Anti-Bernouilli Random Variable, Time Reversal Invariance and the Quantum exp(-Et)

 In (1), a type of probability conservation of an anti-Bernouilli variable linked to diagonal motion along box diagonals in an x-t space with 1-mdt being the probability that x direction motion changes leads to a probability of exp(imt).  This approach seems to involve a kind of time reversal symmetry as (1) proposes a t-x space consisting of grids of spacing e and possible motion along diagonals with both +/- e motion in t and x. In particular, (1) considers motion in the positive t direction from the origin to a some T (with +/-e motion possible) and then retraces the direction to return to the origin. In other words, (1)’s m is not an arbitrary number, but is linked to a conserved quantity, energy, i.e. probability steps are in terms of energy if one compares with the quantum exp(-Et), which (1) does. (1) obtains a kind of Dirac equation, but although we provide analysis of this, we only consider the form exp(imt) as relevant. Furthermore, exp(iEt) is consistent with conservation of energy and conservation of momentum follows from the x form of the solution exp(ipx) and so the probability approach of (1) is somehow linked to a change in direction, i.e. collision, which ultimately is associated with a conserved m, i.e. E (energy).

    In previous notes, we suggested that one may obtain the free particle quantum wavefunction exp(-iEt+ipx) by suggesting that a probability exists in Newtonian two-body elastic scattering. We argued that given initial (e1,e2) and (p1,p2)  (energy, momentum) particles, any (ei,ej) (pi,pj) vectors has the same product probability (equal to the product probability of the (e1,e2) (p1,p2) case) has the same value. In (2), we showed that one cannot use exp(ip) for the momentum probability because it is not unique and violates time reversal symmetry. We suggested using a Lorentz invariant result exp(-iEt+ipx).

    The argument we make in this note is that the approach of (1) and (2) seem to be the same. At first this seems to be surprising. (1) obtains exp(-iEt) strictly through a “special” anti-Bernouilli somewhat-random walk approach. There is no concept of energy or momentum, hence no concept of their conservation and no concept of special relativity. (2) on the other hand is based on all of these ideas. We suggest that given that probability is introduced in (2), the notion of physical paths and changes is common to both approaches. Furthermore, we suggest that both are based on time reversal symmetry. We have already noted that the number m in (1) is equivalent to energy. Thus, energy is linked with the anti-Bernouilli probability motion in time (and p with motion in x). This, in a sense, is the same as a conservation of energy and momentum approach, we argue. If one considers that energy and  momentum follow from special relativity by Lorentz boosting a rest mass, then one really has the concept of conservation of the rest mass which is the same as conservation of a certain kind of probability. Thus, we argue that (1) is an interesting approach to flushing out the probabilistic nature of the Dirac equation which we argue is also directly linked to introducing a dynamic directional probability which describes energy and momentum conserving interactions.