Loss of Information and Classical Free Particle Action

  Classical mechanics is often concerned with the variables x and t, especially their relationship x(t) in describing a moving particle. Change in p(momentum)= Integral Force(x) dx and Change in KE =Integral dt Force(t) also appear, however, and each of these integrate out one of these variables, so p is linked with x only and KE with t only. Furthermore there is information loss as KE=pp/2m is the same whether the particle moves to the left or right and p=m1v1=m2v2 so with time gone one does not know the velocity. The association of KE with t for a free particle and p with x may be made stronger if one thinks in terms of probabiliites i.e. P(p1,x)P(p2,x) = P(p1+p2,x) and P(p,x1)P(p,x2)=P(p,x1+x2). We argue this leads to P(x,p)=exp(ipx) and a similar result of exp(iEt) or exp(-iEt). One may then note the similarity of these two terms px and +/-Et with the free particle action=Lagrangian*t = -Et+px which holds for both a relativistic and nonrelativistic particle, where v=x/t. 

     Does the loss of information appear in nature? The (p,x) loss i.e. p=m1v1=m2v2 does in a two-slit experiment whose solution does not require time and is calculated using exp(ipx). Thus we argue that interactions are based on p and do have a loss of information. A bound quantum state, however, has interactions with V(x) and so exp(ipx) with its information loss should be present. A bound state, classical or quantum mechanical, however, has the same Etotal at each x, so (Etotal, t) is again a pair with x gone and loss of information means that an energy description (i.e. the time-independent Schrodinger equation) does not distinguish between forward and backwards motion. This is not needed to calculate energy values, but physically this distinct motion may exist, especially for high energy values.