Quantum Gravity 368.3: HSP Approximates Quantum Physics and Cosmology via Integers as "Atom" Analogs, including Casimir Force
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From: OsherD on 1 May 2010 23:51 From Osher Doctorow Since Probable Causation/Influence (PI) is subtractive (up to a constant): 1) P(A-->B) = 1 + y - x it has a similar simplicity to simple additive functions, of which the simplest arguably generates the nonnegative integers: 2) f(n) = n + 1, n = 0, 1, 2, 3, .... But what is the next simplest additive function? It is arguably the "almost useless" harmonic series: 3) H = 1 + 1/2 + 1/3 + 1/4 + ... However, the simplest way to convert H to a "useful series" is the definition of the Riemann Zeta Function z(s): 4) z(s) = sum 1/n^s, where Re(s) > 1, s complex (including reals), sum for n = 1 to infinity The quantity z(s) in turn generates the Bernoulli numbers and Euler numbers, which in turn have as their generators secant and tangent function sec(x), tan(x), and their hyperbolic versions sech(x), tanh(x), and ratios of exponentials (times t) such as texp(xt)/(exp(t) - 1), which in turn relate to Logistic Differential Equation subtypes of Riccati Differential Equations, all of which involve quantities that are monotone (one-directional) either increasing or decreasing without cycling either on the whole real line or asymptotically "changing phase" at infinity or -infinity such as tan(x) which goes to infinity and -infinity in successive equal portions of the real line. This contrasts with harmonic motion such as sine and cosine waves which keep reversing direction continuously and continually so to speak on the whole real line without phase changes. Let us state this as a (provisional) principle: 5) HSP (Human Scale Physics) is either Monotone (one directional, either increasing or decreasing but not both) or constant up to a point in time when a phase transition occurs. The Zeta Function in turn applies to the Casimir Force and also approximates Quantum Physics by "Regularization" - it is used to convert divergent sums into convergent ones. In my interpretation, the Regularization using the Zeta Function works because HSP is more "real" than Quantum physics, so that what is really the approximating quantity is the Quantum quantity! Look in arXiv for a list of "Zeta" papers, including 24 papers under "Zeta quantum", 5 papers under "Zeta physics", 12 papers under "Zeta Casimir", 3 papers under "Zeta cosmology", 2 under "Zeta gravity", 2 under "Zeta inflation". Osher Doctorow
From: OsherD on 2 May 2010 00:49
From Osher Doctorow There might be exceptions to (5) involving functions like sin(t), cos(t), etc., if the physics of the problem involves a probability term since for example |sin(t)| < = 1 just as for probability |P(A)| < = 1, and then set P(A) = |sin(t)|. Osher Doctorow |