Numerical experiments in relativistic phase generation through time reversal

G. N. Ord

R. B. Mann

E. Harley

Qin Qin Lin

Andrew Lauritzen

Sep 28, 2006

Abstract

In quantum mechanics, `particles' are represented as wave-packets. We reverse this picture and generate waves from the classical particle paradigm using time reversal to effect subtraction. This is done in the relativistic regime in 1+1 dimensions in order to simulate the `carrier' wave eimt for a particle of mass m.

1  Introduction

Both the Schrödinger and Dirac equations are wave equations describing the propagation of wavefunctions in time. Whether or not wavefunctions directly represent elements of an external reality is still an open question, however their use as elements of a probability calculus is unambiguous.
At a classical level, it is customary to think of an electron as a particle, and on measurement the electron appears to exhibit particle properties. We commonly try to reconcile this wave-particle duality by constructing localized wave-packets and supposing that measurement of position `collapses' diffuse waves into these localized objects.
While this arrangement works "for all practical purposes"1 one might ask "Can we reverse the picture and construct waves from particle-packets?" The most well-known and successful such reversal is the path-integral formulation of quantum mechanics. In Feynman's spacetime approach, the propagator for a particle to move between any two spacetime points is constructed by adding a complex phase factor for every spacetime path between the source and target. The phase of the complex weight is a smooth function of the classical action associated with the path, however there is no known analog of the phase itself in classical particle mechanics. Feynman's phase, exp[i(h/2p)L dt] is an elegant, but formal way of grafting wave mechanics onto the particle paradigm.
Feynman came closer to relating phase to the classical particle paradigm in his "chessboard model" []. This was a path integral description of a relativistic electron in 1+1 dimensions. Paths in this model differed from those in the non-relativistic integral in an essential way. The relativistic paths had an inner scale (the Compton scale (h/2p)/m) over which path-segments were portions of light-cones x(t)=x0 ct. This feature made the paths a finite length and, interestingly, allowed the introduction of phase in discrete units. In the chessboard model, the phase contribution of any path is one of the elements of the set S={1, -i,-1,i}, each element being associated with a specific spacetime direction. The effect of the sum-over-paths is to interpolate this discrete phase into a high frequency `complex Boltzman factor' exp[i m t]. From his unpublished notes[], it is clear that Feynman thought of the four `intrinsic' phases in S in terms of particles and antiparticles(Fig.1).
That is, if 1 and i corresponded to right and left-moving electrons respectively, then -1 and -i corresponded to positrons (ie. electrons with reversed time orientation). If this association holds true beyond the chessboard model then the origin of wave propagation in quantum mechanics can be traced back to the time reversal symmetry of classical particles.
The discrete version of phase in the chessboard model allows one to construct a wave propagator through a `particle-packet'. However, the discrete version of phase in the model requires that a single `physical' particle like an electron be associated with a whole ensemble of chessboard paths. Thus chessboard paths cannot be thought of as individual histories of real electrons, they remain a formal devise for constructing a propagator. This is a peculiar feature of the wave-particle duality of quantum mechanics. Note that there is no such problem in the strictly classical domain. The Wiener paths associated with the Wiener integral of Brownian motion are idealizations of the actual paths of diffusing particles.
The qualitative difference between the roles of classical paths in diffusive systems and Feynman paths in quantum systems comes down to what exactly is being propagated in time. In classical systems only addition is being propagated ... densities are always just proportional to the number of paths in an additive fashion. In quantum systems both addition and subtraction are being propagated. This is illustrated in Fig. 2. The particle density for the first excited state of a particle in a box has a node at the centre. A classical propagator could only maintain such a node by excluding paths from the nodal region via a repulsive potential. Quantum propagators can maintain nodal regions by balancing positive and negative path contributions through the use of phase. At a basic level, quantum propagators create nodes by superimposing positive and negative contributions. For every path with a positive contribution, there has to be a path with a negative one to create the node. Although the continuum phase used in quantum mechanics cannot by itself be associated with a single classical particle moving forward in time in 1+1 dimensions, the subtractive effect of adding two paths of opposite phases together does have a classical counterpart. Figure 3 shows a sketch of a single continuous spacetime curve with an associated charge density as seen by an observer moving forward in t.The area where the forward and return paths coincide corresponds to a primitive node in a charge density arising from the path.
The investigations in this paper graphically illustrate how a single spacetime path may be used to create a coarse-grained version of the phase factor eimt in 1+1 dimensions. We make no attempt at rigour, rather the exercise is to demonstrate that there are a variety of numerically robust subquantum dynamical processes that maybe used to essentially `write phase on spacetime' using a single path. This reversal of the usual picture, while still very incomplete in its coverage of quantum propagation, none-the-less has potentially large advantages over the wave-packet picture. The existence of a single underlying trajectory for the propagator holds out the possibility that the difficult questions of measurement, locality and classical limits may be more easily addressed with this paradigm.
In the following we report on a series of numerical experiments designed to illustrate how entwining the forward and reversed paths allows one to reconstruct a physical analog of eimt from the geometry of the path alone. The first experiment shows how to count oriented squares in an entwined path.

2  Experiment 1

Here we illustrate the relation between a simple deterministic entwined path, and the counting of oriented areas that leads to the construction of phase. In Fig. 4 a simple entwined path is sketched. The horizontal axis is a space axis and the vertical axis is `observer' time. The particle has its own `time' parameter which is not a single valued function of observer time. IN the figure, portions of the trajectory in which both observer and particle time is increasing is coloured blue, when observer time is a decreasing function of particle time the path is coloured bue. Note that the rectangles bounded by the entwined path have alternating orientation in that their boundaries are traversed in opposite senses.
The form of the entwined path is motivated by special relativity in which we take c=1. In the figure the slopes of the line segments are all 1. Choice of this characteristic speed is motivated by the uncertainty principle which forces high momentum and hence velocity on small scales. Since c is the maximum physical velocity allowed we construct paths with this velocity on the smallest scales. In counting the oriented areas in the figure the first rectangle may be recorded as a plus one assuming counterclockwise is positive and the second rectangle is a minus one, and so on. Notice we can get the sign of the contribution by looking at the path formed to the right of the crossing points of the entwined path. Although this is not the trajectory taken by the particle we call the path an `enumerative path' because it is this path that helps us enumerate the oriented areas. In one dimension the enumerative path is made of links that are either parallel to the (1,1) direction or perpendicular to it. We thus count the oriented contribution of a path in two components, a (1,1) component or a (-1,1) component.

Footnotes:

1See J.S. Bell's article "Against Measurement []