Prob(e1)Prob(e2) = Prob(e3)Prob(e4) and Quantum Spin Part 2

    In Part I, we argued that one may create a dynamic probability which ensures that an elastic collision of two particles has equal probability for any (ei, ej) pair with the same sum and any (pxi,pxj) pair, also with the same sum (and the same condition for py and pz). This suggests a power law, i.e.  number power ei or number power px or exp(e), exp(px). We then argued that probability should be a scalar and should not depend on the direction of the x-axis. This means that px and -px should have the same probability so exp(px) must be changed to a function with px multiplied by a linear measure which compensates for a change in the direction of the x axis. Furthermore, there is no equilibrium of free particles (like in a Maxwell-Boltzmann gas) and so we proposed exp(i px delta x), i.e. a complex probability. There is no real weight to adjust for fre particles . To be more general, one may consider an argument which is invariant to frame changes (Lorentz transforms) and this leads to exp(-iEt+ip dot r). In other words, we created a dynamic (imaginary) probability to allow from a frame independent probability which ensures conservation of energy and momentum with no distinguishability between (ei,ej) or (pxi, pxj) pairs with the same sum.

   Noether’s theorem also deals with conservation, but argues that there should be a symmetry in the action (Integral Lagrangian dt). In particular, (1) shows that dL/d variable partial = the conserved quantity because the action does not contain x. If this notion is linked to the above ideas, then -Et+px should be linked to Noether’s theorem.  One may show that -Et+px is in fact the action Lt in both nonrelativistic and relativistic cases if v=x/t. In this case, i d/dt partial and -id/dx partial make use of the symmetry variable which then brings out the conserved quantity, which is a little different from dL/dv=p.

    We then focused on operators (matrices) which take the place of basis vectors for the vector p and more generally for the 4-vector (pc,E). These matrices act on vectors, but should play the role of d/dt partial or d/dx partial if in fact there is a conserved quantity linked with spin. In particular, we tried to show that spin should be linked to the exp(ip1x)exp(ip2x) =exp(ip3x)exp(ip4x) statistical property ((1)) if p1+p2=p3+p4. We note that the “spin” operators have the same group theoretic algebra as those of angular momentum and argued that there was a connection with exp(i m phi), where m hbar is the Lz value. We try here to argue why there should be a conserved quantity and thus why ((1)) should also be linked with spin.