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

 The photon spin matrix eijk (Levi-Civita symbol with i representing the matrix identifier and jk, its elements) follows from the continuity equation:   d/dt (partial) {aEl dot El+bB dot B) + grad dot  c El x B  (where El and B are electric and magnetic field  vectors). The strategy (as shown in (1)), seems to be to replace the vector operator x (cross product) with a matrix operator which acts on (El+iB)  (vector). This matrix also has eigenvectors and so represents an operator. Thus -id/dx partial and id/dt are operators which act on a probability distribution exp(-iEt+ipx) which has physical implications and spin seems to be associated with an operator which also may act on a physical state. In other words, a physical state or degree of freedom gives rise to vector operator properties and does not simply pull out a number like id/dt partial yielding E for a free particle.

    We try to apply this idea to the Klein-Gordon equation  -d/dt d/dt parital W(r,t) + grad dot grad W(r,t)= -momo W(r,t)  (c=1). We argue that one wishes to replace the vector operation: dot product between grad and grad with an operator (matrix) which acts on a physical state (spinor). In other words, we argue that the vector operation of the dot product (like the cross product ) in the photon case, is caused by an operator (which acts on a physical state). We note that one has quadratic type equations, so the one linearizes, with the operator, i.e. matrix, appearing in each expression. Thus one obtains anti-commutation relationships when creating the quadratic form by multiplying the two linear expressions without using any vector operations.

 In other words, it is the presence of a physical state or degrees of freedom which automatically create the constraints introduced by the vector operator.