Quantum Interaction Probabilities Versus State Probabilities

Quantum Interaction Probabilities Versus State Probabilities Francesco R. Ruggeri Hanwell, N.B. Nov. 13, 2025 In (1), it is argued that the Schrodinger cat paradox arises due to a superposition state being considered “physical” instead of an “an expectation over possible future outcomes” . This is consistent with ideas linked to a quantum directional or interaction probability versus a state probability which we have discussed in various notes (2). In particular, classically there exist states, like heads up for a coin, a certain number of a die, an incident, reflected or refracted state for a photon, etc. We have suggested in (2) that exp(ipx) is a directional or interaction probability in that this probability is linked to the probability to have a p in a two-body scattering experiment and is involved in conservation of momentum. A state probability should not depend on x as the state is determined by p (and mo). For example, an incident, reflected or refracted photon probability does not depend on x, rather it is specifically given by AA/c, BB/c or CC/c2, where c is the speed of light in n1=1 (index of refraction), c2= c/n2 and AA,BB,CC are the fluxes of the incident, reflected and refracted photons. Thus, exp(ipx) is linked to the state through p, but through x it is linked to position. This means that for a given p there is uncertainty in x quantified by the wavelength hbar/p. This means there is position uncertainty with respect to an impulse hit making this an interaction probability and a directional one because p and -p have different effects for a conservation of momentum equation. As argued in (2), interactions are based on exp(ipx) because they involve impulse hits and so even though classical state probabilities exist, the physical problem is solved through considerations of exp(ipx)s. In particular, sums or OR situations of exp(ipx)s arise due to uncertainty in x. It is the directional/interaction probability exp(ipx) which predicts the x uncertainty and also appears in so-called superposition equations which represent different possible outcomes which may all exist in a tiny dx during the interaction. One should not think in terms of a state vector, but rather possible interaction probabilities exp(ipx). These are OR or sums of possible probabilities which interact in x and not a mixture of states. In other words, one does not have a dead cat mixed with a living one or an incident photon mixed with a reflected one. Rather, one adds directional/interaction probabilities for different states because one is uncertain which state exists in dx. This leads to math conditions on the exp(ipx)s as discussed in (2) which then ultimately govern the classical probabilities for one to have a particular state or another. The point we make is that exp(ipx) is not a state probability, but an interaction one which is consistent with the ideas of (1). State Probabilities Versus Directional/Interaction Probabilities Classical probability seems to deal with state probabilities such as the state of a coin (heads or tails), a number on a die, an incident, reflected, refracted state of a photon. Certainly interactions occur, e.g. a coin or die is tossed or a photon reaches an n1-n2 index of refraction junction, but state probabilities simply describe the state of the object. The question then becomes: Can there be a probability which is not a state one? In (2), we discussed the case of Newtonian two-body elastic scattering. Given an initial (e1,e2) energy set and a (p1,p2) momentum vector set, one may ask: What is the outcome set of energies and momenta? In general, one cannot answer this question in a deterministic manner as one only knows that energy and momentum are conserved. In such a case, one would argue that any (ei,ej) (pi,pj) (vectors) set which conserves energy and momentum, has the same probability. This means that there should be a P1(ei)P2(pi) such that: P1(ei)P1(ej) = P(e1)P(e2) and P2(p1)P2(p2) = P2(pi)P2(pj) ((1)) Now, the state of a particle is given by ei and pi. Nonrelativistically, one may have various p values linked to different energies, i.e. P = m1v1= m2v2 … but e1=.5m1v1v1 e2 = .5m2v2v2 … ((2)) P1(ei) seems to hold for different v1 values and is separate from P2(pi) it seems.The notion of a state mapping to a single probability is already challenged by ((1)). A particle’s state is defined by ei and mo nonrelativistically, ,or ei and pi and there are two probabilities linked with the latter, not one. We conclude that P1 and P2 are not state probabilities, but in fact a new kind of probability. These probabilities are used to ensure conservation of energy and momentum which is something that classical probabilities do not do. To see this, one must first find the explicit form of P1(ei) and P2(p). In (2) we show that the probabilities are: P1(ei) = exp(-i ei/T) and P2(pi) = exp(-i pi dot r) such that P1(ei)P2(pi) = exp(-i eit+ i pi dot r) ((3)) ((3)) is Lorentz invariant. If one has a time-independent problem in quantum mechanics and represents a free particle by exp(ipx), this is not a state probability. One does not know the energy of the particle and so one does not know its full state. Furthermore, even though a p describes a set of states (ones with momentum p), x also appears. exp(ipx) is linked to a momentum state (which represent many different energies) and x, which represents a classical position, but the form exp(ipx) actually implies uncertainty regions in x given by: Wavelength = bhar/p ((4)) We argue that a specific p related impulse hit may occur within a wavelength range with the exp(ipx) probability which is not the classical viewpoint. A state probability should have nothing to do with uncertainty in space. Even if one states that all particles have rest mass mo and so p represents the state, there should be no uncertainty in x or even x value appearing. Classically, p and x probabilities are independent as: P(x)dx = dx/L for a given length L ((5)) ((5)) holds for an p, i.e. constant v, and so exp(ipx) is an entirely different probability than a state one. We call it in (2) a directional probability and here we note that it may also be considered an interaction probability as it ensures conservation of momentum i.e. Integral dx exp(-ip3x)exp(-ip4x) exp(ip1x)exp(ip2x) = 0 if p1+p2 not= p3+p4 ((6)) A similar result holds for the energy probability exp(-iEt). Superposition and the Directional/Interaction Probability We note that the quantum concept of superposition occurs for the directional/interaction probability described above. It does not occur for a state probability. In fact, in the case of exp(ipx), it occurs because x is present (there is a superposition in space) and exp(ipx) implies an x uncertainty of hbar/p. We try to explain why superposition should occur for a directional / interaction probability. Given that exp(ipx) implies an uncertainty in space of hbar/p, it is not possible to use the Newtonian reasoning of an interaction occurring at a point x. Rather there is an uncertainty region dx. In an interaction, one has an initial state and possible final states. The uncertainty region is due to the exp(ipx) type probability and so one should consider this kind of probability in the region dx. To give a concrete example, one may consider a photon which undergoes 1-dimensional reflection-refraction at an n1-n2 index of refraction junction. As noted in (2), at the interaction “point”, one has a dx uncertainty region and one does not know whether one has: Aexp(ipx) incident photon Bexp(-ipx) reflected photon Cexp(ip2x) refracted photon ((7)) A.,B,C are simply math weights. As a result, as argued in (2), one must somehow obtain an equation relating the terms in ((7)) and this relation actually solves the problem because one does not use state probabilities, but rather directional or interaction ones. In (2), we noted that choosing x=0 as the interaction point suggests that one should use continuity of the directional probabilities in ((7)) and their first derivative d/dx i.e. Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) at x=0 and Apexp(ipx) -pBexp(-ipx)= Cp2exp(ip2x) at x=0 ((8)) This leads to: AA/c = BB/c + CC/c2 ((9)) which is the classical state probability equation 1=P(reflect)+P(refract) for AA/c set to 1. It is ((8)), however, which allows one to obtain the values of B and C in terms of A, i.e. solve the problem. ((8)) does not use state probabilities and one must distinguish between state and directional/interaction probabilities. We argue that in traditional quantum mechanics one does not make this distinction and so calls exp(ipx) a state vector when it is a directional-interaction probability which may be used to find a state probability for p, e.g. AA/c, BB/c, or CC.c2. The use of superposition is then a probabilistic calculation which allows one to consider directional probabilities for all possible outcomes in a reaction and this is consistent with the statements of (1). A superposition is not a physical mixture of different states, e.g. two states of cat, or a mixture of an incident, reflected and refracted photon. One simply adds directional probabilities in an OR scenario. Conclusion In conclusion, in (1) it is argued that a superposition is not a physical state, but rather “an expectation over possible future outcomes”. We argue that this conclusion of (1) is consistent with the ideas presented in (2). In particular, we argue here that even though classical probability seems to deal with state probabilities (e.g. a coin is heads or tails, a photon is an incident, reflected or refracted one), there exists a directional or interactional probability associated with interactions, i.e. conservation of energy and momentum in elastic 2-body collisions. We showed in (2) that this leads to a directional-interaction probability of exp(-iEt+ipx) which is Lorentz invariant and each piece exp(-iEt) or exp(ipx) may be used separately. We argue here that these are not state probabilities. If they were, they should depend on t and x. Rather, if one considers exp(ipx), then a given p (which represents an impulse) is associated with an uncertainty region in x of hbar/p. Thus, exp(ipx) is an interaction probability. Given an interaction, there is an incident object and possible outcomes. In terms of interaction probabilities (which are responsible for the uncertainty in x) one should have an OR situation. One should create equations which involve a superposition of exp(ipx)s and this is the origin of superposition, we argue. It is not a statement that one has a mixture of different physical states (e.g. states of cat etc). In the case of 1-dimensional reflection-refraction at an n1-n2 index of refraction junction at x=0, one should have Aexp(ipx) for the incident photon Bexp(-ipx) for the reflected and Cexp(ip2x) for the refracted in a dx region. To solve the problem, one may argue for continuity of these superpositions and their derivative d/dx at x=0. This is what solves the problem, but this does not mean that one has a new physical state which is a mixture of an incident, reflected and refracted photon. Rather one adds probabilities as in an OR scenario. The state probabilities are then: AA/c = BB/c + CC/c2. References 1.Wang, W. Expectation-Realization Interpretation of Quantum Superposition (2025) https://www.semanticscholar.org/reader/9abd10fdc4c61d57a1c7212cfee7201fde282e6e 2. Ruggeri, Francesco R. Speculation on Quantum Free Particle Probability and Spatial Equilibrium Parts 1-5 (preprint, zenodo, 2025)