Crackpot Spammer Ch 09
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Disclaimer: The following short story is fictional. No resemblance to any person, living or dead, is intended or should be inferred.
*
Faustus begins his lecture with some miscellaneous facts about the mystery of complex variables.
‘We first think of Infinity as a symbol. It is often thought of as the ‘end of the line of integers,’ 1, 2, 3, and so on. And for the real number line there is a positive Infinity to the right and a negative Infinity to the left. The Texan would call the Infinity symbol a ‘lazy eight,’ which is the number eight turned on its side. The medieval scholar would call it a lemniscate, now called the Lemniscate of Bernoulli.’
‘Jump from the real line to the complex plane and many marvels occur. The complex plane may be viewed as a stereographic projection. This may be visualized as a ball on a flat table. The point of contact is zero. The ‘top’ of the ball would be complex Infinity. Pass a line from the top of the ball through any point on the surface of the ball, the sphere, and it uniquely determines a point on the complex plane. Shortly we will see that the table could be round or square, there is but one complex Infinity. A mathematician would say that the Riemann Sphere is the complex plane with an added symbol.’
‘In classical geometry, the stereographic projection is a mapping that projects each point on a sphere onto a point in the complex number plane. The projection is defined on the entire sphere, except at one point—the Infinity point. The stereographic projection is a way of depicting the sphere as the plane plus an Infinity symbol.’
‘So, suppose we are on a complex plane and decide to march along the x-axis say. Recall from high school algebra that a complex number z is given as x + i y. Here the symbol i is the square root of minus one and each of x and y is a real number. So march on and approach Infinity. March to the left and approach Infinity again, not negative real Infinity, but complex Infinity. Take a long, deep breath. This is a real quantum jump. There is but one complex Infinity, regardless of the path taken towards it.’
‘I will now consider time as a complex number. Call it ‘t.’ The part we dwell in, in our 3D reality, is the real component. But wait, the imaginary component may explain something very fundamental. As Re[t] approaches zero, we experience the ‘Big Bang.’ What if the imaginary component Im[t] maps to Infinity as it approaches zero? That is complex Infinity, a pole. So only in a infinitesimal neighborhood of zero need we be concerned about the value Im[t] maps to. At any time Re[t] >, 0 we have ‘time’ defined in the usual manner.’
‘Take a region in space itself. The Schrödinger wave equation and Maxwell’s equations assume that their underlying solutions are analytic or, at worst, ‘Real Analytic.’ In the physical world solutions are bounded. If space were infinite, then by Louisville’s Theorem ‘bounded entire functions are constant.’ This is undesirable. Space must be finite.’
‘The Big Bang theory is very well accepted. There are other theories of creation, but the Big Bang theory is well-substantiated and consistent with reality as we understand it. For convenience, let ‘u’ stand for the real component of time Re[t] and let ‘v’ stand for the imaginary component of time Im[t]. I am assuming that time ‘t’ is a complex number. Then t = u + i v. The Big Bang starts at u = zero. Let’s suppose that when u = 1, the fundamental particles—protons, electrons, and photons—are formed. Then in the time half plane we might say that for all u >, 1 that the universe is analytic out to its boundary. It is like some expanding region. So matter exists and photons and all that we experience for Re[t] and so that Im[t] is finite for all values of v in the half plane u >, 1. But in the half plane u <, 1, there is a singularity at v = zero wherein Im[0] is undefined. (Im[0] maps to Infinity.)’
‘Now I present the major result. From the Schrödinger wave equation and the use of complex time, I derive Maxwell’s equations. This will apply to quantum regions and does not take into account relativity considerations. But that will follow.’
29 Jun 2011 Taunus Trumbo
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