Free Particle Action, Resolution and Invariance

The nonrelativistic classical action of a free particle is A=.5mt vv is equivalent to time multiplied by a constant. v=dx/dt =constant for free motion and infinite x resolution is assumed. One may write v=x/t so that A=.5mxx/t.. Assuming x and t are independent which they are not in Newton’s scenario yields dA/dx partial = p where p is momentum. A=.5mt vv with v=constant, however, leads to dA/dx partial = 0, thus there is a dilemma. To resolve this dilemma we assume a finite unknown resolution to x, but let t have infinite resolution.Thus we write v = Dx / delta t. where Dx is the unknown x resolution distance as the smallest x interval distance allowed. In other words Dx does not tend to 0 so v is really an average speed within this resolution scheme. One may take a number line and mark off Dx points from the origin, but the origin may be shifted. In other words there is x invariance in this scenario which suggests a constant in this direction related to this invariance. The question is: What is this constant? One might consider either velocity or momentum (which is nonrelativsitically m (rest mass) multiplied by v. One, however, wants a theory which is the same in a lab frame as in a frame moving with a constant velocity and (p,E) and (x,t) are 4-vectors in special relativity, while v is not part of a 4-vector. Thus we consider a p as being linked to an invariance in the x direction for nonelativistic motion. Alternatively one may note that Fermat’s principle associates d/dx i.e. spatial invariance in the x direction with constant momentum in this direction. Fermat’s principle, however, applies to light so perhaps there is a similarity between light and a free particle with rest mass. We also suggest that spatial invariance is linked with probability i.e. the probability to be at one x versus another. Both Lagrangian and Hamiltonian theory deal largely with time derivatives. One may note that dv/dx=0 for a free particle, but we are interested in a function different from v, one that yields the constant of motion associated with invariance. Weseek a function A such that dA/dx partial =p where dx is smaller than Dx i.e. may tend to zero. The classical action written with v=Dx/Dt written as x/t for simplicity with Dx and Dt being independent within the resolution ranges. In Newton’s scenario the resolution ranges tend to zero so one never has x,t independence, but with fixed resolution one does. Thus dA/dx partial = p describes the x invariance. In particular holding t constant (say t=0) yields A=px such that the invariance in x is described by nb/p where b is a constant and n=1, 2,3 … This suggests a resolution scheme for the invariant x i.e. Dx = b/p. Then v= Dx/ delta t. dA/dt partial = E which is a constant and so this also suggests an invariance in time. For x=0, A=-Et suggesting a periodicity in t of b/E. This is independent of b/p except for the constant b which ensures that A=0. We focus in this note, however, on resolution b/p in the x direction and its link to b/p length units which define Dx. Thus by forcing the notion of a finite resolution in the x direction, unlike the Newtonian scenario, one is forced to think of p (momentum) as not simply being mv (nonrelativistic), but as related to an invariance in the x direction. The function defining this invariance, namely A the classical action, also serves to define Dx, the x resolution suggesting a periodic scheme in space similar to that found in light i.e the periodicity in the electric and magnetic fields cos(-Et+px). Thus even though A = .5m Dx Dx/ delta t, and Dx/delta t = v, one may vary Dx within the resolution range by a tiny amount dx which does tend to zero and leave delta t unchanged. This demonstrates the invariance in the x direction of shifting Dx points which is associated with the conserved quantity momentum as shown by dA/dx partial = p. These same results hold for a relativistic free particle with A= -Et + px, but E = mocc/ sqrt(1-vv/cc) and p = mv/ sqrt(1-vv/cc)