Importance of Newton's Action-Reaction for Free Particle Quantum Mechanics

Importance of Newton’s Action-Reaction for Free Particle Quantum Mechanics Francesco R. Ruggeri Hanwell, N.B. Nov. 30, 2025 In a number of previous notes, we argued that free particle quantum mechanics, i.e. exp(-iEt+ipx), follows from the Lorentz invariant A = -Et+px. For t=0,x=0, A=0, but we argued that there is another solution, dx = hbar/p and dt=hbar/E (we use intervals because x=0, t=0 is a trajectory point). We then noted that this solution is linked with the probability exp(-iEt+ipx) which is Lorentz invariant and creates the above intervals in t and x. Finally, we remarked on the fact that exp(-iEt) conserves energy and exp(ipx), momentum, when one uses these for 2-body collisions, i.e. AND situations. Here we point out that there are numerous dt= f(p,E) and dx=g(p,E) (f, g are arbitrary functions) which yield A=0. For example, dt=p and dx=E is an example. Only dx=hbar/p and dt =hbar/E, however, lead to a probability form which conserves momentum and energy and this is something which we did not stress clearly in previous notes. Thus, there seem to be a number of issues involved in obtaining a free particle probability exp(-iEt+ipx) and these are independent requirements, which then lead to a particular solution. We argue that these requirements are: ((1a)) One wishes to preserve Lorentz invariance ((1b)) One desires discrete dx, dt intervals instead of Newton’s intervals which tend to 0. ((1c)) These intervals need to be linked with a particle related probability which defines them. This probability is critical physically because if the theory is to define photons and particles equally well ( as special relativity ((1a)) applies to both), one must account for the probabilistic effects of 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction. ((1d)) One must have conservation of momentum and energy. We argue that it is the combination of these requirements that ultimately leads to the free particle probability exp(-iEt+ipx). Special Relativity We have argued in a number of previous notes that one may obtain the free particle probability exp(-iEt+ipx) from the Lorentz invariant: A = -Et+ipx ((2)) If one considers t=0,x=0 as a trajectory point which creates A=0, then one may introduce intervals (not trajectory points because they don’t respect x=vt): dx=hbar/p and dp=hbar/E ((3)) such that A=0. We then noted that one may create a probability function exp(-iEt+ipx) which physically “creates” these intervals and noted that this probability is linked to conservation of energy and momentum, i.e. P(p1)P(p2) = exp(ip1x)exp(ip2x) = exp(i (p1+p2) x) ((4)) if one writes: exp(-i (p3+p4) x) ((5)) for the final products, then ((4))*((5)) = 1. The point we make here is that it is sufficient to have dx and dt intervals which keep A=0 as there are numerous ones, i.e. f(E,p) and g(E,p). For example, dt= constant * p and dx =constant * E ((6)) is another example, and there are many more. As a result, one may ask: Why was dx=hbar/p and dt=hbar/E chosen? There must be further requirements. We first stop to note that introducing intervals above has an inherent physical reason. In Newtonian mechanics dt→0 and dx→0, but this is a mathematical approximation which cannot really be correct at all length scales. Physically, one does not have a particle accelerated by Force = -dV(x)/dx in a dx→0 region. This simply means that the region is small and that one approximates it as tending to 0. We wish to finite physical dx and dt regions. The fact that photons interfere at 2-slits suggest that this length scale is not extremely tiny, i.e. beyond experimental tests. A second physical consequence is that finite dx and dt must be enforced physically. One cannot simply have an external ruler with special gradations. These must be internal gradations, i.e properties of the particle, we argue. A third consideration is that special relativity applies to both a particle with rest mass and a photon and so the requirements/ consequences listed above must apply to both. It is known that a photon interferes at a 2-slit apparatus and that probability is associated with 1-D reflection-refraction at an n1-n2 index of refraction junction. This probability has to “arise from somewhere” and we argue that this probability needed to physically create dt and dx may also serve as the same probability in these physical examples. (This is in fact the case in calculations using exp(-iEt+ipx) which reproduce 2-slit interference and 1-D reflection-refraction results.) The final requirement which we impose together with the idea of special relativity, dx,dt intervals and probability is that of conservation of energy and momentum. This implies that one cannot use any arbitrary functions of p,E for dt, dx which leave A = -Edt+pdx=0. One must use special intervals such that the resulting probability is one which conserves energy and momentum. One may see that dx=hbar/p and dt= hbar/E requires a product function which creates periodic dt and dx intervals, i.e. exp(-iEt+ipx). Thus, we suggest that it is the set of requirements ((1a))-((1d)) which together lead to the free particle probability (wavefunction) exp(-iEt+ipx).. Features of the Reaction in A = -Et+px In the above discussion, we argued for conservation of energy and momentum. These are linear in E and p and so probabilities p(E) and p(p) should seemingly be linear in E and p. To make the expression Lorentz invariant, one would need to take a dot product (minus metric) with some other 4-vector, e.g. (x,ct). This leads to a function which is the sum of a strictly linear E term and a strictly p term. Given that probabilities multiply in AND situations, this allows for -Et+px to be the exponent of a base number and so conserve probability. To obtain the required intervals, one must have periodicity i.e. base number (i (-Et+px)). We point out that the separation of E and p (which after all are math functions in special relativity mocc/sqrt(1-vv/cc) and mov / sqrt(1-vv/cc) and might not be considered as describing different ways in which a particle may interact, actually appear separated in -Et+px. In other words, the two different physical manners of interacting, i.e. losing kinetic energy of E in time or an impulse hit p at a specific x appear in -Et+ipx. The separate conservation of both E and p suggests that one really has a P1(E) and P2(p) and that there may be some problems which require only one, e.g. P2(p) = exp(ipx). Such is the case for 2-slit interference and 1-D reflection-refraction at an n1-n2 junction. Conclusion In conclusion, we point out that in previous notes we suggested that free particle quantum mechanics seems to follow from special relativity, i.e the Lorentz invariant A = -Et+px. Here we suggest that there are really four requirements: Lorentz invariance, dx,dt intervals, a probability expression P1(E)P2p) which create these dx, dt intervals, i.e. they must be periodic functions, and finally conservation of energy and momentum. It is possible to consider A= -Et+px and note that x=0, t=0 yields A=0. x=0, t=0 is a trajectory point and so one might consider dx=hbar/E and dt=hbar/E as actually being an interval which leaves A=0. We argue that one actually requires dx and dt intervals physically because the Newtonian dx→0, dt→0 is an idealization which ultimately must break down on some length scale. (It is fortunate that it breaks down already for photons on fairly large length scales which were seen in the early 1800s through 2-slit interference.) In order to manifest dx, dt, one must have the particle physically create these so to speak, i.e. there must be a P1(E) periodic function and another P2(p) one. Finally, we note that dx=hbar/p and dt=hbar/E is not the only way to have A=0. There are numerous other ways such as dx= E and dt=p. We argue that one requires a dx, dt set such that the resulting probability conserves energy and momentum and it this requirement, together with the others which ultimately leads to exp(-iEt+ipx).