Minimal Functional Forms in Statisical Mechanics Part 4

In Part 1-3, we essentially argued that one may try to create math probability distributions which may possibly appear in nature by seeking the simplest functional invariants of certain groups (rotational group, Lorentz group etc) and then seeking a minimal solution in this invariant to p(ei)p(ej)=p(ek)p(el), e.g. ei+ej=ek+el (Maxwell-Boltzmann case, or ei*ej=ek*el for e/T, Tsallis and Kaniadakis cases.) We applied these ideas to the Maxwell-Boltzmann case where a*v dot v or b*p dot p is the simplest invariant of a rotational group and then used this variable to solve for p(ei)p(ej)=p(ek)p(el) using the simplest form of the variable in the solution, i.e. ei+ej = ek+el for the MB case. We note that the MB distribution is linked to a gas of N particles with Etotal. This is a deterministic problem, not a statistical one, in principle although one might argue that one does not really know the details of particle-particle scattering. The problem is simplified by losing information and trying to describe the system in a simpler way using probabilities. For instance, the feature of an upper ei tending to infinite and p(ei)’s being independent are approximations which are not really true, but are fine to the degree of the approximation. Here, we consider the Lorentz group in a similar manner. We utilize a simple invariant of this group -Et+p dot r (instead of p dot p of the rotational group in the MB case) and then consider p(invariant1)p(invariant2) = p(invariant3)p(invariant4) ((1)). The Lorentz invariant is composed of E and p which should be conserved in interactions (according to classical physics) and so we consider a solution ((1)) which uses the simplest form of the Lorentz invariant. We mimic the conservation solution of the MB case and so introduce exp(-iEt+ip dot r). Unlike the MB case, one must use an imaginary i because one cannot have the probability grow or shrink indefinitely. One might ask: Where is the a priori probability for a particle moving at constant v? This is analogousl to asking: Where is the probability for an ideal gas? In the MB case, one describes the problem in a probabilistic manner to simplify it following the math procedure described above. Without having any a priori probability for a particle with constant v, we follow the same math procedure and create a probability. It is possible that this probability approximates the fully detailed description of motion (but then so does Newtonian mechanics with its dx->0, dt->0 which is also pure math). In this case, E and p are fixed, they represent no uncertainty and so it seems one must conclude that there is probability or uncertainty in t and x. Newtonian mechanics, however, which describes the classical world very well does not require any such uncertainty except in the case of one dimensional reflection-refraction which has been well known to people for a long time. Thus, we suggest that a math approach to creating a possible physical distribution exp(-iEt+ip dot r) does seem to manifest itself, like the MB case, in the physical world.