Quantum exp(-iEt+ipx) and Classical Action

Quantum exp(-iEt+ipx) and Classical Action Francesco R. Ruggeri Hanwell, N.B. Dec. 9, 2025 It is well known that the quantum free particle wavefunction is exp(-iEt+ipx) (one dimension) and that -Et+px is a Lorentz invariant. It is also known that the classical free particle action A = Lagrangian*t may be written as A= -Et+px in both the relativistic and nonrelativistic cases if one uses x=v/t. One may ask whether it is a coincidence that A is the same Lorentz invariant as -Et+px in exp(-iEt+ipx). We argue that it is not. Quantum Free Particle Wavefunction We have argued in previous notes that one may obtain exp(-iEt)exp(ipx) as two probabilities which are involved in conservation of energy and momentum and in the removal of all possible bias (consistent with conservation) in two body elastic scattering. First of all, free particles should all carry a weight of 1 because there is no reason to favour one p, E over another. In an elastic collision, however, one must distinguish between (E1,E2) (p1,p2) and (Ei, Ej) (pi,pj)s for which E1+E2 = Ei+Ej, p1+p2 = pi+pj (one dimension) and Ei,,Ej’s and pi,pj’s for which this does not hold. One cannot have P1(E)=1, P2(p) = 1. E and p dependence are required and a weight of 1 must be maintained, suggesting a complex number. One might propose exp(i C1 E) and exp(i C2 p), where C1 and C2 are constants to make the arguments dimensionless. The problem, as we have noted several times before, is that for p=m1v1=m2v2 (or the relativistic analogue), both p’s may satisfy a conservation of momentum equation in one frame, but not in a boosted frame because: p,E1 → g(v) p + v g(v) E1 while p, E2 →g(v) p + v g(v) E1 ((1)) This is unphysical and one must seek a Lorentz invariant which is linear in E and p to allow for conservation of energy and momentum. Hence: exp(-iEt+ipx) ((2)) is the probability. The presence of x and t, however, with x=vt representing the classical trajectory now involves space and time as well as E and p. It is well known that Newtonian formalism may be expressed in terms actions, Lagrangians, Hamiltonians etc. Furthermore: H = pv - L = Energy for a free particle ((2a)) and dL/dv = p ((2b)) These math constructs are linear in the variables E and p which also appear in the Lorentz. invariant in ((2)). In general, L(x,v, t) and H(p,x) ((3)) If, however, one uses the constraint x=vt = p/E t relativistically, then in a Lorentz boosted frame, one sees: x’ = p’ / E’ t’ (c=1). Now, x=vt follow from: d/dt dL/dv - dL/dx = 0 if there are no constraint ((4)) v is not part of a 4-vector, but one may recast L in terms of the special relativistic variables E.,p,x,t. To see this, consider: L = -mo sqrt(1-vv/cc) ((5)) We consider E and p as constants which are linked with the derivatives in ((4)). We consider: H= E = pv - L and dL/dv partial = p to see what consistent form works. The action is t * L. Thus: Ht = Et = pvt - Lt = px - action or: action= -Et+px ((6)) ((6)) is a Lorentz invariant. If one considers x = p/E t, then: L = action/t = -E+ p x/t so dL/dv partial = p with E and p held constant. ((7)) Furthermore, Lt = -mot sqrt(1-vv) = -Et (1-vv) = -Et + Exv = -Et+px ((8)) Thus, action= -Et+px if x/t = v in the relativistic case. One may show that the same holds in the nonrelativistic case. Now, the above formalism should hold in any Lorentz boosted frame. If Action = a Lorentz invariant, then the above arguments hold in all frames. The next point is that this Lorent invariant is linear in E and p because H=E and dL/dv = p. There is only one choice for the Lorentz invariant: Action = -Et+px ((9)) The stochastic non-biased energy and momentum conserving interactions, however, also require a Lorentz invariant linear in E and p and so there is only one choice if x and t are used, namely: -Et+px ((10)) It does not seem that there are variables other than x and t which may be used in general. Both the non-bias, conservation obeying stochastic collision and the action theory must lead to the same Lorentz invariant because they must both contain x,t, E and p. This suggests that conservation of momentum must somehow be part of the action, Lagrangian, Hamiltonian formalism which seems to be the case because these yield Newton’s equations which introduce conservation of momentum and allow for elastic (conservation of kinetic energy) collisions. By introducing Lorentz invariance and linear E,p,x,t one cannot separate free particle motion from stochastic behaviour. Thus, each point x,t on a trajectory is linked with a stochastic: dx= hbar/p and dt= hbar/E ((11)) Conclusion In conclusion, we argue that it is possible to obtain the free particle complex probability (wavefunction) exp(-iEt+ipx) from free particle non-biased collisions which conserve energy and momentum. As we argue above, one is forced to use a Lorentz invariant linear in E and p and x and t are seemingly the only other variables available. This leads to exp(-iEt+ipx) and one might think that this has nothing to do with x=vt motion, but that is not the case. The relativistic free particle action is: L*t = -mo sqrt(1-vv) t = -Et + px which is a Lorentz invariant. We show above other ways to obtain this result. The point is that the Hamiltonian, Lagrangian theories deal with linear terms of p and E and should be part of a Lorentz invariant theory. Thus, action = -Et+px is the only possible such Lorentz invariant quantity and must match the stochastic Lorentz invariant -Et+px. Thus, both stochasticity about the x,t trajectory point described by exp(-iEt+ipx) (dx=hbar/p and dt=hbar/E) and the trajectory x=vt are both linked with -Et+px.