Question

How do different dirac matrices give same results?

Answer 1

The Dirac equation is associated with a specific transformation behaviour: although it has four components, the Dirac wave function y is not transformed as a spacetime vector. Instead, it is subjected to the so-called spinor transformation: y Sy, which arises because the Dirac matrices gm are supposed to stay invariant after a Lorentz transformation L. The spinor representation: L S = S(L), is defined (unambiguously up to a sign) for L Î SO+(1,3), the proper orthochronous Lorentz group, but it cannot be extended to the group of general linear transformations, GL(4, R) [1-3]. This means that the use of the genuine Dirac equation is limited to Cartesian coordinates. Thus, for instance, the genuine Dirac equation cannot be used to describe the situation in a rotating frame, which is relevant to Earth-based experiments. In such non-inertial frames, one has to use [4-8] the extension of the Dirac equation proposed independently by Weyl [1] and by Fock [9], hereafter the "Dirac-Fock-Weyl" (DFW) equation. However, the DFW equation does not transform as the Dirac equation under a coordinate change: for the DFW equation, the wave function y stays invariant after any coordinate change, while the (ordered) set (gm) transforms as a mixed object which is only partially tensorial [10, 11]. Now one lesson of relativity is that the physical consequences of an equation may depend, not only on the equation itself, but also on its transformation behaviourfor instance, the Maxwell equations do not describe the same physics, depending on whether Galileo transformation or Lorentz transformation is used. Therefore, it is not a priori obvious that, if one neglects gravitation (thus assuming a flat spacetime), the DFW equation is physically equivalent to the Dirac equation.

It turns out to be possible [12] to transform the usual Dirac equation covariantly, with the wave function transforming as a spacetime vector.