Pauli Matrices Versus A Vector Dot Product/ Square Root Combination Part II

  In Part I, we argued that the square root of a positive real number bb has a degeneracy, i.e. -b and +b are solutions. Usually one adds +/- by hand, but we argued that there may exist a mathematical formalism which automatically carries this degeneracy. We suggested this formalism involves the three 2x2 Pauli matrices, each of which has eigenvalues -1 and +1 associated with this degeneracy for 3-vector components.

    Here we suggest that a positive number bb upon which the square root acts, may be thought of as a vector dot product. In other words, b is the vector magnitude, but in a rotated frame one would have (bx,by,bz) and each of these carry a degeneracy of +/-. A dot product is formally defined as:  Sum i,j  bi bj delta(i,j), but this equation forces one to add +/- by hand when taking the square root. It simply ensures that there is no mixing of i with j (for i not equal to j). 

  The relationship for Pauli matrices:   { s(i), s(j)} = 2 delta(i,j) I  ((A))  (I = identity matrix and {} means anticommutation) may be used to show that the dot product has removed information related to the +1/-1 degeneracy..In other words, information inherent to a Pauli matrix, namely its eigenvalues of +1 and -1 disappear in ((A)) and so I the 2x2 identity matrix has 2 eigenvectors with the same +1 eigenvalue. This is related to the product of two identical Pauli matrices so +1 * +1 = -1*-1=1. In other words, delta(i,j) used in the dot product explicitly loses degeneracy (+/-) information.