Speculation on Temperature in Pure Quantum Bound States

In (1) it is suggested that Newton’s second law for constant acceleration, i.e. F=ma, may be derived using the first law of thermodynamics (with dE=0) and the special relativistic idea of a constant acceleration being linked to a temperature as shown by Unruh i.e. T=a C1 (where C1 is a constant given in terms of hbar, c etc).

    In this note, we consider the two Lorentz invariants EE = pp + momo  (c=1) (and its generalization (E-V(x))(E-V(x)) = pp + momo )and -Et+px. The former becomes Newton’s energy conservation law in the nonrelativistic limit i.e. E= pp/2m +V(x) which is equivalent to Newton’s second law (upon taking d/dx), but contains a variety of accelerations. 

  The Lorentz invariant -Et+px suggests t and x are independent and that t is linked with an eternal clock with frequency hbar/E and x with an internal ruler with spacing hbar/p. Thus even though a particle with constant speed moves as x= p/m t (as measured using an external clock and ruler)  there is an internal wavelength hbar/p in which there is a probability for the particle to be at various x points (as seen with an external ruler) and no internal time. 

   Given that one may consider externally x = p/m t, one may also consider externally the situation of acceleration. Each constant p, however, is associated with exp(ipx (an eigenfuction of the translation operator -id/dx) so at a given x, one has a superposition of exp(ipx)’s. This suggests a different “probability” distribution for p at each x or in other words a temperature linked with  {Sum over p a(p) pp/2m exp(ipx)} / {Sum over p a(p)exp(ipx)}.. The temperature changes from one x point to another. If one considers that E is constant at each x. then applying the first law of thermodynamics yields Fdx = dx d/dx (-1/2m d/dx dW/dx)  which should be equivalent to TdS. Using Uruh’s results, this leads to the expression for entropy given in (1).

     This, however, seems to be an entropy which is different from one constructed using Shannon’s entropy equation. We consider some of these ideas in this note.