spacetime

The path in yellow is multiplied on both sides by all 16 combinations of the basis vectors on the left and the right. An Italian physicist named DeLeo figured out how to map the gamma matrices (also referred to as the Dirac matrices) to the triple product ("Quaternions and Special Relativity", J. Math. Phys., 37:6, 2955-2968, 1996). A team in Mexico, José López-Bonilla, L. Rosales-Roldán, and A. Zúñiga-Segundo detailed the process - and made me aware of the connection via email. The gamma matrix machinery can be hard to appreciate, there being all kinds of combinations of matrices and spinors that play roles. With quaternions the story is much more straightforward: multiply on the left and the right by (1, i, j, k).

Let's look at 1 triple product, i (t, x, y, z) k = (z, -y, -x, t). The algebra is simple, but the results are odd. This function swaps the value of time into the z position, and visa versa. The values of x and y trade places and signs. While you and I might like to consider values for time and space to be solid, in relativistic quantum field theory, Nature has a need to take what ever numbers are in the house and systematically shuffle them so that the sum of all possible paths can be calculated. This animation shows four points coming in form four directions, all paths possible with these 4 numbers. The 16 paths can be seen. Now the work done by the 16 Dirac matrices does not seem so utterly abstract.

Complex numbers have some freedom to move in space. The 3 straight lines indicate a choice of Cartesian basis vectors, but other basis vectors could have been chosen. Complex numbers are used extensively in quantum mechanics. Yet the obvious limitations in the animations suggest we should rebuild the foundations of complex-valued quantum mechanics. Sounds like a lot of work!

The real number -1 in blue is added to the real number 3 in yellow to generate the real number 2 in green.

The global economy is fueled by real numbers. This is somewhat frightening when you realize how excedingly dull real numbers are in an animation. They are paralyzed, able to exists in only one place, the origin 0, 0, 0. Their variation comes from blinking at different times from now. Minus real numbers are in the past, positives in the future, and only one point may exist at the origin at time zero or now. This set of blinking lights is totally ordered: a real number at the origin will either be before, after or at the same time as another point.

The input is in yellow, going from (-5 -5 -5 -5) to zero. The space reflection in blue, (-5, 5, 5, 5) to zero, is a mirror operation around the origin. The time reflection, (5, -5, -5 -5) to zero, in green requires you recall how the yellow input collection of events came onto the stage, so the green back it out. The reflection of both time and space, (5, 5, 5, 5) to zero, in red looks like a continuation of the yellow. Deep in quantum field theory they tell the odd tale of antiparticles going backward in time looking like particles going forward in time. Such an animated story now looks more reasonable.

Time requires memory, space needs mirrors. These do not look the same animated. If animation starts off like it ends up, then time reversal is in play. While time reversal can involve a minimum of one event on a screen, space reflection requires matching pairs of events. In math, imaginary basis vectors are represented as a 90 degree rotation in the complex plane. There is no difference except in label between real and complex numbers. The complex plane misleads. Real numbers have a graph that is undirectional - one times one is one - so real numbers can live without the imaginaries. Imaginary graphs have a directional graph - 1 times i is i, while i times -i is 1 - and there is no way to do multiplication with only an imaginary basis.

The look of time reversal is familiar. It requires a linear sequence of ordered events to be played back in exactly the reverse order. Time represents the real numbers, a totally ordered set. This set property grants the ability to run things backwards.