Lagrangian and Fisher's Information Extremization (EPI) and Information

For the case of a free particle, it is known that one may extremize a Lagrangian L=.5mvv, where v=dx/dt, to obtain:  d/dt dx/dt = 0. In this classical case, one is interested in x and t or x(t). One may think of moments i.e. powers of derivatives of x(t). Given the first moment is equal to a constant, all higher moments are 0. Each moment yields a piece of information. Thus there is one piece of information, namely the constant velocity v=dx/dt. A second derivative d/dt dx/dt =0 demonstrates there is no further information. The form d/dt dx/dt, however, is associated with an extremization of a function, namely L=.5mvv or Integral .5mvv dt. Thus we argue that extremization is really linked to information found in the problem for the free particle case.

   Next, we consider .5 Integral dt vv = .5 Integral v dx.  This is proportional to Integral p dx where p represents a physical impulse. The equation d/dt dx/dt =0 shows there exists only one piece of information, regardless of the value of v. In other words, all v values satisfy this equation (where v=constant). In analogy, if one wishes all p values to satisfy Integral p dx, p = constant, without distinguishing p, then x= constant/p. This, however, is the quantum wavelength. If it is repeated, in space, one desires a periodic function in px which has a modulus of 1, i.e. an eigenfunction of -id/dx namely exp(ipx). This does not follow from a Fisher’s information type variation, but rather from the idea of information and two quantities of interest, namely a kind of probability exp(ipx) and x.  The modulus of 1 is equivalent to the single piece of information present i.e. v=constant.

   To see Fisher’s information enter the quantum picture, one may consider a particle in a box with infinite walls. In such a case, exp(ipx) and exp(-ipx) are present or cos(px) if a(p)=a(-p). There is one piece of information, so d/dx cos(px) = function(px). function(px) is the one piece of extra information like v. Finding a second moment cannot yield any further information i.e. d/dx d/dx cos(px) = constant cos(px). If it did not, there would be extra information in the system, but v=constant is the only information. d/dx d/dx cos(px), however, follows from the extremization of Fisher’ s information subject to the constraint Integral cos(px)cos(xp) CC = 1. Thus we argue that in the free particle case, extremization really represents moments (shown by derivatives) and information.

  We also consider the case of a potential V(x) present.