Speculation on Special Relativity and Entropy

 -Et + px is an example of a Lorentz invariant. The two 4-vectors (p,E) and (x,t) transform according to a Lorentz transformation which is linked to viewing a system from different frames moving at constant velocities v. There is,  however, a second solution namely t=C/E and p=C/x  (C=constant) which also keeps the inner product constant. This is not linked to frame transformations because for a given p or E one has a specific x or t value which has to be thought of in a different manner. In particular, it may represent an interval (“pocket”) in space or time i.e. x as 1/p is a wavelength. This may also be connected with a free particle Action. 

     If x as 1/p represents an interval in space, it seems there should be a probability distribution associated with such an interval. If this is the case,  the idea of Shannon’s entropy should hold within that region. Thus increasing energy (and p) leads to smaller wavelength regions i.e. higher resolution. Using ideas from a bound infinite well we argue that  entropy for a crest decreases as the resolution increases. Thus high energy of a constant velocity particle is linked to the idea of low entropy if one uses invariance in special relativity and utilizes a second solution which keeps -Et+px constant. This approach is equivalent to exp(-iEt+ipx) the wavefunction of free particle. Thus the idea of entropy  and its relationship to energy may be linked with both special relativity and the quantum mechanics of a free particle, something that does not seem to be evident from classical mechanics.