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I Dirac Matrices and the Pythagorean Theorem

  1. Nov 8, 2017 #1
    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.
  2. jcsd
  3. Nov 8, 2017 #2
    Think of your A and B as orthogonal vectors. What do you get if you add them?
  4. Nov 8, 2017 #3
    Got it. Thank you.
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