Joint Phase Space (x-p) Probability and Free Particle Quantum Mechanics

Joint Phase Space (x-p) Probability and Free Particle Quantum Mechanics Francesco R. Ruggeri Hanwell, N.B. Nov. 21, 2025 There seems to be much interest in the literature (1) in associating quantum mechanics (in particular the Schrodinger equation) with a classical joint phase space probability F(x,p,t) such as appears in classical Liouville theory etc. Here, we consider that P(x)dx = dx/L for a particle at rest and analyze the extra physics which appears when one considers a single particle bouncing back and forth in a box, which is what a person in a frame moving with -v would see (albeit with a Lorentz contracted L.) We suggest that the constraint L is important as it is linked with physical forces which appear when the particle moves. In other words, a particle at rest is linked with a static probability, while for motion, one must have a probability which is linked to more than x and t, but to force considerations in x and t, a dynamical probability. (Experiment bears this out as we discuss.) In particular, we argue that in the case of a moving particle, two physical concepts related to force appear, namely pressure which is an average over time and impulse hit which is only linked with space. Thus, two different particles may have the same p, but different v and pressure values. We thus suggest a physical grouping of (p,x) and (pressure, t). We then note that one needs to somehow preserve the notion of P(x)=1/L, i.e motion does not introduce a bias to x, nor to time. In a rest frame, there is no motion and so there is neither p nor pressure, but P(x)=1/L pertains. If one tries to define a probability with two groupings, one for pressure, t and the other for p,x, then if one multiples a p1 case by a p2 case, the p,x grouping should yield p1+p2, x. Furthermore, a p,x grouping should imply mixed terms, e.g. px etc, otherwise one does not really have an association between the two variables. This suggests a pf(x) form linked to the p-x grouping. Given the notion of special relativity, it seems that one may replace pressure with the notion of energy and create a Lorentz invariant probability. This would then lead to the free particle quantum wavefunction (probability) exp(-iEt+ipx), but we argue that the roots are in classical physics and a desire to physically describe the physical information present for a particle bouncing back and forth in a box. Joint Phase Space (x-p) Probability It seems that there are many attempts (1) to link quantum mechanics (i.e. the Schrodinger equation) to a joint phase space probability: F(x,p,t) ((1)) This type of probability applies to many particles and appears in Liouville classical mechanical theory. Quantum theory, however, should apply to a single particle. We further note that a quantum wavefunction is complex exp(-iEt+ipx) and no complex probability appears in Liouville theory. As a result, approaches which consider F(x,p,t) tend to mathematically force a complex number into the scenario. Here we suggest the following. For a particle at rest in a constrained length of L, the probability to find the particle in dx is given by the uniform distribution (as the maximum lack of information): P(x)dx = dx/L ((2)) A person in a frame moving with -v, however, would see the particle as moving and given the constraint on L (now Lorentz contracted, but still a constraint), would consider this a problem of a particle bouncing back and forth in a box. The question then becomes, How would such a person analyze this situation? Particle Bouncing Back and Forth in a Box We argued above that ((2)) formally applies to a particle at rest in a constrained length L. For a moving particle, this constraint immediately becomes associated with force because the particle must be stopped from moving outside the L region. In the stationary case, the L constraint is not linked with force as one simply places the particle somewhere within the L region. We stress, however, that in both cases L is a constraint. In the moving case, this constraint is linked to force and so one should consider the forces in the problem. We suggest that there are two considerations one may make. (A) There is the impulse hit proportional to p which occurs at x. (B) There is a time average of the hit against the wall which is called pressure and so linked to time and velocity. We suggest that there are two distinct groupings in this problem and both must be considered in order to capture the relevant physical information of the problem, i.e. (A1) p must be grouped with x. Mathematically, this presumably means terms with mixed p and x factors (B1) pressure should be mixed with time We next note that even though a particle is moving in a box, there is no special weight given to any x or t value and this notion must be present in a mathematical probability scheme. In fact, this consideration suggests a real valued weight: F(x,p,t) =constant ((3)) As a result, all information about p and pressure is lost in such a probability prescription. This approach is equivalent to ((2)) with a similar expression existing for t which is consistent with ((2)). This begs the question: Even though p (impulse hit) and pressure exist physically, is it really necessary to involve them in a probability scheme? ((4)) We have noted above that L is a constraint which appears in ((2)) even for a particle at rest. It cannot be eliminated. This same constraint must involve the notion of force for a moving particle and we suggest that if the constraint cannot be eliminated, then neither can its accompanying force. In other words, the constraint and all of its physical consequences must be considered. This suggests that the answer to ((4)) is Yes. This is surprising because one is used to the particle at rest scenario. By answering ((4)) with a yes, one is forced to include the p and pressure information in a manner which does not violate ((2)), i.e. for p=0 , pressure=0, and one must obtain the real value ((2)). For p and pressure not equal to 0, the same notion of equal weight for all t and x still applies, thus it seems that one must use a complex number for a probability which includes pressure and p. We suggest that this is how the notion of a complex number appears in the free particle situation. In other words, it has to be introduced in order to “hide” pressure and p within a phase which seems to be a departure from classical physics. Thus, one might be wary of approaches which use classical Liouville theory which should not have a complex probability and then introduce complex functions. ((2)) is a probability linked with a particle at rest which does not interact. A moving particle is a dynamical one and one is not simply interested with position in space, but with interactions in space and time and this requires a dynamical probability. Thus, one cannot separate x and t from the interactions with which they are associated, we argue. We note that in (1), a marginal probability: Integral dp exp(i p dx) F(p,x,t) ((5)) is introduced a priori, but it is not clear why a complex value should be used. (1) notes that one may then write ((5)) as W*(q-dq/2, t) W(q+dq/2, t), but this is mathematics. We suggest that the complex value nature arises because one must include the physics of the forces linked to the constraint L in a probability, but must at the same maintain the equal weight in a t,x situation. Thus, the p, pressure information is hidden (mathematically) in the phase of a complex function. The Link With Special Relativity We started above with the notion of a particle at rest and ((2)). We then considered an observer in a moving frame as seeing the particle in motion. Such a frame scenario is linked to special relativity. We suggest, as we have in previous notes, that a probability linked to p, pressure, t, x should really be Lorentz invariant. We note that pressure is really linked to p and v and that v=pc/E for a particle with rest mass. Thus, the Lorentz invariant: -Et+px ((6)) is a candidate for the probability exp(-iEt+ipx) ((7)) ((7)) has an additional feature. If one has a p1,x grouping AND a p2,x grouping, then probability math implies multiplication and the result must be equivalent to p1+p2. The form of ((7)) allows for this and also for E1, E2→ E1+E2. In other words ((7)) is consistent with conservation of energy and momentum. Link with Physical Reality We have argued for a probability which contains E linked with t and p with x. In ((4)), we considered the possibility of having probability which does not include p and E as a physical possibility as p and E are not important in the rest mass scenario. The fundamental question then becomes: Is there any physical reason one requires the presence of a p,x grouping in a probability? We argue that there is. First, exp(ipx) and exp(-iEt) show that in Newtonian 2-body elastic scattering, any (ei,ej) (pi,pj) (momentum vector) outcome has the same product probability. Secondly, one dimensional photon or particle reflection-refraction at an n1-n2 index of refraction junction leads to a probabilistic result and we argue that this is linked to the values of p as experiment shows. This points to a probability associated with p and x, i.e. a dynamical probability. Conclusion Our main conclusion is that given a particle at rest, P(x)dx=dx/L, where L is an arbitrary length, L is often seen as linked to normalization, but we suggest it is a physical constraint. One cannot place a particle at rest outside the L region. If one views a particle at rest from a moving frame (-v), the particle moves with v, but if the constraint L (or its Lorentz contracted value) is taken seriously, the particle must now bounce back and forth in a box.. This implies that forces now appear such as p (impulse) linked to x and the time average pressure, linked to t. These forces, however, do not change the basic idea that no x or t carry any special weight. No information about p and pressure is needed to reach this conclusion and one might suggest that P(x)= 1/L and P(t) = 1/T. In such a case, all information about p and pressure would be dropped even though it is physically relevant information. The idea would be that it has nothing to do with probability in x and t. We suggest, however, that finding a particle at x is associated with an impulse hit p and time considerations are linked to v for a moving particle. These notions follow from a Lorentz transformation of a particle at rest. Thus, we replace pressure with E and still use p and try to create a Lorentz invariant probability with the extra force information hidden in a complex phase, i.e. exp(-iEt+ipx). This suggests that one has the notion of conservation of energy and momentum contained in such a probability which is physical and is something that one does not need to consider in a rest particle case. In other words, a particle at rest is linked with a non-dynamical probability as no interactions occur while a particle in motion is linked with a dynamical probability which may handle interactions/forces which are linked with x. References 1. Silva Filho, O. and Ferriera, M. A Bird’s Eye View on a New Stochastic Interpretation of Quantum Mechanics (Oct. 2025) Mathematics 2025, 13(21), 3571; https://doi.org/10.3390/math13213571 https://www.mdpi.com/2227-7390/13/21/3571