Extremization of Fisher's Information, Free Particle and Conservation of Energy

    The extremization of Fisher’s information subject to constraints usually containing V(x), the classical potential, appears in the literature (1), (2). V(x), however, is a concept introduced by Newton and associated with conservation of energy for a conservative potential. We argue the appearance of V(x) in an extremization approach to obtain an “equation” bypasses the already existing conservation of energy equation. If V(x) is used, we argue that one cannot dispense with conservation of energy and that it should be written a priori. If that is the case, what does extremization of Fisher’s information guarantee? 

   In (1) a probability distribution P(x) is set equal to W(x)W(x) its square root. In (2), a complex function is defined:  W(x) = sqrt(density(x)) exp(i S/hbar) where S is the classical action. Here density(x)=P(x). In both cases, the square root of the density appears.

The problem, we argue,  is that for a free particle with density(x)=P(x)=1, there is no Fisher’s information to extremize, yet the free particle is a key part of quantum mechanics. Its probability (square root probability) is the solution of an eigenvalue problem of the translation operator -id/dx and it is consistent with Newtonian mechanics i.e. conservation of energy with E= pp/2m. From this one may create an ensemble W(x) = Sum over p a(p) exp(ipx). The average kinetic energy is -1/2m d/dx/ dW/dx / W which happens to be the extremized value of Fisher’s information for P(x)=W(x)W(x). 

   In other words, we argue that conservation of energy must occur and that one needs a solution for a free particle and this does not follow from Fisher’s information extremization. Extremization of Fisher’s information produces the form d/dx dW/dx which is taken to be linked to <pp> which already follows from -d/dx d/dx exp(ipx). 

   One argument  made in the literature (3) is that Newtonian conservation of energy also follows from Fisher’s information extremization i.e. the Lagrangian  for a free particle  .5m dx/dt dx/dt is of the form of a Fisher’s information with xx = probability (which is unusual). The strategy, it seems, is that one may apply extremization of Fisher’s information directly to a quantum problem by assuming a P(x)=W(x)W(x) and use d/dx  instead of P=xx with d/dt. An issue, as noted, is that for a free particle W(x) exists, but extremization of Fisher’ information does not exist. Even if E=pp/2m (nonrelativistically) and this follows from the extremization of Fisher’s information, W(x)=exp(ipx) does not, but it is consistent with E=pp/2m. The reason for this consistency is that it contains p (momentum) which by its very definition must ensure E=pp/2m (nonrelativisitcally). Fisher information extremization, however, yields the bound state solution i.e. may be used to derive the Schrodinger equation. In (1) this is done for a bound state, in (2) for the more general case, but (2) defines W(x) = sqrt(density) exp(iS/hbar) where S= the classical action. For a free particle, the definition of W(x) is exp(-iEt+ipx) which is the solution of the free particle problem without using any equation. Thus the definition is the solution itself for the free particle case in (2). 

   It seems that Fisher’s information, which uses density(x), does not include a scheme for describing motion in one direction in space without time present. exp(ipx) i.e. cos(px) and sin(px) distributions in two dimensions allow for this. This motion is associated with the translation operator d/dx which we argue is key in solving for the square-root probability of a free particle in keeping with E=pp/2m, which (3) argues follows from the extremization of Fisher’s information. Thus, even if one accepts the argument that Fisher’s information is responsible for the energy conservation equation, using Fisher’s information = Integral dt dx/dt dx/dt constant, does that allow one to immediately switch to a problem with P(x)=W(x)W(x) and the derivative d/dx and apply Fisher’s extremization there, we ask? We argue that there is an implicit assumption that:

C1 dx/dt dx/dt  =   C2 d/dx dW/dx / W  where C1 and C2 are constants

This implies that for a free particle which treats each x point the same i.e. P(x)=constant, W(x)W(x) cannot be P(x). It also implies that d/dx dW/dx = pp W ((1))  where p=mov nonrelativistically. ((1)), however, is the solution to the free particle problem without performing any Fisher information extremization i.e. exp(ipx) is a solution. This solution may be used to create an ensemble W(x)=Sum over p a(p)exp(ipx) which may be used in a conservation of average energy equation. 

   Thus we argue that Fisher extremization used to create the Schrodinger equation is based already on the idea that W(x)=exp(ipx)  for a free particle and simply produces the conservation of energy equation which is already known to exist from Newtonian mechanics or Fisher’s information extremization which is argued to be the same as Lagrangian variation in (3).