Quantum Spin as an Operator on a Physical State which Replaces Vector Operations? Part II

In Part I, we considered spin as arising from a desire to replace vector operators (cross product, dot product) in a quadratic equation (continuity equation for a photon, Klein-Gordon equation for a spin .5 particle with rest mass). Here we focus on spin .5 particles and consider the Lorentz invariant: A = -Et+px. If the x-axis lies along the direction of p, then E and p appear as two scalar numbers. If one, however, rotates the co-ordinate system, E->E, but p-> (px,py,pz), and t->t, but x-> (x,y,z). The Lorentz invariant becomes: A= -Et + p(x) x + p(y) y + p(z) z. In other words, p(x) is multiplied by x etc, there is no mixing of p(xy) with say y. In the case of -Et+px with p along x, one has a sum of two number -Et and px, each being the product of two numbers. Under rotation, one still has the scalar -Et, but px is now p dot r. One may ask: If E multiplied by t remains a scalar composed of the product of two scalars, why does one need to introduce a vector dot product, i.e. px → p dot r. Why does not one still have a product of two numbers, just like the Et or px case? In other words, if one wishes to have symmetry between Et and px, a rotation should not break this symmetry. A rotation does, however, occur and p→ (px,py,pz) and x-> (x,y,z). One may suggest seeking generalized numbers (showing the rotation) which keep the form px i.e. the form of two generalized numbers multiplied together. (These arguments apply just as well to the Lorentz invariant -EE+pp =-momo (c=1)), but -Et+px shows the coordinate system explicitly in that it contains x and we focus on it here). In other words, we argue that symmetry arguments applied to -Et+px suggest one should look for generalized numbers sx, sy,sz such that: p-> A+px sx+ pysy +pz sz and x->B= x sx + ysy + sz z with A multiplied by B yields p(x) x + p(y) y + p(z) z. The set sx,sy,sz cannot be whole numbers, but matrices do solve the problem (in particular the Pauli matrices (1)). In other words, p(x) is really associated with a matrix sx and x as well. Lorentz invariants are quadratic and so if sx sx=sysy = sz sz =1, one does not see the presence of the s matrix. If one really has p(x) sx (etc), then given that px is a number, there is a lack of symmetry because sx should be a property as well. A matrix, however, does not seem to be a property, but rather an operator. One may thus postulate the existence of an operator P, which yields the px numerical value, associated with sx which is also an operator. Given that sx iss introduced to deal with co-ordinate rotations, one might expect its associated state to be related to rotations, although it is not clear specifically how. Experimentally, it is seen that a particle with spin .5 couples to a Bz magnetic field, suggesting a degree of freedom associated with current (i.e. with some kind of rotational or angular motion of charge) in one direction or the other, consistent with a 2-vector eigenvector of sz (the Pauli matrix sz)