Fisher Constrained Extremization in Quantum Mechanics and Directional Motion

 In classical mechanics, a free particle moves through x and t with dx/dt=v=constant. Given that x and t may be shifted, it is dx and dt which are important. For fixed dx, dt(x) yields information about the “probability” for the particle to be in dx so to speak. Thus one does not really distinguish between deterministic and probabilistic descriptions in this scenario. A problem with the probabilistic picture, however, is that different v (or p values) have the same constant probability P(x)=constant for free motion. In other words, P(x)=constant does not distinguish either the magnitude or direction of v. Thus, information is lost. 

   Is it possible to restore this information and still maintain a probabilistic picture? The answer, given by free particle quantum mechanics, seems to be yes. The probabilistic approach is to assign a probability P(x) to each x point. This probability must treat each x point as the same and yet give information about v or p. (We use p because it represents impulse and is linked to special relativity i.e. a 4-vector.) The key idea we argue is to consider the first order change in x space because one assigns a P(x) to each x and different p values are linked with different motion from one x to another.. This is the way to distinguish the direction and magnitude of p, but nothing else we argue. 

   In such a case:   d/dx f(px) = Constant p f(px) ((1)) ,we argue, where f is an unknown function. We show that this function, which is odd in x,  cannot be real i.e. one needs at least a pair of real functions comprising f. We note that the derivative only gives information about p, thus df/dx /f = p.  We argue  this implicitly means each x point has the “same weight” even though the pair of functions f are not constants.  We next analyze the behaviour of the two pairs of functions comprising f and conclude that in order for d/dx d/dx f = pp f, each function of the pair must be converted into itself. Thus:   d/dx d/dx f1 = -pp Constant f1 and a similar equation for f2 where f1,f2 are the pair. These, however, are periodic equations. Thus the implicit idea of each x point carrying the same weight is equivalent to f1(x) and f2(x) being separately periodic. This is equivalent to  constant modulus problem.

   How is this linked to Shannon’s entropy or Fisher information extremization? We argue that ((1)) means that ln(P) does not carry enough information in this situation, while for a Maxwell-Botlzmann gas it does. It is d/dx ln(P) which carries information about p and its direction. If one wishes to create a function which does not distinguish the sign of momentum, it still has to incorporate the information of motion in one direction  i.e. d/dx ln(P) d/dxln(P). One may take the average of this (called Fisher’s information) instead of the average of ln(P) in Shannon’s entropy. To link with the above ideas, P(x)=f1(x)f1(x) if one considers a bound problem with infinite potential walls. Then Fisher’s information becomes df1/dx df1/dx. The extremization of Fisher’s information subject to Integral f1(x)f1(x)= 1 leads to the periodic condition described above. This means, we argue, that for the case of pp, one cannot sense any change in either f1(x) or f2(x). f1(x) and f2(x) are used to indicate differences to an odd power of d/dx and so for an even, there is no informational change at the f1(x) and f2(x) level. This we argue is what Fisher’s information extremization with Integral f1 f2 =1 gives i.e. periodicity.