Relative Information and Quantum Free Particle exp(ipx)

Relative Information and Quantum Free Particle exp(ipx) Francesco R. Ruggeri Hanwell, N.B. Oct. 28, 2025 In (1), the idea of relative information is introduced and applied to quantum mechanics, albeit in the form of state vectors. Here we try to apply the idea directly to the notion of exp(ipx) as a free particle probability. We consider an example presented in (1), namely that one may not know if a person Robert is at home or at his office. The notion of relative information is introduced in the form of another person, Mary, knowing Robert’s whereabouts. We suggest that in classical physics, momentum is measured by an impulse hit if one does not know the mass. One cannot use velocity alone or energy considerations if one does not know mo. This, we argue, is like knowing Robert’s whereabouts through direct measurement. In previous notes, we tried to introduce probability into Newtonian 2-body elastic scattering. In particular, given initial energies and momenta (e1,e2) and (p1,p2) vectors, we argued that any (ei,ej) (pi,pj) vectors outcomes, which conserve energy and momentum, should have the same product probability. This means, we argue, that one is unaware of what forces, i.e. impulses act in the collision. It is impulse, however, which is used to measure p. We argued that one would at first expect that momentum probability be given by exp(i p ). P, however, is measured relative to the x-axis and so changing the direction of the x-axis should not change the probability value. We then suggested using exp(ipx) (and generalizing the overall probability to exp(-iEt+ipx) which is Lorentz invariant). The point we make here is that x now becomes a relative information source for p. Thus, even if one cannot find p from an impulse hit (direct measurement of Robert’s location), one may find it from x measurements using exp(ipx), as in a 2-slit experiment (relative information from Mary). We argue that a probabilistic relative information approach is introduced which allows one to find p in a second manner and it is this second manner which is linked with quantum mechanics. Relative Information In (1), it is argued that quantum mechanics is a theory based on relative information. A particular example is given, namely that of a person Robet who might be at home or in his office. The problem is that one cannot directly observe his whereabouts for some reason. One is, however, told that a second person Mary knows his whereabouts. Thus, one may find out where Robert is, not by direct measurement of his presence, but through a second variable (Mary) so to speak We suggest here that this seems to be exactly what happens in the case of the free particle wavefunction: exp(ipx) ((1)) Classically, one could measure p directly through an impulse hit. This is equivalent to directly measuring Robert’s presence to find his location. In such a case, one would have no need of ((1)). We argue that ((1)) provides an alternative method of obtaining the value of p, namely through the use of the variable x, i.e. the second person Mary. If a particle described by ((1)) interferes with a 2-slit apparatus with a separation of slits about hbar/p, one may measure the distances between minima or maxima and find the value of p using standard formulas which describe 2-slit interference ((2)): D theta(n) = n wavelength ((2)) where wavelength = hbar/p and theta is the angle from the y axis to a maximum on a screen very far away from the 2-slit apparatus. D is the slit separation (center to center) Thus, there is a relative manner using x which yields information about p. In fact, both variables must appear in ((1)) to indicate that relative information relationship. Derivation of exp(ipx) In previous notes, we tried to derive exp(ipx) by arguing that probability is present in Newtonian 2-body elastic scattering. We argued that if one has an initial (e1,e2) energies and (p1,p2) momentum vectors, one may have any outcomes (ei, ej) (pi,pj), which conserve energy and momentum, with equal product probability. This product probability equals that of the initial variables. We suggest here that arguing for different (pi,pj) hits means that one does not know which impulse hit occurs in the elastic interaction. Knowing the impulse hit, however, is a way of measuring p. Thus, it seems that this is how the idea of relative information is creeping into the problem. At first, one might suggest a probability for momentum of: exp(ip) ((3)) Here only p appears and there is no notion of any relative information. Relative information, however, immediately appears if one argues that p depends on the direction of the x-axis and that one should have the same probability whether the x-axis points to the right or left. This then suggests: exp(ipx) ((4)) In other words, a relative variable is now introduced ,namely x. This variable allows one to obtain the value of p without performing the usual impulse hit measurement of p. We have also pointed in previous notes that one may consider a full free particle probability as: exp(-iEt+ipx) ((5)) which is Lorentz invariant. Again, the notion of relative variables appears because one wishes to have linear E and p forms, but these change under a Lorentz transformation and must be coupled with other variables (relative information) to create a Lorentz scalar. Thus, we argue that the above derivation of exp(ipx) and exp(ipx) itself seems to be consistent with the notion of relative information proposed in (1). Conclusion In conclusion, (1) suggests that quantum mechanics seems to be based on the notion of relative information. In (1), the following example is given. A person Robert may be at home or at his office. Classically, one would simply measure his presence to find his location. In (1), however, it is suggested that one might need to find this information from a relative source, i.e. a second person Mary who knows his whereabouts. This implies that one would not need to do a direct measurement to find Robert’s location. We suggest that the free particle wavefunction exp(ipx) (x part) follows the proposal of (1). Classically, one may measure the value of p directly through an impulse hit. This is equivalent to directly measuring Robert’s presence, i.e his location. exp(ipx), however, allows one to use relative information, namely in the form of x, to find p without performing a direct impulse hit measurement. One may simply perform a 2-slit experiment and find the value of p from the interference pattern on a screen as described above. We suggest that the notion of relative information emerges from the manner in which exp(ipx) may be derived. We suggest that this probability appears in Newtonian mechanics, namely in elastic two-body scattering. If one has an initial (e1,e2) energies and (p1,p2) momentum vectors, than we argue that any (ei,ej) (pi,pj) pairs which conserve energy and momentum have the same product probability as the initial set and are equally likely. This means that one is not measuring the impulse hit which occurs in the interaction, but is rather using probability. At first one might propose exp(ip) as a probability. An imaginary number is used with modulus 1 because there is no real value weight for a free particle. P, however, depends on the direction of the x-axis (i.e. a relative variable) and we argue that exp(ip) and exp(-ip) yield different results and this is not physical. Thus, we suggest exp(ipx) which immediately introduces the relative variable x. To generalize, we consider the form exp(-iEt+ipx) which is Lorentz invariant. Given that one wishes to have a form linear in E and p (for product probabilities to be the same), then one must introduce relative informational variables t and x in order to achieve Lorentz invariance. References 1. Di Biagio, A. and Rovelli, C. Relative Information, Relative Facts (2025 https://www.semanticscholar.org/reader/c0ccc246970146c31bd6a60cee5cc53f7f6117e3 2. https://en.wikipedia.org/wiki/Double-slit_experiment