Classical Action and the Bohm Aharonov Effect

The Bohm Aharonov phase shift for a free quantum particle with momentum p moving through a magnetic potential A(x) with no magnetic force is well known as exp(i Integral dx1 (a,x) e/cA(x1)dx1. In the literature this has been obtained in two very different ways. The first way (standard) (1) is to use the classical equation for a charged particle in an electromagnetic field with electrostatic potential i.e.  (p-e/cA)(p-e/cA)/2m = E with p=-id/dx representing an overall momentum, not just that of the particle. In such a case, there is coupling at the momentum level, but E is the energy of a free particle with momentum p1=p-e/cA (classically) i.e. p1p1/2m. In other words, momentum considerations are treated differently from energy ones. The time dependent wavefunction is exp(-iEt) where E shows only a noninteracting free particle.

    The second approach (2) is the opposite. It ignores the Schrodinger equation and the momentum picture, but retains the idea that for a nonacclerating free particle p/m = velocity=dx/dt. It then considers an interaction which changes E namely  delta E = p e/cA(x)/m. This leads to Integral pA(x)/m dt = Integral pA/m dx/v which yields the same Bohm Aharonov phase shift.

    In a previous note (3) we considered a free particle classical action as A=-Et+px. We argued that Et and px should be kept separate leading to exp(-iEt) and exp(ipx) (not combined) if one creates an eigenvalue equation. We argued that scaling exists in both time (by E) and space (by p). At the same time, the particle moves classically with a speed v=p/m, but has an associated exp(ipx) momentum wave (and independently an energy wave exp(-iEt)). Even if one uses v=dx/dt as in (2) these waves are different i.e. Et= Integral E dt = Integral .5pv dx/v = .5 px  while px is the phase from the momentum consideration. Thus one might argue that it is unusual that a change in E i.e. delta E = p/m A(x) from (2) applied to exp(-iEt) yields the same Bohm Aharonov phase as the Schrodinger equation with p, A coupling, but E left untouched

    In this note, we consider the classical action A= -Et+px and try to explain this feature directly from this expression.