Dirac Matrices and the Pythagorean Theorem

[Bob Dylan]

I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross terms cancel because the matrices square to one and anti-commute. I can follow the mathematics; however, I don't understand this at a more visual, intuitive level. Is it possible to retain the understanding of these terms being on a triangle? If so it seems like A^2+B^2=C^2 has gone to A+B=C and I don't see how that could describe any right triangle.

Please help me understand, as visually as possible, what's happening to Einstein's triangle as the Dirac matrices are applied.

 

[mikeyork]

Think of your A and B as orthogonal vectors. What do you get if you add them?

 

[Bob Dylan]

Got it. Thank you.