SemiQftSchwarzschild

[WrathfulSpatula]

(This is entirely speculative, per earlier work by the same author, but it's optional to this fork of OpenRelativity.)

For a long time, we've had the thermodynamic limit of my hypothetical black hole evaporation in this fork, optionally enabled. Recently, in pull request #46, I added a quantum mechanical treatment, proceeding from the same concerns.

If all we want is the permutation basis expectation value, from the quantum mechanical treatment of my hypothetical quasi-scalar gravity waves, (which I argue arise from removing the assumption of metric tensor symmetry from the theory of general relativity,) we might dispense with a full quantum computation simulation of it, and talk about the statistical distribution of the expectation value directly, at each quantum mechanical step.

In the quantum mechanical limit, I hypothesized in #46 that our exponential "folds" of the size of the space follow the quantum Fourier transform ("QFT") with the successive addition of a uniformly random qubit at each step. Losing exact state information, but attempting to model the expectation value directly, from the first (or "zero-eth") fold of the simulation, we have a series like this:

r_f is the radius at any fold, f, numbered from 0. \sigma_n is a different random real number in the interval [0,1]

Assume this holds to the limit of a Riemann sum becoming an integral:

The \sigma(n) function is a uniformly random distribution scaled to integrate to a real number in the interval [0,1] over any 1 "fold." The MonoBehaviour Update() loop proceeds via finite difference of the partial derivative:

The total time after each "fold", f, is 2^f times the Planck time. The change in the "fold" size is the change in the state.TotalTimeWorld divided by the fold size. We can "semi-classically" simulate in Unity, with this.

This quantum mechanical treatment works up to the thermodynamic limit; eventually, naive quantum mechanics predicts faster growth/evaporation than the Second Law of Thermodynamics allows, (i.e, the quantum treatment can reduce the global system entropy, whereas it must actually increase or stay the same). When the quantum treatment exceeds this limit, we assume that macroscopic thermodynamic considerations of state dominate, and we replace the naive quantum estimate with the macroscopic thermodynamic prediction.

[WrathfulSpatula]

Sorry, there's one problem with my graphics, above. Note that \sigma_0 on the first step is renamed \sigma_1 on the next, and a new step is inserted. These are better, to communicate the idea:

and

[WrathfulSpatula]

I've been thinking about this since I left the keyboard: in code, we're actually sampling "\omega(f)," as defined or implied below:

Critically, "\omega(x)" implies this "exponentially expanding permutation eigenstate reordering" concept I confused myself on above, to map to "\sigma(y)". The map is smooth, but the distributions are not. Therefore, this (smooth) map is not a diffeomorphism, because it is not between smooth functions.

Either "\omega(x)" or "\sigma(y)" can be sampled independently with the same statistics, but they are effectively time-reversals of each other, as the space expands or contracts exponentially over folds.

I think the values implied by sampling either direction of these distributions is any random real number greater than or equal to 0, with uniform probability at any parameter value. By the way, this random number at parameter value has no meaning except under integration; these only qualify as distributions, effectively defined as context-specific inverse histories. Any and all sampled intervals integrated over any domain length 1 must produce a real number in the interval [0,1].

[WrathfulSpatula]

(Pardon my arbitrary and odd aesthetic, but I'd probably prefer to call "\omega" as rather "\chi":

I'll think about it.)

[WrathfulSpatula]

(Call them φ and χ, and I think they correspond to generalized momentum and position field operators.)