Caduceus and the Causal Thread

Swinton Roof

Dec. 9, 2001

"The caduceus emerges as a snake in time which bipartitions itself into a real and imaginary half, conjoined at the waist, but free to fly with the wings of an eagle. At the end of its epoc it consumes itself and enfolds back into the oroboreal cleavage from whence it came, complex and chaotic yet organized and creative. It's alive! The sheer monstrosity of its infinite reflection lies deep within the heart of darkness" ... rasputin11

Previous papers have suggested that all of physics boils down to spatial measurements. Specifically, motion always involves a correlation between two spatial measures, one of which is deemed temporal. The correlation is nothing other than the fact that measure A coincides with measure B every time the two are performed together.

Imagine Galileo with a ball dropped from the Tower of Pisa and a ball dropped on an inclined plane. Galileo decides to call the inclined plane a clock to measure the time it takes for the first ball to fall from the top of the tower to the ground. He scribes a ruler on the inclined plane to measure time and optimistically says his clock always keeps reliable time. He then proceeds to measure the time it takes the first ball to fall to ground. To his astonishment it always takes 10.5 tick marks on his inclinometer clock. Eureka, he has measured motion!

He is so excited that he goes about using his clock to measure all sorts of other motions but soon discovers there are slight discrepancies. Eventually he realizes his clock isn't so regular and comes up with a dripping pail of water. It seems to be much more constant. Now his measurements do indeed improve in accuracy but in the process, by dropping balls from different heights, he discovers that his measured motions are not linear. They seem to accelerate. He still insists all of his clocks and their subsequent time keeping are linear and regular. His tick marks come for him to represent actual fixed size snippets of time. Always, however, he can find no other way to measure time other than some mechanical measure of a spatial quality.

With these advances under his belt Galileo tells Newton to come up a math for these results. Newton invents a calculus of change and math for describing motion as a differential ratio of two values - a derivative of space in time. He further elucidates acceleration as the second derivative of space over time. By then, clocks were mechanical beasties gear and spring or pendulum ridden and most assuredly very accurate and regular. A tick was always the same tick. Everyone forgot that at the heart of those beasts were little spatial rulers marked as countable teeth on gears or swipes of a pendulum in its swing. In fact, didn't time really move in constant repeatable cycles like the sun, moon, and stars. Time was real and it was LINEAR, or at least that was the idea.

The trick of course was coming up with really good clocks because just about all the motions one is ever about to run across in real life are decidedly not linear. Motions almost always speed up or slow down in unpredictable ways. That's why really good clocks were needed to begin with. Newton decided that motion in itself, left to its own devices, actually preferred to stay the same and he called it inertia. He discovered that inertia was somehow associated with heavy objects. It was the light ones you had to worry about changing too quickly. The mountains and towers of pisa tended to not move fast. But if they ever do get moving, look out, for they are really hard to stop! Since inertia correlated with size he came up with the idea of mass which correlated with force which correlated with acceleration and so.

Then Hamilton came along. He is mostly known for his equations which formalized Newton's mechanics into a universal system invariant to those nasty coordinate transforms (using other people's clocks and rulers). Hamilton was a bright guy with an obsession for quaternions, though, so he invented the math of imaginary numbers. When imaginary numbers were added to real numbers you could do all sorts of neat things unthinkable before. They were downright disturbing in the complexity of their behaviour and everyone decided to call them 'complex numbers'. These new numbers made it possible to describe periodic behaviours with algebraic type functions and wave theory was born. Schroedinger came along and took Hamilton's basic particle mechanics stuff and stuffed it with complex waves. I'm sure Hamilton would have been joyed to be one-upped in this fashion. He might have even said, "I'm glad you finally put it all together grasshopper! You did me in by stuffing my foot in my own mouth"

Eventually everybody got together and invented quantum physics. So long as you used complex numbers you could encode imaginary waves and real measurements on particles into the same equation. The catch was, you could only get a statistical reliability. The answers had to be probabilities. Feynman, another really bright guy, came along and said that those complex numbers were like little clocks whose hand spun round and round. The length of the clock pointer or hand represented the probability of a quantum event. The angle the hand makes was simply the confluence of the imaginary and real producing a periodic ever changing direction. You can never guess the actual angle or absolute time for such a tiny little clock so it's basicly running own it's own drumbeat. You could, however, surmise the frequency or period and most importantly the size of that arrow hand.

Thus we have some complex number P which represents say a photon. The complex wave equation for P is P = P*e^i(2*pi*f*t - phi) or alternatively P = |P|(cos(theta) + i*sin(theta)) where theta is the angle or time reading on this little complex clock. |P| is the length of the arrow hand. One can imagine that the imaginary component skews time sideways and backwards until it comes back around to it's starting position ready for another cycle. Following this imaginative idea, one could say that the little photon clock is moving around in imaginary time with no real value except at it's real number pivot point. This gives one a clue as to why things are probalistic. With a near infinite number of tiny Planckian clocks spinning around in unmeasurable asynchronous epocs, there simply is no way to make any consistent measurements except as a statistical endeavor. If we attempt to pin down an individual clock and stop its hand from moving, the whole universe would have to collapse into one grain of sand. That's quantum physics folks!

So what are we doing when we make measurements? We are defending Galileo before the Inquisition of Truth and declaring that there IS a correlation. Things correlate. There are invariances. Generalities are possible. Somehow or other some of those tiny clocks conspire to act together. When they do this, a pattern emerges. When the pattern is big enough we can measure it with our big clocks and big rulers. Well, what the devil organizes those little clocks? Feynman said that the length of the arrow could be given by the sum of all the possible little arrows that could have been in all possible other places and times. Somehow they all add up to this unique moment with this particular little clock. Strangely, each little clock is the result of all the other little clocks adding up - a hologram if you will. Depending on where and when it is, any individual little clock will correlate more strongly with some than others, hence the patterns. The clocks are recursively or self-referentially defined! Goedel be damned; full speed ahaed!

Now at each moment the superposition of little complex clocks yields the best guess for the size of a particular arrow (spatial measurement !!! probability only though!!!). The instantaneous patterns are emergent. What happens next though? Feynman said that when the little complex clocks happen in steps to get to some pattern you multiply them instead of adding. Remember that the adding of all possibilities above was to get a one time probability reading for the 'now'. When sequential steps are connected you multiply probability clocks to get the final 'now' reading. This equates to multiplying complex numbers.

That's where rasputin11 gets in on the action. He surmises that multiplicative chaining of complex numbers is actually what the causal thread running through events is. The instantaneous additive superpositions provide the convergence of measure but the multiplicative scaling and rotation over time determine the causal connections between those instantaneous pattern measurements. Now scaling and rotational invariances are what fractals are made of. That's how repeating patterns emerge to begin with. The whole thing is non-linear and chaotic if you will, but there is a thin edge where the patterns are indeterminate. These curvilinear lines of balance between convergence and divergence form the pathways of causal correlation. Strangely, the real and the imaginary conspire and intertwine to produce an indeterminate yet organized pattwern of behaviour. It's all in the numbers.

Z is a traditonal complex number but let's use P instead to highlight the fact that the absolute magnitude squared equals the probability of an event ala Feynman. Mandelbrot discovered a most wonderful set of complex numbers. He discovered that all you had to do was multiply a number by itself and add a constant. Then take that number and repeat. Do this a given number of times and check its final probability (absolute magnitude). If it is less than a given maximum, add it to the collection. Do this for all the numbers in the complex plane and you get the Mandelbrot set. Feynman might have said that Mandelbrot was exploring all the possible causal threads (iterations for an given starting number). A causal thread could thus be seen as:

P -> P*P + C where C is some additive extraneous factor. This extra term must surely be the superpositions available as a starting parameter for that particular moment in space and time. In fact, if we take a one dimensional view of a complex number, we might say that the real component is time and the imaginary component is space. This inversion of interpretation allows for the Mandelbrot's bilateral symmetry as a spatial phenomena and the progression from tail to body as the temporal exegesis. Julia set iteration taking C from the Mandelbrot as the starting factor produces an infinite succession of causal territory maps. The Mandelbrot in turn is the master, or should I say monster, map which organizes the whole collection.

The above is merely a one dimensional view however. A full three dimensional mapping would reveal perhaps the full reality of the physical world at our disposal. Plotting the individual causal trajectories (instead of the final convergence value) of the main map yields an image strikingly like the human brain in horizontal cross section.

Does form follow function here? Does iterative expansion of causal pathways attract structural acretion over time? Where do we go from here? Can we map the superpositions too? That will soon be explored. Stay tuned.

Any ideas about what I might mean by this and actually how to do it in my Frax program would be appreciated.

R.O.M. dec, 2001

Issue 2 of the Caduceus Papers

by S.Roof

Dec 9,2001

More on the Mandelbrot-Quantum Theory Connection

The information below was gleaned from the Wickipedia HomePage

...the instantanous state of a system is described by an element ? of some complex Hilbert space which encodes the probabilities of outcomes of all possible measurements applied to the system. The state of a system in general changes over time,  = (t) is a function of time, and the Schrödinger equation describes this change quantitatively. The equation is therefore of central importance in quantum mechanics. The general, time-dependent equation reads

i h d/dt  = H  ... and ... (t) = e^{-iE(t - t) / h}

where i is the imaginary unit, h equals Plancks constant h divided by 2p, and H is a self-adjoint linear operator on the Hilbert space, known as the Hamilton operator. The Hamilton operator describes the system under consideration and corresponds to the total energy of the system.

@@@

From the above we see that the Schrödinger equation basically equates the derivative of the wave equation with respect to time as being equal to the energy of the system. The wave function itself is unknown and has to be solved for, given a set of constraints. (t) = e^{-iE(t - t) / h is one such solution. This is just a complex wave equation as described in previous Caduceus papers. Psi() therefore is the same thing as our Z or P i.e. the complex probability amplitude which we iterate.}

^{Now, remembering that we discussed causal trajectories as iterated products and sums of complex numbers in the Mandelbrot set, it is interesting to notice the derivative:}

d/dt

Thinking in terms of discrete iteration, we see that d is just the difference between the last and current iteration, while dt equates to the time interval between iterations, which in the context of plotting the Mandelbrot we simply call it 1. When I originally designed my fractal program, I incorporated an unusual color selection feature in addition to the traditional convergence value method. I called it 'Trajectory Gradient'. This method arrives at the pixel color by taking the gradient or difference between the last two iterations. An interesting conjecture is to imagine that this form of plotting would reveal something about the Hamiltonian Energy since d/dt equates to H energy in the Schrödinger equation above.

Below is a plot I did of the Mandelbrot using trajectory gradient. I converted to grayscale to simplify the interpretation, so the color represents the absolute energy value (so to speak). Black represents convergence or low energy. White represents divergence or high energy.

It is tempting to speculate that this picture represents a potential well. Remember that we made the assumption in Issue 1 that the case we explore in the simple Mandelbrot is a one dimensional waveform where the real part of the complex number is space and the imaginary part is time, or vice versa. At any rate, if the above picture is of one-dimensional phase space, and the other two dimensions are equivalent under symmetry, we may be looking at the energy map of a solitary charged particle. If we plot the actual iteration trajectories at low iteration values for clarity we get

It appears that the orbits spiral in a fashion similar to what a negative charge might do in the presence of a positive source. There are some rather odd alterations or inconsistencies, however, so what's going on?

Well, consider that the particle has a size and that it is say a hydrogen nucleus. Below a certain point, an electron has to cross a forbidden boundary and merge, forming a neutron. It has fallen into the potential well, so to speak, and it can't get up. Just outside the indeterminate boundary of the 'mandelwell', however, the energy is striated and particulate with other seeming wells of lesser depth. Might these represent permissible stable energy shells esp. along the real axis? Remember, this is a 1dim view in phase space so the extra stuff apart from the levels exposed along the real axis reside in complex imaginary territory admixture. Are they transitional states? Another consideration is that the buds on the Mandelbrot produce Julia iterations that have simple number symmetries (rotational) with values of 3,4,5 etc. Allowing for the bilateral symmetry, we surmise 6,8,10 which are some of the shell numbers for electron orbits. Might this extra transitional stuff relate to the spatial topology of the shells?

Now for the kicker. The orginal complex waveform listed at the beginning is a solution derived for an actual hydrogen atom.

CONCLUSIONS:

Quantum entities seem to recursively define their own states or identities as causal threads in time. Complex mutitiplicative iteration yields quantized energy wells which become organizing attractor basins. The additive factor at each iteration is an outside complex superposition of complex waves i.e. the enviroment. Z ->Z*Z+C seems to say it all. Even the iteration arrow (->) is the same as the logic "implies" operator ->

A -> B -> C -> D ...

Without the ouside enviromental additive constant, a causal thread simply explodes or dies a quick death. The conjoining of superposition and multiplicative iteration yields structure - quantum structure! Entities thus map out their own identity as they explore every possibility in the complex Hilbert plane. The temporal beat of this iterative causal mapping must surely be governed by light speed considerations. Superposition guides the next best step, but MandelMath guides the overall destiny. Walking the fine line between catastrophy and chaos seems to be built in.

R.O.M.

PS. The next installment will attempt to grapple with that additive superposition thing. Preliminary thought suggests simply taking the sum of the trajectory amplitudes as they occur and then normalize by dividing by the iteration depth (if required). I am curious to see the result, but it will take some time as I have to reopen my dormant Frax programming project and do some more coding.

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"Junah finally learned to stop thinking without falling asleep" ... Bagger Vance
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Swinton