## Tuesday, September 11, 2012

### Current understanding of physics

I thought I'd share my current interpretation of how I think relativity and quantum theory work, from a high level. Maybe I'll get time to study them more in future, but here are my current thoughts.

### Special relativity

Time is simply an imaginary spatial dimension, so its square value is negative. This is a sufficient premise for describing Minkowski space, time dilation, constant speed of light etc. The rest of the theory is just making mass/electro-magnetic equations etc consistent with this single premise.
Why don't we have galilean (old fashioned) relativity? perhaps because a finite communication speed is more general than a single infinite one, or maybe unbounded communication speed leads to circular dependencies, so cannot be consistent.

### General relativity

Think of space-time as a 4d space populated with some 4d vectors (with units of momentum). The distribution of these vectors requires a matrix to define them for each little volume of space-time. Much as a stress tensor defines the stress on an object, this 4x4 matrix defines how dense the vectors are in each direction. If you think of the volume of space-time as a tesseract (4d cube) then the matrix defines the total vector amplitude for a given axis against each cubic side of the tesseract.
This density matrix defines the curve of space-time at that particular point. The curve is the second differential in space of the volume of the space. This curve is represented by a matrix, called the Ricci tensor, Einstein removes a proportion of the trace of this matrix in order to prevent any torsion of space. I believe this is a principle of least work.
So, if we know our distribution of 4d vectors, we calculate the curvature matrix at each point and we have a space where the second differential is known. An iterative method could take these curvatures on a flat Minkowski space and converge to the warped space-time, which is defined by a matrix at each point (metric tensor). This matrix is 4x4 and looks like a -1,1,1,1 identity in Minkowski space. Bent space-time will deviate from this.
I think of this as like the 4d vector equivalent of a thermal distribution, where most of the space wants to simply be the mid temperature between its neighbours. That is a scalar example, for a 4d vector you need to work with matrices as this does.
So what are these momentum vectors? they generalise force and mass, if they point in the time direction they are like mass, which moves objects due to gravity, if they point in a spatial direction they are like a pressure and they move objects like a force. Since the curvature matrix is just proportional to the density matrix, a time-pointing vector (a mass) will cause a bend in time, and a space pointing vector (a force) will cause a bend in space. Though it is more of a mixture than this, due to subtracting the trace from the Ricci matrix, I think.

Now that we have a metric for the whole of our space-time, we need to iterate to move the mass distribution, it should shift to only be along the geodesics (locally straight lines in the bent space-time). I'm not so clear how this is best done.
Of course this moves the mass distribution so you have to re-iterate the whole thing continually until it converges to a valid Einstein universe.

### Quantum theory

I personally don't see anything weird about quantum theory. It is a set of differential equations (much like most physics theories), the main difference being that it operates on complex numbers rather than the reals. This small difference means that values don't just accumulate, they can also annihilate. The path-integral formulation shows the wave-particle duality in action, a single interaction on a particle is integrated over all space-time and all possible interactions to produce a field (the wave aspect). This is really no different than classical physics formulas using calculus, since differential equations are defining the rules at a single point and requiring they be integrated over space-time to form a field. So no real difference there (apart from the use of complex numbers).
The remaining aspect that people say is different is that the integrated values only define a probability of an effect, rather than an amplitude. This is not actually the case in my opinion. When a simple object like a photon hits a macroscopic object like a recording device, the complex waves decohere and average out to effects that are more classical (amplitudes), the resulting state of the universe contains a spectrum of versions of the same scientist that see the detector light up at different points. Hence it is only natural that any single observer thinks the results are random after repeating the experiment many times. In other words the many-worlds theory seems most sensible and (apart from state-space not occupying a single time-line) the physics of quantum theory is just another set of equations like all the others.

### Combining

Since complex numbers are complete, it seems to me that general relativity will need to include complex numbers to work together with quantum theory. This was tried to some degree of success in Twistor Theory, but this theory uses twistors which can't handle arbitrary 4x4 deformations, so can't really get beyond special relativity.
It is true that a 1,-1,-1,-1 metric space is just as valid for general relativity as a -1,1,1,1. This is a rotation by 90 degrees of having i,1,1,1 as the components of time and space. It should follow that any rotation is equally valid, so we have a free parameter in general relativity where quantum objects can be stored... perhaps.

### Extending

One problem with these theories is they assume the existence of real numbers, and/or continuums. An improvement would be to define each point in terms of larger and smaller points, so that a continuum emerges in the integration. This is different than normal calculus, but similar to covolution mentioned in a previous post. It also allows certain rules to _not_ produce a continuum but rough (as in fractal) distributions.