Not quite; the values at every point are boolean, not floats.

(From the link) The algorithm is explained here: http://www.youtube.com/watch?v=iyTIXRhjXII

It's confusing because several algorithms are explained, but I believe they're actually floats in this version.

Also it's not a normal PDE because ∂f/∂t is permitted to be an arbitrary functional of f restricted to a fixed-diameter neighborhood of the point, not just an infinitesimal neighborhood.

That's the case for our universe as well, for example d^2x/dt^2 due to gravity for a piece of material is dependent on all other material in the universe.

Not instantaneously: https://en.wikipedia.org/wiki/Speed_of_gravity

So the Schrodinger equation isn't quite accurate then?

Does the Dirac equation take this into effect?

You can think of modern physics as a bit like a cube:

      1/3_____1/2/3
      /|      /|
     / |     / |
    1------1/2 |
    |  3____|__2/3
    | /     |  /
    |/      | /
    0-------2

0) At the bottom left corner you have "classical mechanics" - the theory that explains pendulums, bouncing balls and spinning bodies.

It branches out along the three axes:

1) Add mutual gravitation, giving you the theory of Newtonian celestial mechanics (in which gravity acts instantaneously)

2) Add relativistic effects, giving you Einstein's theory of special relativity (there is a finite upper limit to all communication, the speed of light)

3) Add quantization, giving you 1920s era quantum mechanics, as described by the Schrodinger equation.

We know how to combine any two of these:

1/2) Combining gravitation and relativistic effects gives you Einstein's theory of General Relativity. In this theory, gravitational effects travel at the speed of light.

2/3) Combining relativity and quantum mechanics gives you quantum field theory and the Standard Model. This encompasses the Dirac equation, which is a quantum relativistic theory of fermions (i.e. matter particles).

1/3) Combining gravity and quantum mechanics gives you... well, it's kind of boring and we don't talk about it much, but you get a quantum theory with gravitation, but no relativistic effects. No one really studies this.

Combining all three is the holy grail of physics:

1/2/3) Often called `quantum gravity` or the `theory of everything`, this is the as yet nonexistent theory that can explain both very small and very massive (in the sense of having a large mass) objects, like black holes or the early universe.

Could you elaborate a bit on your point 1/3? How come the little interest in this subject? Even though it contains no relativistic effects it would seem to have some importance in filling out the complete 'cube' of theories?

The equations you end up with are called the Schrodinger-Newton equations [1]. These are the same as Schrodinger's equation, but with an extra term that couples the wavefunction to a global gravitational field via its mass.

I'm not an expert, but possible reasons for the relative lack of interest in these equations include:

1. It doesn't produce many interesting predictions (possible exception: it might be useful for explaining how gravitational effects can induce wavefunction collapse, but this appears to be highly speculative.)

2. There isn't a natural domain of applicability. For example, combining 1/2 (gravity and relativity) has a natural applicability to things that are heavy and move fast (i.e. stars, galaxies, the universe). Combining 2/3 (relativity and quantum mechanics) applies to things that are small and move fast (electrons and other fundamental particles). The domain of applicability of 1/3 would be things that are small and heavy, but move slowly. I can't think of any examples of things that fit the bill (note that 1/2/3 applies to things that are small, heavy and move quickly, i.e. black holes).

When I say "move quickly" here I don't necessarily mean that the object you're modelling must be moving quickly - just that there are speeds in the problem that are appreciable fractions of the speed of light.

[1] http://en.wikipedia.org/wiki/Schr%C3%B6dinger%E2%80%93Newton...

There are a few experiments that combine Newtonian gravity and quantum mechanics (1/3).

For example you can split a ray of neutrons, direct each beam throu a different path with different height and then make them collide and see the interference pattern. (The details are in the book of Sakurai "Modern Quantum Mechanics" pp127-129, with data from an experiment of Colella, Overhauser, Werner (1975).)

It is possible to create systems that combine 1/3, but they are almost corner cases and most of the time the other combinations are more useful.