Association of X and P in Classical Physics and Information/Invariance Part II

In part I we argued that Newton’s first law implies that constant momentum (i.e. momentum conservation) is associated with spatial invariance i.e. a lack of information in a certain direction. This is equivalent to stating there exists no force in that direction. We argued that one may apply the generator for translation in the spatial direction i.e. d/dx for x and apply it to a function f(x) such that  df(x)/dx = (px). The simplest form is f(x)= (px)x. We also argued that it is precisely this function which appears in Fermat’s least time principle because  time = distance/speed and one may multiply time by  |p| = momentum magnitude. 

    Given that momentum and displacement are parallel vectors for a light ray,  p dot r = p distance = (px)x + (py)y for a two dimensional problem (e.g. reflection or refraction). “Distance d” is the key function to use because dd= xx + yy implies that for fixed y, d d/dx = x/d = sin(theta1)  (angle measured from y axis) so p d d/dx = (px) which is a conserved quantity if x is invariant. Thus conservation of momentum for two rays is equivalent to extremization.  Extremization, we argue, suggests there may be a built-in physical mechanism which allows a free particle to correct itself if perturbed slightly  just like a Maxwell-Boltzmann gas which extremizes entropy subject to constant average energy does. 

   This built-in mechanism should be linked to motion in space. Writing (px)x as ln(exp(ipx), one not only obtains an eigenvalue equation i.e. -d/dx exp(i (px) x) = (px) exp(i px x), but the notion of a relative change arises if exp(i px x) is interpreted as a kind of probability. A relative change is natural because an absolute one ignores the idea that probability may differ with x even though (px) is constant. 

   Thus from the idea of invariance in space found in Newton’s first law, which is equivalent to conservation of momentum in that direction, the equation d/dx f(x) = (px) leads directly to -id/dx exp(i px x) = (px) exp(ipx x) where exp(i px x) represents a physical probability type of wave which stablilizes the constant (px) motion because this motion is equivalent to extremizing time or |p| time or  p dot r / speed of light in Fermat’s principle. In other words, the quantum mechanics of a free particle is associated with stabilizing wave behaviour exp(ipx x) which was not seen experimentally at the time of Newton or Fermat, but seems to be part of their formalisms, with Fermat’s principle derivable from Newton’s first law.