Quantum Free Particle exp(-iEt+i p dot r) Based on Relativistic Path Probability

Quantum Free Particle exp(-iEt+i p dot r) Based on Relativistic Path Probability Francesco R. Ruggeri Hanwell, N.B. Dec. 15, 2025 In a recent paper, (1) introduces a path related probability scheme in order to derive quantum mechanics probability. This probability is a real positive number between 0,1 and so one is classical, but the fact that it is linked to paths and not simply a position, time x,t suggests that this approach differs from strictly classical probability. In particular, the assumptions used by (1) are path related. (1)’s path based probability is P(nm) ( f(x,t)), where nm is a path and f(x,t), a Lorentz invariant. Bayesian composition is allowed and P(nm) = P(n)(path1) P(m)(path2). As a result, one may break a probability into a product of probabilities based on paths. For example, path1 might be from a starting point to a point P and path2 from P to a detector. For a linear path along x, this suggests the form exp(-iEt+ipx)=W(x,t). To obtain a real probability, one needs (1)’s assumption of time reversal invariant, i.e. P(-Et+px)=P(Et-px), so P = W*(x,t)W(x,t). We argue that (1)’s quantum form follows from the last two (of three assumptions). Formally, one suggests that quantum mechanics arises from (A) “Kolmogorov additivity restricted to one or two path contributions”, (B) time reversal symmetry and (C ) Bayesian composition. We go on to argue that the path related approach of (1) is equivalent to a momentum-energy conserving probability approach of (2). Probabilistic Theory of (1) In (1), a probabilistic theory is developed using three basic assumptions: (A) “Kolmogorov additivity restricted to one or two path contributions” (B) Time reversal symmetry (C) Bayesian composition The point seems to be that there exists a real-valued classical probability that is associated with a path in space, i.e. a dynamical situation. Usually P(x,t) is not linked to a path, it is simply a probability at x at t regardless of what trajectories the particles take. We argue that to assume that a path is involved in the probability is already a departure from classical probability theory. In (A), (1) mentions the idea of a path and later insists on Lorentz invariance. It is noted in (1), that: -Et + i p dot r ((1)) is the simplest Lorentz invariant. We argue, however, that it is the Lorentz invariant in keeping with assumption (C ) Bayesian composition. In particular, (1) states that: Pnm(f(x1,t1)) = Pn(f(x2,t2)) Pm (f(x3,t3)) ((2)) Here f(x,t) is a Lorentz invariant and Pnm represents a path from say the origin to n and then on to m. We argue that if ((2)) is to hold, one may consider a path along a single line, say along the x axis. Then n is the distance from 0 to n and m, from n to n+m. If the product is to hold in ((2)), then one expects a power law in the linear x exponent. Probability, however, should not be a real function of distance and so one may use a complex i. exp(i constant x) ((2)) ((2)), however, is not Lorentz invariant and so ((1)) is not just the simplest Lorentz invariant, but the one which corresponds to a linear x and so ((2)) becomes: exp(-iEt+i p dot r) ((3)) ((3)), however, is complex and (1) wishes to find a real-valued probability. This is where the assumption of time reversal enters. (1) defines it as: -Et+px → - (-Et+px) with Pn(-Et+px) = Pn(Et-px) ((4)) To find a scheme consistent with ((4)), one may consider dt → -dt and dx→ -dx. Then v→v and p→ p and ((4)) holds. If ((4)) is the case, then one may simply make the probability a product of: exp(-iEt+ipx) and exp(iEt-ipx) ((5)) As a result, using (1)’s assumptions (B) and (C ), one may obtain exp(-iEt+ipx) without any need for (A). To see what assumption (A) entails, one may note that Kolmogorov additivity is introduced as: P(A union B) = P(A) + P(B) - P(A intersection B) ((6)) From ((6)), (1) derives (with Si = -Et+px for path i) P(n) (Si) = Sum 1<i<j<n P(2) (Si,Sj) - (n-2) Sum i=1,n P(1) (Si) ((7)) Assumptions (B) and (C ) are then used with ((7)) (i.e. assumption (A)) and (1) states that the only function which satisfies all three assumptions is: Pn(Si) = | Sum over i=1,n exp(i k Si) | power 2 ((8)) ((8)) is the quantum result which holds for various paths, but we have argued above that one only needs assumptions (B) and (C ) to determine exp(-iEt+ipx)= W and Probability = W*W. Given that the path approach holds for W, a single path becomes an OR multiple path through: Sum over i exp(-iEt+ipx) (path I) ((9)) As a result, it seems that one may obtain the result of (1) by only using assumptions (B) and (C ). We next wish to compare the path oriented approach of (1) with a momentum-energy conservation probability approach given in (2) Probability Associated with Conservation of Momentum and Energy In (2), we suggested that one may introduce a probability into Newtonian elastic scattering to deal with the notion that given an initial e1,e2 (energies) and p1,p2 momentum vectors, any ei,ej and pi, pj (vectors) have the same probability weight. We assume that such a weight is given by P1(ei)P1(ej)P2(pi)P2(pj) (i.e. the assumption of Bayesian composition of (1)) and is also associated with conservation of momentum and energy. This suggests a linear form in E and p for P1 and P2, possibly: exp(i C1 E) and exp(i C3 p) (p magnitude along the x-axis ((10)) The reason that the probability is complex is that there is no real valued distribution of any conserved quantity (energy or real value probability) among the free particles. Any E, p is as likely as the other, but there are probabilistic considerations when an elastic collision occurs. If one considers exp(i C2 p) (C2=constant for units), then changing the direction of the x-axis (for p along x) means that exp(i C2 p) and exp(i C2 (-p)) have different values when they should have the same. We thus suggest that one should have a variable linked to the x-axis so that xp is invariant under p→ -p and x→-x. Dt remains the same because we only change the direction in which the x-axis points. It is possible, however, to provide an explanation for exp(ipx) and then exp(-iEt) based on special relativity. In particular, if one has p=m1v1=m2v2 in one frame, either p,E1 or p,E2 may be used in a conservation of momentum equation. Such is not the case in a Lorentz boosted frame because: p (from E1) → g(v) p + v g(v) E1 and p(from E2) → p g(v) + v g(v) E2 ((11)) where g(v)=1/sqrt(1-vv/cc). The two p’s in the boosted frame are no longer equal and cannot be substituted for each other in a conservation of momentum equation. We thus suggest that one must have: exp(-iEt+ipx) ((12)) as the complex probability for a free particle. A point we wish to make is that Lorentz invariance of ((3)) introduces x and t as well as E,p. Thus, one starts with adding two E’s and 2 p’s at a given x, t in a two-body elastic collision, i..e exp(-E1t+ip1x) exp(-iE2t+ip2x) ((13)) Given the math form of ((12)), one may also consider it as being linked to paths. For example, if a particle is considered as moving to x,t from x=0, t=0, then this may have happened through an intermediate point, i.e. one may hold E, p fix and consider exp(-i E dt1 + i p dx1) exp(-i E dt2 + ip dx2) such that t= dt1+dt2 and x=dx1+dx2 ((14)) Thus, an analysis of probability which conserves momentum/energy at a point x,t ,together with Lorentz invariance automatically introduces a probability which may be seen to be linked with paths. Thus, we suggest that the momentum-energy conservation probability approach we use in (2) and the path approach of (1) are essentially equivalent. Conclusion In conclusion, (1) introduces a real valued classical probability which depends on a dynamical path. We suggest that this is a departure from classical probability because it is path dependent and the three key assumptions (1) uses are related to the notion of paths. We further argue that because (1) introduces Bayesian composition as the third assumption, i.e. Pnm( f(x,t)) = Pn(f(x1,t)) Pm(f(x2,t2)), where f(x,t) is a Lorentz invariant and path nm (0 to n to n+m) is equivalent to traveling from 0 to n and then from n to n+m. This immediately suggests a math function of exp(-iEt+ipx). In (1), -iEt+ipx is called the simplest Lorentz invariant, but for Bayesian composition along a straight line, one requires a linear x and hence -Et+px. Thus, -Et+px is the Lorentz invariant to use. The time reversal assumption of (1), namely that P(-Et+px) = P(Et-px) suggests that P = exp(-iEt+ipx) exp(iEt-ipx) for a simple path. For a more general path one may use: W(x,t) = Sum over paths exp(-iEt+ipx) and P= W*(x,t)W(x,t). Thus, we suggest that one may obtain the results of (1) if one only uses the last two of the three assumptions. We then argue that the path approach of (1) is essentially equivalent to the approach of (2) in which a momentum-energy conserving probability is derived for two body elastic scattering. In such a case, exp(i C1 E) and exp(i C2 p) cannot hold in all frames because for p=m1v1=m2v2, p with E1 and p with E2 both satisfy a momentum conservation equation. In a boosted frame the two p’s are no longer equal and cannot satisfy the same conservation equation. Thus, we suggest that one must use a Lorentz invariant and arrive at exp(-iEt+ipx). Given this form, one may hold E,p fixed and consider t and x as broken into parts and thus obtain the path approach of (1). To obtain a real-valued probability one may write W(x,t) = Sum over path exp(-iEt+ipx) and then take W*(x,t)W(x,t). References 1 Sajnok, K. and Debski, K. and Dragan, A. The Relativistic Reason For Quantum probability Amplitudes (2025) .https://www.semanticscholar.org/reader/5badadce20a117c21bd7e6768144b026db89c1ef 2. Ruggeri, Francesco R. Joint Phase Space (x-p) Probability and Free Particle Quantum Mechanics (preprint, zenodo, 2025)