Lagrangian/Action Formalism As a Flow Scheme and Spin

Date on paper should be 2022 not 2021.

 In a previous note, we argued that for a free particle there exists a function A(x,t) such that d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)). In particular, we used p=E(v) v for the relativistic case and  p=mov. We showed that A(x,t) is the classical action for a free particle  Integral dt L = T L with v=x/t before the derivatives are taken. After, x/t is set to v. As a result, there is a nonrelativistic A(x,t)= T .5mvv and a relativistic one A= -mo sqrt(1-vv) (c=1). We argued that ((1a)) and ((1b)) are in the form of flow equations. Energy E flows through space out of a dx region (i.e there is an energy flux Ev) so if x is held constant, the change in time is -E. We also noted that one may obtain both the Schrodinger and Klein-Gordon equations from ((1a)) and ((1b)) (see reference in (1)). As a result, A(x,t) is linked to both the nonrelativistic and relativistic free particle actions, yields free particle quantum mechanical equations and may also be used to obtain the form of special relativity (as discussed in the reference in (1)).

    In this note, we wish to consider spin which is not present in ((1a)) and ((1b)). For both relativistic and nonrelativistic classical dynamics which are linked to A(x,t) one deals with numbers (scalars) and so no notion of spin arises. In obtaining both the relativistic Klein-Gordon equation and nonrelativistic Schrodinger equation from ((1a)) and ((1b)), one creates an eigenvalue equation i.e.  id/dt exp(i Action) = E exp(iAction) ((2a)) and -id/dx exp(i Action) = p exp(i Action) ((2b)) which only hold for the free particle. One may then use E and p in expressions EE = pp + momo (where p=Ev= dAction/dx partial) and E = pp/2m + V(x) where p=mov=dAction/dx (partial). For the free particle, one has ((2a)) and ((2b)) without establishing the Klein-Gordon equation. Given that p=Ev is a vector, one may try to write ((2b)) as a vector, but given that one has operators acting on exp(i Action) one does not use a basis vector ei (ei dot ei=1), but rather a “basis” operator or a spin matrix. Thus, we argue that spin emerges in this way without having to linearize the Klein-Gordon equation. Interestingly, this approach only holds for fermions. For the photon, which is a boson, a Dirac-like equation may be obtained directly from a d/dt (partial) EL EL + BB + b grad dot  (El x B) = 0 as shown in (2) (with a,b being constants). This again follows from a flow scheme.