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Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 2 Francesco R. Ruggeri Hanwell, N.B. Nov. 1, 2025 In Part 1, we argued that for one-dimensional reflection-refraction at an n1-n2 junction at x=0, the equation 1 = P(reflect) + P(refract) has nothing to do explicitly with the physical direction in which the incident photon moves. We then noted that this equation may be written in terms of a pressure balance equilibrium one using dynamic variables, i.e. AA/c = BB/c + CC/c2 -> AAp - BBp = CCp2, where AA, BB and CC are the fluxes of the incident, reflected and refracted photons and E=pc, with E being the same for all three. We stressed in Part 1, that such a pressure balance suggests a classical equilibrium, but that in classical physics such equilibria involve motion in both directions. Clearly, however, a physical reflection-refraction experiment involves an incident photon moving in one direction. We thus suggested that one requires equations which capture probability in a directional setting, i.e. a directional probability to match the physical direction.. As a result, we wrote AA = Aexp(ipx A exp(-px), i.e introduced the directional probability exp(ipx), and then required that it and its first derivative be continuous at x=0. As a result, free particle quantum mechanics seemed to emerge from forcing a probability scheme to have direction. Here we argue that this same principle seems to be at work when considering spin. In a previous note, we argued that one may linearize -EE + cc p dot p = - momocccc ((1)) (using E→ id/dt partial and p → - i grad partial, hbar=1) to obtain an equation involving linear d/dt and grad acting on exp(ipx), for a free particle. This introduced spin matrices (described in detail by Dirac). In the case, of a continuity equation for a photon based on Maxwell’s electromagnetic equations, i.e. d/dt partial (.5eo El dot El + .5/uo B dot B) + grad dot 1/uo (El x B) = 0 ((2)), we linearized to obtain an equation linear in El and B (the electric and magnetic fields), i.e in exp(i p dot r). In other words, in both cases, we wanted an equation linear in exp(ip dot r) and d/dt, grad. Again, spin emerged through the Levi-Civita symbol eijk, with i denoting the spin component and jk, matrix entries. Here, we argue that the above math approach is linked with writing an equation describing the directional probability and its motion. This is why one requires both a linear d/dt, grad and linear exp(ip dot r). The Klein-Gordon equation has a linear exp(-iEt+ipx), but quadratic operators p dot p, EE, while the photon continuity equation has linear d/dt and grad, but quadratic terms in El, B, where El and B both go as exp(ipx). Thus, the directional probability is not linear in the continuity equation. As a result, we suggest that free particle quantum mechanics, including spin, arises out of the desire to have a directional probability and information about its direction. Unfortunately, equations such as ((1)) and ((2)) which are commonly used do not give this and one is forced to find another equation which yields exp(i p dot r) and d/dt , grad, from which one may later recreate the equations ((1)), ((2)) which are not based on physical direction of motion.
Physical Direction of Motion In Part 1, we stressed that a physical problem such as reflection-refraction at an n1-n2 index of refraction junction has a physical direction. This physical direction, however, is not of essence in the probabilistic equation: 1 = P(reflect) + P(refract) ((1)) In ((1)), one simply has two states with probabilities just as in a coin toss. It is somewhat irrelevant that the incident and reflected photon move in the region of space and the refracted, in another by viewing ((1)) alone. Physically, however, it is important that the problem is directional. We note that one may write ((1)) as: AA/c - BB/c = CC/c2 → AAp - pBB = p2 CC at x=0 ((2)) Here AA,BB,CC are the fluxes of the incident, reflected and refracted photons. ((2)) is a pressure balance equation, but such an equilibrium is not based on directional motion. Classically, a pressure balance involves motion in both directions in x. Physically, the problem represents a single direction, however, and so it seems that one requires a probability scheme which is directional. This is something that does not seem to exist in classical physics. In Part 1, we argued that such a scheme may be built from the observation that: AA = Aexp(ipx) A exp(-ipx) ((3)) In other words, AA includes both momenta p and -p, i.e. both directions of motion. One may then establish an equation or equations based on exp(ipx), a directional probability linked to a specific direction of motion. These equations then may be combined to create ((2)), as shown in Part 1. Here we ask: Can this reasoning be applied to other examples? The Case of Spin and the Klein-Gordon Equation In previous notes, we argued that spin arises from taking commonly used equations and breaking them down into equations linear in d/dt, grad and exp(ipx). In particular, we considered the Klein-Gordon equation: -EE+ p dot p c c= -momoccc → d/dt d/dt partial exp(-iEt+px) + cc d/dx d/dx exp(-iEt+ipx) = -momocccc exp(-iEt+ipx) ((4)) This equation includes the directional probability exp(ipx) in linear form, but does not show physical direction of motion because one uses p dot p. One must have an equation linear in p and a linear expI(ipx) in order to see physical directional motion. It is not sufficient to use a directional probability exp(ipx) if its direction of motion is not described. This leads to the
linearizing of ((4)) which in turn introduces spin matrices as discovered by Dirac. We thus suggest that motivation for linearizing ((4)), which leads to the appearance of spin ½, is the desire to have a directional probability in an equation which demonstrates its physical direction of motion, i.e. one linear in -i grad. This is the same reasoning which applies to the n1-n2 junction reflection-refraction case. Photon Spin We next consider an electromagnetic equation describing the spin of the photon. d/dt partial ( .5eo El dot El+ .5/juo B dot B) + grad dot 1/uo El x B = 0 ((5)) In this case, El and B are the electric and magnetic fields and both go as exp(ipx) if the photon moves in the x direction. The El and B fields are perpendicular to each other and the direction of motion. Thus, ((5)) does not contain exp(ipx) in linear form, i.e. the directional probability in linear form. It does, however, have d/dt and grad in linear form. The goal is then to create an equation linear in aEl + b B because these are proportional to exp(ipx). This leads to: d/dt (El + iB) (i) + eijk d/dxj (El+iB)(k) = 0 ((6)) ((6)) is linear in the directional probability exp(ipx) as well as in d/dt and d/dxj which shows direction of motion. The Levi-Civita symbol eijk is the ith spin matrix with elements denoted by (j,k). Again the same principle of finding an equation linear in the directional probability exp(ipx) and linear in grad (which shows direction of motion) describes quantum mechanics. This equation may then be used to construct ((5)) which does not show the directional probability. Conclusion In conclusion, in Part 1, we argued that probabilistic equations exist in physics which do not explicitly show a directional probability and direction of motion. In Part 1, we considered a single example, namely that of reflection-refraction at an n1-n2 index of refraction junction at x=0. The equation 1 = P(reflect) + P(refract) or its equivalent pressure equilibrium form: pAA - pBB = p2CC, where AA, BB,CC are the fluxes of the incident, reflected and refracted photons, do not show physical direction of motion. They are involved in classical equilibrium-probabilistic scenarios which allow for both directions of motion. Physically, only one direction appears and we argued in Part 1, that there must exist a directional probability and corresponding equations which clearly show direction of motion. This, we argued, is how the n1-n2 reflection-refraction problem is solved and it introduces the idea of free particle quantum mechanics, we suggested. Here we consider two further examples, both linked to spin. We note that the Klein-Gordon equation does not show direction of motion due to p dot p → -grad dot -grad even though this acts on the directional derivative exp(-iEt+ipx). We suggest that one must have an equation linear in exp(ipx) and d/dt, grad to demonstrate the direction of motion. This then is the reason why Dirac linearization of the Klein-Gordon equation is needed, we suggest, as it gives rise to spin ½ as shown by Dirac. The second example is the energy -Poynting momentum equation
for a photon based on .5eo El dot El and .5/uo B dot B and 1/uo ElxB. Even though the equation is linear in d/dt and grad, it is not linear in the directional probability exp(ipx) with both El and B (electric and magnetic fields) going as exp(ipx) with the photon moving in the x direction. Thus, one must transform the continuity equation into one linear in exp(ipx), d/dt and grad, i.e. d/dt (El+iB) (i) + eijk d/dxj (El+iB) (k) =0 and so eijk, the ith spin matrix (jk denoting elements) emerges. In this two further examples, which show the existence of spin, one must obtain an equation linear in the directional probability exp(ipx) as well as in d/dx which shows the direction of motion.
