Lagrangian/Action Formalism As a Flow Scheme and Spin Part III

 In Part 0 (Parts I and II deal with spin) of this note, we argued that one may begin with two flow equations:

 d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b))

and find that A(x,t) is the classical action for a free particle i.e A(x,t)= L(v) t where L is the classical Lagranian.  In particular, after taking the partial derivatives in ((1a)) and ((1b)) one may set x/t=v. Alternatively, one may link ((1a)) and ((1b)) to eigenvalue equations i.e.

  id/dt exp(i A) = E exp(iA) ((2a)) and -id/dx exp(iA) = p exp(iA) ((2b)). 

Thus operators appear, in particular one has the momentum operator -id/dx associated with px etc.. These two equations may be incorporated in EE=pp+mm or E= pp/2m  (although E and p have different meanings) to obtain the Klein-Gordon and Schrodinger equations. 

   At this point there is no spin. We argued in Part I that spin may be brought in by considering p in ((1a)) as flowing in 3 directions px, py and pz. In such a case one would normally have pi ei where ei is a basis vector, but with the operator equations being used ((2a)) ((2b)), ei is replaced by a basis operator ai (a matrix) such that when pi ai is squared, ai disappears and one obtains pipi = -d/dx d/dx - d/dy d/dy-d/dz d/dz As P. Dirac has shown this leads to 4x4 matrices containing the 2x2 Pauli matrices. As a result, the independent px,py,pz flow is only linked to spin 1/2 . In part II we discussed other spins. Other spin matrices follow different constraints.

    In this note, we wish to link the idea of basis operators ai with the idea of angular momentum i.e. rxp. In (1), the spin of a fermion is calculated using current from the Dirac equation J (vector) together with r x J i.e. the classical form of angular momentum. In this note, we wish to show the link between this approach and the basis operator one,  which does not  on the surface, bring in any idea of angular momentum, are the same.