Special Relativity, Spinors, and the Dirac Equation

The crucial Lie group for special relativity is the Poincaré group: all transformations of Minkowski 4-space (spacetime) that preserve the Minkowski pseudo-metric. The Lorentz group is the subgroup that leaves the origin fixed, and the proper Lorentz group is the subgroup of orientation preserving Lorentz transformations. The proper Lorentz group in turn contains the rotation group of 3-space, $SO(3)$.

Just as $SU(2)$ is the double cover of $SO(3)$, so $SL(2)$ is the double cover of the proper Lorentz group, where $SL(2)$ is the group of unimodular $2\times 2$ complex matrices.

Say $A$ is in $SL(2)$ and $v$ is in ${\bf C}^2$. It turns out to be important to pry the mapping $vv^*\mapsto Avv^*A^*$ apart into $v\mapsto Av$ and $v^*\mapsto v^*A^*$. The vector $v$ can be pictured as a geometric object consisting of a vector in space (rooted at the orgin) with an attached ``flag'', i.e., a half-plane whose ``edge'' contains the vector. Moreover, if the flag is rotated through $360^\circ$, $v$ turns into $-v$! (Recall the earlier remarks on untangling threads.) Such an object is called a spin vector. And just as one can create tensors out of the raw material of vectors, so one creates spinors out of spin vectors.

Dirac invented spinors in the course of inventing (or discovering) the Dirac equation, the correct relativistic wave equation for the electron. As it happens, the $\sigma$ matrices are not enough to carry the load; Dirac had to go up to $4\times 4$ matrices (called the Dirac matrices). The $\sigma$ matrices are imbedded in the Dirac matrices.

I won't repeat the story of how Dirac discovered antiparticles. Nor the story of how he rediscovered knitting and purling (see Gamow's Thirty Years that Shook Physics.)

© 2001 Michael Weiss

home