Pauli Matrices Versus A Vector Dot Product/ Square Root Combination

 After the discovery of id/dt partial and -id/dx partial as energy and momentum operators in quantum mechanics, Dirac linearized the Klein-Gordon equation to obtain 4x4 matrices based on Pauli 2x2 matrices. As a result, a quantum spin degree of freedom emerged from theory, and matched experimental results.

     Given the Lorentz invariant form: =  -EE + p dot p = -momo  (c=1), one may mathematically write:  E = +/- sqrt( p dot p + momo), which is standard procedure. One may ask: What is wrong with this expression? We argue in this note that it is not incorrect, but rather incomplete mathematics for the reason that one has to insert +/- by hand. We suggest that a complete mathematical description of the square root of a dot product p dot p should automatically give rise to two solutions associated with +/-. Thus, Dirac’s linearization is not simply a procedure for finding spin, but, we suggest, is a more complete mathematical approach to describing the square root of p dot p, which holds even if one does not consider physical situations. Interestingly, nature seems to choose this more complete mathematical description rather than the looser one using the sqrt function and the manually inserted +/- signs.