Speculation on Lorentz Invariant Information Associated with Quantum exp(-iEt+ipx)

 In information theory, ln(probability(i)) equals  information. The approach of maximization of Shannon’s entropy - P(ei) ln(P(ei)) subject to a constraint ei P(ei) yields:  ln(P(ei)) = -ei/T where ei is a conserved piece of information associated with an interaction and -1/T, a Lagrange multiplier. In the case of tossing  a die or coin, the constraint is P(i), so the Lagrange multiplier is not important, but P(ei) is the conserved quantity and is associated with interactions i.e. trial runs.

   In the case of a quantum free particle, one may ask: Where is the probability and where are the trial runs or interactions? A free particle, classically, has an energy and momentum as well as a time and x associated with v=constant=x/t. There do not seem to be any reactions so it is not clear what constraint exists, nor is it clear that there is any probability present as E,p and v=x/t are deterministic.

   In this note, we speculate on the relevant information and the presence of probability for a free particle. Such a particle does not seem to be probabilistic at first glance but, we argue that its intrinsic probability is not defined through interactions (like the classical statistical gas), but rather is inherent due to the various different ways it may have been formed. We argue that it creates other visible probabilities that are not based on maximization of entropy subject to a constraint.