Lagrangian Extremization from Special Relativity? Part II

   In Part I, we focused on the special relativistic Lorentz invariants EE = pp + momo (c=1) and (E-V)(E-V) = pp + momo (c=1). We argued that these together with the variable switching forms (familiar in thermodynamics e.g. E(S,V) but F=E-TS → F(T,V)) i.e. E = -L1(v) + pv and (E-V) = -L1(v) + pv  lead to p=dL/dv (partial). From this, one may show using the variable switching forms that an equation of motion follows from the extremization of L.

   Here we focus instead on the Lorentz invariant -Et+px (for a free particle) which we write as:   L1(v)t = t(-E+pv) and  L1(v)t = t( -(E-V) + pv) for a particle with a potential. We show again that dL/dv partial  = p from which the extremization of L =L1-V follows. Interestingly, this same Lorentz invariant with another partial derivative linked to p, namely Action = Lt = -Et+px → dAction/dx = p. In this case, the independent variables differ.

  Forming an eigenvalue equation:  -id/dx exp(iA) = p exp(iA) leads to quantum mechanics. This time, in the case of a potential one cannot simply use E→ E-V(x) as many exp(ipx)s are stirred up to create the distribution W(x)=Sum over p a(p)exp(ipx). If one, however, uses -id/dx as a momentum operator (which follows from -id/dx exp(ipx) = p exp(ipx)) and applies this to the Lorentz operator:  (E-V)(E-V) = pp + momo (c=1) then one obtains the Klein-Gordon equation (id/dt partial-V) (id/dt partial -V) W = -d/dxd/dx W + momo W. Thus a quadratic form holds with E-V, but the linear form -Et+px only holds for a single p in the quantum (but not the classical) case.