Random walks with step lengths equal to the shortest possible physically meaningful distances are considered from the point of view of special relativity involving two observers moving uniformly with respect to each other. A requirement of statistical equivalence of the probability distributions seen by those observers leads to the Lorentz transformations, provided a randomly moving particle shifts from one submicroscopic cell of uncertainty to a neighbor with a speed equivalent to that of light. Ordinary smooth motion would appear to involve a tremendous amount of submicroscopic back and forth randomness subject to a statistical bias favoring a particular direction. The diffusive nature of the motion naturally leads to a spreading of the probability distribution.
Random walks with step lengths equal to the shortest possible physically meaningful distances are considered from the point of view of special relativity involving two observers moving uniformly with respect to each other. A requirement of statistical equivalence of the probability distributions seen by those observers leads to the Lorentz transformations, provided a randomly moving particle shifts from one submicroscopic cell of uncertainty to a neighbor with a speed equivalent to that of light. Ordinary smooth motion would appear to involve a tremendous amount of submicroscopic back and forth randomness subject to a statistical bias favoring a particular direction. The diffusive nature of the motion naturally leads to a spreading of the probability distribution.
Abstract The transformation from a time-dependent random walk to quantum mechanics converts a modified Bessel function into an ordinary one together with a phase factor e,ir/2 for each time the electron flips both direction and handedness. Causality requires the argument to be greater than the order of the Bessel function. Assuming equal probabilities for jumps ± 1 , the normalized modified Bessel function of an imaginary argument is the solution of the finite difference differential Schrödinger equation whereas the same function of a real argument satisfies the diffusion equation. In the nonrelativistic limit, the stability condition of the difference scheme contains the mass whereas in the ultrarelativistic limit only the velocity of light appears. Particle waves in the nonrelativistic limit become elastic waves in the ultrarelativistic limit with a phase shift in the frequency and wave number of 7r/2. The ordinary Bessel function satisfies a second order recurrence relation which is a finite difference differential wave equation, using non-nearest neighbors, whose solutions are the chirality components of a free-particle in the zero fermion mass limit. Reintroducing the mass by a phase transformation transforms the wave equation into the Klein-Gordon equation but does not admit a solution in terms of ordinary Bessel functions. However, a sign change of the mass term permits a solution in terms of a modified Bessel function whose recurrence formulas produce all the results of special relativity. The Lorentz transformation maximizes the integral of the modified Bessel function and determines the paths of steepest descent in the classical limit. If the definitions of frequency and wave number in terms of the phase were used in special relativity, the condition that the frame be inertial would equate the superluminal phase velocity with the particle velocity in violation of causality. In order to get surfaces of constant phase to move at the group velocity, an integrating factor is required which determines how the intensity decays in time. The phase correlation between neighboring sites in quantum mechanics is given by the phase factor for the electron to reverse its direction, whereas, in special relativity, it is given by the Doppler shift.
Quantum mechanics is an incredibly successful theory and yet the statistical nature of its predictions is hard to accept and has been the subject of numerous debates.The notion of inherent randomness, something that happens without any cause, goes against our rational understanding of reality.To add to the puzzle, randomness that appears in non-relativistic quantum theory tacitly respects relativity, for example, it makes instantaneous signaling impossible.Here, we argue that this is because the special theory of relativity can itself account for such a random behavior.We show that the full mathematical structure of the Lorentz transformation, the one which includes the superluminal part, implies the emergence of non-deterministic dynamics, together with complex probability amplitudes and multiple trajectories.This indicates that the connections between the two seemingly different theories are deeper and more subtle than previously thought.
Abstract Quantum mechanics is an incredibly successful theory and yet the statistical nature of its predictions is hard to accept and has been the subject of numerous debates. The notion of inherent randomness, something that happens without any cause, goes against our rational understanding of reality. To add to the puzzle, randomness that appears in non-relativistic quantum theory tacitly respects relativity, for example, it makes instantaneous signaling impossible. Here, we argue that this is because the special theory of relativity can itself account for such a random behavior. We show that the full mathematical structure of the Lorentz transformation, the one which includes the superluminal part, implies the emergence of non-deterministic dynamics, together with complex probability amplitudes and multiple trajectories. This indicates that the connections between the two seemingly different theories are deeper and more subtle than previously thought.
We argue that continuity and discontinuity in special relativity play an important role in the probabilistic wave properties of quantum mechanics. Special relativity is involved with transformations from a rest frame to a frame moving with a constant -v, for example . The discontinuity appears in the form of a change from v=0 to v, while the continuity arises from Lorentz invariants which indicate one has the same physical state whether it is viewed from a frame with one velocity or another. This invariance applied to E, p (momentum) as a modulus forces a minus sign between the two, in order to allow for linkage with E=mo (c=1) and p=0, i.e. -EE + pp = -momo (c=1). Another invariant, however, is A= -Et’ + px’, and in this case the minus sign and the discontinuity lead to a new x’,t’ (in comparison with the rest frame for a particle with rest mass mo), but also to t’ -> t’+ b/E and x->x’ +b/p leaving A unchanged (b=constant). The discontinuity in v, which leads to a jump in x’,t’ (as seen in the moving frame) also manifests itself in a time-length interval which ultimately leads to exp(-iEt+ px), we argue. In other words, classically one considers constant velocity as smooth continuous motion, but the introduction of two frames, v=0 and -v, leads to a discontinuity which leads to “chunkiness” in the form of a frequency and wavelength. If these cannot be perceived classically (as is often the case), then one considers only smoothness. Relative differences in time or length or in frequency or wavelength, however, can appear in nature for length scales of the order of the frequency/length scale. In the case of n-slit interference-diffraction, a relative difference leads to a “jump” (so to speak) which manifests the wavelength. In particular, a photon moves along the y direction with a phase px (considering a steady state picture). It reaches a two-slit system along the x-axis and there is probability for it to interact with both slits. If one considers two parallel rays from each slit moving towards a point on a screen very far away (as is done traditionally), there is again smooth motion with a rotated p vector (same magnitude), but for a point on the screen to the left of the center, there is a jump in the length associated with the right slit. In other words, there is a jump similar to the v=0, -v jump. This leads to a relative difference in length associated with both rays (linked to an OR probability situation using exp(ip dot r). This relative difference in length displays the phase difference initiated by special relativity. In the case of a bound state, different impulse hits by a potential create different momenta which have different phases exp(ipx) because of special relativity and one has interference. The same phase A= -Et + p dot r, which is linked with a chunkiness due to a discontinuity, is also associated with continuity because A is a Lorentz invariant. Thus A=constant may be used to solve for refraction/reflection problems with or without moving mirrors/interfaces with different indices of refraction. Continuity appears further in a 1-dimensional reflection/refraction problem when applied to the exp(ipx), i.e. Aexp(ipx) + Bexp(-ipx) = C exp(ip2x) where E=pc = p2 c2, p2=n2p and c2=c/n2 with n1=1, together with a continuity equation for the first derivative. Thus, both continuity and discontinuity (or jumps), which follow from special relativity, seem to play an important role in quantum mechanics.
The concept of a "random walk" has been widely used to describe many real-world systems from the Brownian motion of smoke particles to the fluctuation of prices on the stock market. The mathematics is usually explained though the analogy of a person dictating their movements by the toss of a coin: get heads, for example, and they move one step to the right; get tails and they shuffle one step to the left. After many coin tosses, the person's position is random – though, statistically, it is likely to be near the starting point.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). Lorentz invariance for a particle with rest mass seems to be linked with the idea that one ultimately has a rest frame for the particle. One then establishes a relation between the rest mass in one frame and its new E and p as seen by another frame. One does not, however, need to impose Lorentz invariance conditions on the interaction which creates the rest mass it seems. For example, exp(-iEt+ipx) is the Lorentz form used to describe a free particle, but exp(ipx) is the only portion used to describe its interaction with V(x) in the rest frame of the large mass creating V(x). Furthermore, exp(-iEt+ipx) ensures conservation of energy and p for a free particle, but in an interaction case, that does not happen as the potential may absorb energy.