Lack of Information / Invariance / Conservation and Reflection/Refraction

The idea of using a lack of information to solve a physical problem seems to often be linked to statistical mechanics. In particular, if one has a Maxwell-Boltzmann gas in a box with no potential, one may ask: Is there any specific information which would suggest a particle prefers one position point to another? There does not seem to be any hence density is constant in the gas. In other words there is a conservation law namely density(point A) = density (point B) for any A,B. A similar information argument may be made to describe the argument of the probability in the case of a potential. From classical mechanics E= .5mvv + V(x). Given E one cannot determine to which x point it corresponds. For a statistical gas, for a given E constant there is no information suggesting that probability should differ for any v,x pair which produces E. Similar arguments exist for other statistical results. We argue here that an idea of a lack of information is not restricted to statistical mechanics. It may be present even in a conservation law such as conservation of momentum. For example, in (2) it is stated that conservation of momentum along the medium surface (say x) may be used to derive Snell’s law. We argue that this is linked to an invariance in the x direction as the index of refraction n is only a function of y. n(y) carries no information about x and so we argue that this lack of information should manifest itself in a conservation law in the x direction. This idea may be extended to a mirror moving at constant v in the negative y direction. Again, the “medium” now only contains information about motion in the y direction. In (2) we have argued that this problem is like one with an index of refraction in the y direction, but this index does not depend on the medium, but on relative speeds of light and the mirror. Again there should be a conserved quantity. This quantity is the hypothetical velocity described in (2). It was previously shown that it is equivalent to the result obtained using Fermat’s least time principle, but here we argue that it follows from a lack of information. Thus we try to argue that a lack of information may lead to conservation laws and invariances in a wide range of problems. Being able to spot such an invariance may simplify a problem. For example, instead of using Fermat’s idea of least time and taking a derivative of an expression for total time for both refraction and a moving mirror problem, one may simply identify the conserved quantity and write the final result in one line. One should note, however, that one must be careful in creating this conserved quantity. For example, in the case of the hypothetical velocity, it lies along the x axis, but represents a hypotenuse with relative velocities being a projection. It is not a projection of a given relative velocity (i.e. given information)