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Exploring discretized wave equations

The simplest dynamic discrete system involves a single scalar value that changes at each time step or iteration. A simple time symmetric8.5finite difference equation for this is as follows.

\begin{displaymath} f_{t+1} = T(nf_t/d) - f_{t-1} \end{displaymath} (8.13)

$n$ and $d$ are integers (numerator and denominator). $T$ is truncation toward 0 defined in Table 7.1. The equation says that the next value $f_{t+1}$ is obtained by multiplying the current value $f_t$ by a factor ($n/d$) truncating the result and then subtracting the previous value $f_{t-1}$.

The corresponding differential equation is as follows.

\begin{displaymath} \frac{d^2f}{dt^2} = (n/d - 2) f(t) \end{displaymath} (8.14)

The $-2$ comes in because the second order difference equation subtracts $(f_t - f_{t-1})$ from $(f_{t+1} - f_t)$ generating a term $-2f_t$.

For $-2 > n/d > 2$ 8.14 has a solution as follows.

\begin{displaymath} f(t) = cos(t\sqrt{2-n/d}) \end{displaymath} (8.15)

$t$ is in radians8.6. Figure 8.6 plots a solution for $n = 19$ and $d = 10$. The solution increments the angle of the $\cos$ $.3162$ radians or $18.11^\circ$ each time step. It takes about 20 time steps to complete one cycle of the $\cos$ wave.

The solutions to the finite difference equation 8.13 are more complex than the solutions to the corresponding differential equation 8.14. The former are completely described by equations like 8.15. The latter have a rich structure that varies with the initial conditions. Table 8.1 gives the length until the sequence starts to repeat of the solution to 8.13 (again with $n = 19$ and $d = 10$) for various initial conditions.

Could the rich structure of the discretized difference equations account for both the weirdness of quantum mechanics and the fundamental constants of physics including those not derivable from an existing theory?

The remainder of this section is about intuitive possibilities. We need to develop the skills for collective work at the intuitive level. Western culture is good at focusing intellectual talent on a project beyond the capacity of an individual. The same is not true at early intuitive stages. One way to start is intuitive brainstorming in print.

The ultimate goal is to write on a half a sheet of paper a single discretized finite difference equation that explains all of physics and thus all of creation. One suspects this is possible in part because of the universality of the wave equation and in part because of the added complexity and nonlinearity that discretization produces. Such a model would only explain the structure of our conscious experience and not its essence. (See Chapter 3.) But such a model would be the Holy Grail of physics. It is the ultimate explanation Einstein was seeking. So let us brainstorm about this possibility.

The nonlinearity introduced by discretization may produce chaotic like behavior. Chaos theory is the study of continuous nonlinear systems that are so sensitive to initial conditions that an exponential increase in knowledge of initial conditions only allows a linear increase in predictability. For example to predict one second into the future might require an accuracy of $10$, but to predict five seconds into the future would require an accuracy of $10^5$ or $10,000$. Typically computing resources needed for prediction grows exponentially as well. These systems are not predictable in any practical sense. It is not possible to obtain sufficient knowledge of initial conditions and the computing power rapidly exceeds what would fit in the known universe. While the detailed behavior of these systems is not predictable many global aspects of them may be.

Chaotic systems often have attractors. These are states the system converges toward over time. They are like the point at the bottom of a circular bowl. If you drop a marble it will eventually settle down at the bottom of the bowl. One can often determine the attractors in a chaotic system and use these to predict the system's behavior. There may be multiple attractors and it may be impossible to predict which will win out, but one can be sure the system will wind one in one of these states. This is like a double bottomed bowl. If you drop a marble in at point midway between the two bottoms allowing the marble to roll in any direction you cannot predict where it will wind up but you know it will be in one of the two bottom points.

Discrete systems cannot be chaotic. There is an upper limit to the information it takes to fully characterize a discrete system and that alone disqualifies them. They can approximate chaotic behavior just as they can approximate any continuous system. If the universe is discontinuous then no truly chaotic systems exist. The sequences in Table 8.1 are a little like attractors. If a sequence is perturbed by slightly changing the current value it may start a new sequence. Longer sequences are stronger attractors. It is more likely to fall into or stay in such a sequence after a perturbation.

Going from a finite difference equation at a single point in space to one spatial dimension (or a line) greatly complicates matters. A single spatial dimension has an enormous number of states. There must still be loops of repeated sequences of states because the total number of possibilities is finite. However these loops could easily exceed the age of the universe in units of Planck time8.7. A one dimensional line of only $100$ integers between $-100$ and $100$ involves $201^{100}$ possible combinations.

Discretizing the wave equation makes it nonlinear. That is reflected in the varying amplitude of the peaks in Figure 8.6. In larger three dimensional examples it is expected that this can introduce chaotic like behavior that appears to be random. Yet there will be structural conservation laws, if the discretized finite difference equation is symmetric in time. This makes it reversible. In a sense nothing can ever be created or destroyed. The history of the universe is contained in the most recent states. Reverse their order and time will evolve backwards.

I speculate that there will be ``stable dynamic structures'' in these models that are somewhat like attractors in chaos theories. These are state sequences that repeat themselves approximately and that are relatively immune to external disturbances. However these structures can transform into each other either is a result of interactions or spontaneously. They are the model for particles.

Absolute conservation laws and probabilistic laws of observation are characteristics of this class of models. Could that account for the existing experiments? The transformation of particles would be a physical quantum collapse process. But it is a process spread out in time and space. There is no point at which the process is definitely complete. The conservation laws can prevent a transformation from being complete or even cause it to reverse after it seems complete. So the objections raised by Franson[24] make it difficult to know when a measurement is complete.

Envision a microscopic world of attractor like stable states. Occasionally particles are perturbed and transform between states. Time reversibility imposes a strong form of conservation that must be honored in the long run but can be deviated from significantly in the short run because of the nonlinear effects needed to discretize the wave equation. Transformations start to happen and reverse far more often than they complete. Multiple transformations can start at different parts of the same particle but at most one of them can complete. In this model strange things can happen.

Even with such a radically different discrete model, it can be hard to imagine how the more recent experimental results can be consistent with classical locality. In this model quantum collapse is a process of converging to a stable state consistent with the conservation laws. This can happen in many and very indirect ways. Nature may seem to conspire to remain consistent with classical locality and quantum mechanics until every possible loophole is plugged.

We now turn to the problem of explaining gravity in this class of models. The hope is to use only a discretized version of the wave equation. There is a different wave equation for a single particle with rest mass.

\begin{displaymath} \frac{\partial^2\psi}{\partial t^2}= c^2\nabla^2\psi-\frac{m^2c^2\psi}{\hbar^2} \end{displaymath} (8.16)

This is known as the Klein Gordon equation or the relativistic Schrödinger equation. It is identical8.8to the wave equation in Section7.4 except for the term $-\frac{m^2c^2\psi}{\hbar^2}$ where $m$ is the rest mass of the particle and $\hbar$ is Planck's constant or approximately $6.62606891 \times 10^{-34}$ Joule-seconds.8.9The rest mass decreases the rate at which the level of $\psi$ accelerates in time.

How can 8.16 be derived from the same rule of evolution that approximates the classical wave equation? This may be possible if there is a high carrier frequency near the highest frequencies that can exist in the discrete model. The Schrödinger wave equation for particles with rest mass would represent the average behavior of the physical wave. It would be the equation for a wave that modulates the high frequency carrier. The carrier itself is not a part of any existing model and would not have significant electromagnetic interactions with ordinary matter because of its high frequency.

Such a model may be able to account for the Klein Gordon equation for a particle with rest mass. A high frequency carrier wave will amplify any truncation effect. Because of this the differential equation that describes the carrier envelope is not necessarily the same as the differential equation that describes the carrier. If the carrier is not detectable by ordinary means, only effects from the envelope of the carrier will be observable, not the carrier itself. The minimum time step for the envelope may involve integrating over many carrier cycles. If round off error accumulates during this time in a way that is proportional to the modulation wave amplitude, this will lead to an equation in the form of the Klein Gordon equation.

The particle mass squared factor in the Klein Gordon equation can be interpreted as establishing an amplitude scale. The discretized wave equation may describe the full evolution of the carrier and the modulating wave that is a solution of the Klein Gordon equation. However, since no effects (except mass and gravity) of the high frequency carrier are detectable with current technology, only the effects of the modulating wave will be observable. No matter how localized the particle may be it still must have a surrounding field that falls off in amplitude as $1/r^2$. It is this surrounding field that embodies the gravitational field.

If discretization is accomplished by truncating the field values this creates a generalized attractive force. It slows the rate at which a structure diffuses relative to a solution of the corresponding differential equation by a marginal amount. Since the gravitational field is a high frequency electromagnetic field it will alternately act to attract and repel any bit of matter which is also an electromagnetic field. Round off error makes the attraction effect slightly greater and the repulsion slightly less than it is in solutions of the continuous differential equation.

Because everything is electromagnetic in this model special relativity falls out directly. If gravity is a perturbation effect of the electromagnetic force as described it will appear to alter the space time metric and an approximation to general relativity should also be derivable. It is only the metric and not the space time manifold (lattice of discrete points) that is affected by gravity. Thus there is an absolute frame of reference. True singularities will never occur in this class of models. Instead one will expect new structures will appear at the point where the existing theory predicts mass will collapse to a singularity.

Figure 8.6: Simple discretized finite difference equation plot
\includegraphics[height = 6in,width = 6in,keepaspectratio = false]{../fdesimp}

The above is a plot of the solution to $f_{t+1} = T(nf_t/d) - f_{t-1}$ with $f_0 = 100$, $f_1 = 109$, $n = 19$ and $d = 10$. $T$ is truncation toward 0 (see Table 7.1). It completes about 5 cycles for every 100 iterations. It departs significantly from the solution to the differential equation.

Table 8.1: Cycle lengths for discretized finite difference equation
$f_0$ 100 101 102 103 104 105 106 107 108 109 110 111 112
100 154 269 154 154 269 328 328 328 289 309 174 116 116
101 269 77 250 328 250 289 328 77 309 116 309 174 174
102 154 250 154 289 328 309 250 77 77 58 174 289 77
103 154 328 289 77 328 328 328 328 77 309 116 289 289
104 269 250 328 328 77 309 289 250 309 309 289 174 174
105 328 289 309 328 309 328 77 77 289 58 289 77 309
106 328 328 250 328 289 77 116 77 77 116 116 174 77
107 328 77 77 328 250 77 77 58 289 309 174 289 309
108 289 309 77 77 309 289 77 289 309 309 309 289 174
109 309 116 58 309 309 58 116 309 309 289 174 174 289
110 174 309 174 116 289 289 116 174 309 174 250 116 77
111 116 174 289 289 174 77 174 289 289 174 116 174 174
112 116 174 77 289 174 309 77 309 174 289 77 174 116
113 309 135 289 251 309 116 309 174 309 116 309 251 289
114 58 174 58 309 135 135 174 289 77 289 174 135 135
115 174 174 251 174 174 174 251 116 174 58 174 116 251
116 58 174 309 135 77 251 58 77 58 135 174 135 58
117 251 174 251 251 251 58 135 174 309 135 251 174 251
118 58 251 406 58 135 309 251 251 309 174 58 77 135
119 484 368 174 58 58 174 174 58 309 251 174 251 251
120 368 232 484 406 368 368 58 251 58 174 135 251 58
121 58 232 232 406 406 484 58 174 97 251 174 58 174
122 58 58 484 232 484 58 406 368 406 58 174 58 58
123 406 194 484 232 484 368 58 484 484 368 58 406 58
124 213 155 174 368 232 232 58 97 484 368 58 58 368
125 194 155 174 406 97 213 484 58 368 232 484 406 58
126 349 136 213 174 174 97 232 368 58 97 484 232 232

The above gives the length until repetition of the sequence generated by $f_{t+1} = T(nf_t/d) - f_{t-1}$ with $n = 19$ and $d = 10$. $T$ is truncation toward 0 (see Table 7.1). The table is symmetric about a diagonal because reversing the order of the initial two values does not affect the sequence or its length. It reverses the sequence order because the equation is symmetric in time.

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