Locality

AVI posted a link to a Steve Hsu podcast (sadly without the slides) about using AI as a kind of "idea generator" linking concepts in physics together -- untrustworthy, but sometimes says something worthwhile. He wasted time on some of the ideas, and found another useful -- and published a paper on the result.

By using AI, he means using several different AI systems, and then cross checking them. If they converge, there might be something useful there. Or not.

Anyhow, the useful idea was based on one of his own papers which showed that a non-linear version of Schrodinger's Equation was going to be "non-local" too: namely that regions that are distant from each other would be correlated/entangled instantaneously -- before light could travel between them. (To be clear, he works with the field equations, since that's simpler for his plan.)

That sounded curious. Quantum mechanics does seem to be linear--at energies below those where general relativistic effects would matter. We don't know what happens when GR and QM have to play together, but non-linearities seem likely to me (admittedly not an expert in that particular field).

The paper discusses non-linear models that involve powers of the wave function. Recalling that the wave functions are going to be linear in the sense that if A is a solution and so is B, A+B is too. If the wave function enters the equation as, for example a linear term plus a square, that square term will couple near and far components automatically. E.g. if "n" represents the near part and "f" the far-away part, (n+f)^2 will have terms like n*f and f*n, connecting near and far from the get-go.

That's the simplest way to put a non-linearity in, but it doesn't seem the most likely, if only because it will automatically ruin locality. Physically, you'd expect something more like a "back-reaction" non-linearity, where the energy of the wave pushes on the vacuum, which "pushes back." For example, an electric charge in space results in an electric field in which there's a non-zero probability of pair-producing (temporarily/virtually) an electron and a positron, which briefly interact with the original charge. Hawking showed that this can be non-trivial for gravity and black holes.

That would give a non-linearity restricted to the effects local to the history of the wavefunction. If one electron has been sitting here and another on Alpha Centauri, if they haven't been there long enough for light to reach from one to the other, the local volume that light can have reached and returned would represent, in my naive model, the volume of the wave function that could contribute a non-linear effect to the electron "here." The Alpha Centauri's contribution is nil until enough time has passed. (And of course, at such a distance the effect is utterly trivial, but it's the principle of the thing.

Now you will ask if I will "put my money where my mouth is" and write an equation for an example. Let me get back to you on that. You'd think in the simplest case one could add in a term like $\alpha \int_{t_0}^0 \int dA \psi(\vec{x}, t-t_0)$ where $A$ is the shell about the given point at radius $c(t-t_0)$, $\alpha$ is some small constant, and $t_0$ is the creation time of the wave function. But I can see that's likely to be bit messy, especially with inserting a "creation time" boundary condition.

I'll play around with it a bit and see what happens.