Hi, Rasmo suggested that I let him continue the previous conversation about the Transactional Interpretation in Quantum Physics. I was hestitant, but he said he had a point to make. I apologize for his long-windedness, but he said he wanted to be sure everyone understood the basics along the way. Before he begins, however, I'd like to tell you about my new program I am almost finished with. I used it to illustrate this paper. It is a function parser that lets you compose your own functions in algebraic notation and then graphs them in either cartesian or polar coordinates. It can do iterated recursive function calls as well as normal graphs. To be added will be superposition, heterodyne, parameterization, and composite functions. Line draws and text annotation are included. It's a nifty little tool for exploring ideas. I'll have it on the web site before too long. It's called 'TinkerGraph'.

Swint

Through the Transactional Looking Glass

"New interpretations shed light on quantization and gravity!"...Ras

Mass is most universally symbolized by the circle. From planets to breasts to atoms, the circle epitomizes perfect completion - a bisection of space into inner and outer. A circle is the locus of all points equidistant from any given locus. Rasmo has discovered that a circle is not a function, however. Recall from Calculus and Analytic Geometry that a function is a one-or-more to one mapping of points in the plane i.e. for every independant value (X) there exists one and only one dependant value (Y). This means that graphically one gets a normal graph where no points lie directly above or below each other. The circle, however, circles about itself such that the top half semicircle above the x-axis lies over the bottom semi-circle below the x-axis. See Fig.1

One can see why this must be so by analyzing the relation between points on a circle. The Pythagorean Theorem tells us that :

R*R = X*X + Y*Y

Solving for Y to get in functional form y = f(x), gives Y = +- sqrt(R*R-X*X) The square root being a quadratic has two roots, one positive and one negative. The positive root graphed as a function yields the top semicircle, while the negative root yields the bottom semicircle. We are talking second-order non-linear equation here.

Now nature is most fond of non-linear clothing. Linearity is a most simple kind of order found only at equilibrium or in tightly regulated systems. When physicists model nature and attempt to get her measurements, they tend to view only her front side in a positive fashion, forgetting the negative inner curvature. Their eyes just can't penetrate that far, so they do the next best thing - They parameterize their way around her total body making positive measurements along the way. This is the only way they can get umambiguous readings. An typically male instrument that reads more than one value for a given dip of the stick presents one with a rather dark conundrum about indeterminancy. This is nature's tease for the physicist. Ok, so what the bejeez is parameterization. Well, its an analytic tool for breaking the complex whole into simpler parts - functional parts - that give only one answer at a time. The chinese puzzle box experiment gives one thought to ponder on this subject too. Well, how does one parameterize the full circle?

Easy: X = R cos(theta) and Y = Rsin(theta) where theta is the angle of a given radius R. We are only part way there though, because we need a way to sequentially navigate a way around her voluptuous form. That's easy too. Simply set our speedometer to angle (theta) = frequency (omega) * time (t)

X = Rcos(wt) and Y = Rsin(wt)

The terms are now parameterized over time, a pseudo variable we introduced which we will vary, at our own rate, to systematically step our way round the circle. Note that the sin() and cos() are true single valued functions! Now, traditionally, positive time in this sytem is counterclockwise as angles go. One can just as easily step clockwise around her beauty, as she is indifferent to such games. The important thing is that our happy physicist uses his two functional parameter probes, one in each hand, to take measurements at his leasure. Each probe, sin and cos, yields precise determinate measures at each location. Now everyone is happy. We have a complete map of her nature and no ambiguities. We had no need of negative time, but it was available as an equally acceptable alternative approach. The important thing is that the logistics of real knowledge have been preserved. The negative time thing was just a figment of protocol or point of view anyway. No extra information will be gleaned by taking that trip even if it is consistent.

Once one gets used to the idea of parameterization, he realizes that time is an immanently circular affair. It's sole purpose of invention is to denote cycles of measurement. A timepiece that was linear like a yardstick would be of very limited use indeed. Imagine waking up to a tiker-tape time alarm that was twenty miles long after a few years. Again negative time is an artifact of protocol, and has no impact on synchrony.

Physicist use time in equations to parameterize the action because those dang non-linear equations are so difficult to manage. One can take even very simple non-linear stuff like R*(1-X)*X and get really wildy chaotic results at certain values. Modeling nature's non-linearity is a wild ride. Non-linearity involves adding higher dimensions to the picture. There are ways to produce dimensional flattening. One is the one we have just accomplished. Read BVector.txt on the web site for additional thoughts about how information can be transmographied(?).

Next Rasmo, as promised, wishes to give one a deeper peek through the looking glass. A curious mathematical physicist might question, can I solve for t and get real time values, just like I did for x and y? Ok, you asked for it.

X = Rcost(wt) :: angle wt = arccos(X/R) :: t = (1/w)arccos(X/R)

The problem is that there are an infinite number of angles that lead to a given value of X. The arccos function is a function only in the sense that mathematicians and computers throwout all but the first resultant inorder to be functional. This all goes back to the fact that the time variable is a circular parameter that cycles out to infinity. Give up? No? Well ok, here's something that will blow your mind.

Did anyone notice that so far we have confined ourself to values of X & Y that are less than or equal to the radius of mother's circle? How curious. What happens if we attempt to measure mother with our probes outside the radius of her garment? Well, things sort of blow up in our face. Remember the y =f(x) form Y = sqrt(R*R-X*X). If X > R we have a negative inside the radical, and this is radically unsolvable! Well, never fear, mathematicians have lots of imagination for imaginary things, so they came up with a name i for imaginary number or squareroot of minus one. If we don't get upset we can use this to our advantage. Now factoring out minus one, we have Y = sqrt(-1) * sqrt(X*X-R*R). This can be written as Y = i * sqrt(X*X-R*R). This gives us 'i' times a real value we can probe! All we have to do is set some protocols. Let +i represent points in the positive Y direction and -i represent points in the negative Y direction.

Now we have two forms of the equation, one for X <= R and one for X >= R involving the imaginary number i.

Y = f(x): {for X<=R| Y = sqrt(R*R-X*X) and for X>=R| Y = i * sqrt(X*X-R*R) }

This gives us something we can really measure and graph for all real X!

Here it is: Fig.1b: The Apparent circle.

This is the circle as it appears to the dweeby physicist running from side to side with out using his dual parametrization probes. Its a mass view from the front (or top if youre going to get picky) The two wings on either side are quite interesting. Closer examination reveals that they are assymptotic to 45 degree lines going through the origin. This is all valid function, baby, but there is still a conundrum. The zero pts on the X axis have infinite curvature ( curvature being the second derivative of a function ). They present what is called a singularity in some 'circles' eh, eh. You see we have discovered a primitive quantization inherent in mother's dress- you're either in or out so to speak, or you're mighty confused. Confused in an experimental physicist's case is when his readings go off the scale and don't mean anything. So what can we do to eliminate this interpretive conundrum?

Easy. Just remember that the outer wings are imaginary and our protocol was to use positive i on the first attempt. Also remember that a circle represents inner space and outer space, topologically distinct islands of mathematical truth. Why dont we adjust our protocol to negative i and flip the outer wings over? Ok, here we go:

 This sucker is pretty easy to interpret if you give it a chance. It is still the facing-forward mass view, mass being the hidden space inside the marble. It has strict limits of extension but is now differentiable. The outer space has a reverse curvature- just what one would expect as a compromise between space and not-space so to speak. Put more simply, a circular (spherical.. n dim) mass cordons off space into two opposing curvatures. The inner curvature exhibits inertia, while the outer curvature exhibits gravitational curvature or field force. Field force is imaginary in the sense that it is active only in the presence of another mass. That's why 'i' entered the equation. This force is inverse square by the way because mother's form is circular second-order non-linearity.

Well, dont leave yet folks. We are only half way there. Remember we still have some protocols to experiment with. What if we graph this:

Y = f(x): {for X<=R| Y = -sqrt(R*R-X*X) and for X>=R| Y = i * sqrt(X*X-R*R) }

This time we are looking at the bottom (back) half. Here's what we get:

This time we have an interpretative view of force on a mass sitting in a potential gravity well. It is important to notice the 45 degree assymptotes in both cases. These lines mark the natural light-cone limits of relativity.

Ahhhh so; very interesting.

Ok. Let's put it all together now in one complete picture of mother and her relations.

Below we have the total circle parameterized, and freed of infinities.

Valla