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Some problems only require a certain amount of information for their solution. For example, a photon refracting from a medium interface along the x axis problem may be solved by noting that momentum and speed in medium 1 are pn1 and c/n1 (where p,c are momentum and speed in a vacuum and n1 is the index of refraction). In medium n2, one replaces n1 with n2. From these classical smooth values (deterministic values) one may obtain Snell’s law which may be verified experimentally. What happens if one takes this same information and tries to solve for the relative probabilities of reflection/refraction for a photon moving along the x axis in a medium with n1 and hitting an interface along the y axis at x=0. The medium for x>0 has n2>n1. Can this problem be solved simply using E=pc, and pni , c/ni? It seems it cannot. Furthermore, the smooth pni and c/ni appear to be approximations because the speed of light in a medium changes due to interactions with atoms. One might expect some type of stochastic behaviour, as a result, with pni and c/ni being averages. As shown in (1), one may maximize Shannon’s entropy subject to periodic constraints to find an unusual two-dimensional probability exp(-iEt+ipx). This probability allows one to find relative probabilities for reflection/refraction and also predicts/solves a n-slit interference-diffraction problem. Thus the smoothed over pni and c/ni information is not wrong, but only solves certain problems. More information is given by exp(-iEt+ipx) which may be used to solve further problems. Given that we suggest exp(-iEt+ipx) follows from two maximizations of Shannon’s entropy, one for P(x) and one for P(t), one might expect there is no further information. We suggest, however, that for t-> t+ constant/E and x-> x+constant/p, -Et+px does not change and this result should hold as seen in any frame moving at constant average v (velocity). By mentioning average v, one already takes a deterministic view and approximates the stochasticity of exp(-iEt+ipx). In (1), we argued that one may find the specific transformation involved in viewing E,p,x,t from a moving frame and that not only is -Et+px invariant, but also -tt+xx and -EE+pp. In fact, -EE+pp=momo (c=1) is a relation between E and p as well as being an inner product. We argue here that an inner product, in general, loses information as it is a relative relationship between two vectors. A point we wish to make is that exp(-iEt+ipx) is a solution of -EE+pp=momo (c=1) if one replaces E with id/dt partial and p with -id/dx partial. Thus, we suggest some information is lost. There is no reason to require a second order differential equation when a first order one yields p and E, although the second order equation is not wrong, it just removes some information as does an average. We thus suggest that Dirac’s approach to finding a linear equation in id/dt partial and -id/dx partial is an attempt to retrieve this extra information. This approach applies to finding extra information called spin, which again, is only useful in certain problems (for example, when a magnetic field is present). In (2) we examined a continuity equation with the Poynting vector and energy density .5(C1 Electric field Electric field + C1 Magnetic field Magnetic field) and showed it may also be broken into two linear equations which yield spin 1 information. Thus we argue that certain equations (quadratic in nature) may represent a loss of information, especially those that represent a kind of coarse grained average as in the case of energy density or even classical probability to reflect or refract. In such a case, one may try to linearize in order to find this lost information. |
