Gamma Matrices


The gamma matrices are a tool to systematically look through all possible paths through spacetime given 4 numbers.


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.

q_graph -out gamma -dir gamma -box 2.5 -loop 0 -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000' -color yellow -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000| gamma -almost' -color blue
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Outside video: 

( 1 |  i |  j | k) (t,x,y,z) ( 1 |  i |  j | k)

Physics Tag: 
quantum field theory
gamma matrices
Programming Tag: 
command line quaternions