses, like after a measurement, in the quantum Zeno effect, the coupled observable also changes. If position uncertainty changes, so does momentum uncertainty. One can decrease as long as the other increases, which to me seems like the second law of thermodynamics, entropy again.
But what I wonder is how does mass density change? Mass is conserved, but mass density changes if the position uncertainty changes. A changing mass density is highly nonlinear, and the linear Schrödinger equation would not model a collapse into the most likely mixed state of mass density.
Maybe I am drawing too much of a parallel between probability density and the likely mass density of a particle or ‘mass-energy’ density of a photon.
A little less Kafkaesque now.