Continued Fractions and Quanta .
2 January 2009
We try to establish a more natural computational method for quantal systems and a putative quantal tensor .
See www.mathworld.com " Continued Fractions" et al for the mathematical nitty-gritty . It is a well-researched field . We will concentrate on what it means .
We already have a decimal designation for a number based on the continuous and differentiable principles as established by Weierstrass and Dedekind . The well-known decimal x = 0.123456789… etc .
The number x is expressed as a set of integers assembled in a defined order of the base (ie an algorithm)
The Continued Fraction Algorithm .
x = a(0) + 1 / ( a(1) + 1/ ( a(2) + 1/ ( a(3) + …
Note the left-hand brackets . It will be clearer if you write it out under the division lines .
The number is then expressed as x = ( a(0) , a(1) , a(2) , a(3) , …)
Where a(n) are integers .
Why is this important ?
1. Because the continued fraction algorithm embodies hierarchical , hyperbolic power principles .
A term far to the right of the expression has little effect on the value of x . But the effect is also of the form y=1/z (hyperbolic , not linear) .
We have thus a computational method embodying non-linear and hierarchical principles .
2. Khinchin's Constant . K= 2.68545…
See www.mathworld.com " Continued Fractions" for the definition . It means that there is a mathematical Invariant lurking around .
Invariant Quantal Tensors can be defined from this (see para 3 below) .
Co-variant quantal tensors from Khinchin-Levy Constant ?
3. The Quantal bit .
The Cantor diagonal proof is just as applicable here .
Orders of Randomness can thus be adduced .
But because the Continued Fraction algorithm incorporates division , we can generate an infinity (actually at least aleph(1) ) number of negative numbers using the diagonal method , each one causing a discontinuity . (ie 1+ a(n) = 0 ) These discontinuities define the quanta .
This makes most of formal differentiation or integration unworkable without some major re-timbering .
But actually , regularized discontinuities can be incorporated into a formal framework without busting the bank .
Sub-Planck Catalysts become a real possibility , even if only using pico- or nano metamaterials .
But numerical methods can (and has been ) used to great success . But pesky summation-to-infinity near discontinuity boundaries will persist .
Beth(x) systems can also be defined from this basis .
(See http://andreswhy.blogspot.com "New Tools : Orders of Randomness" )
Particles , humans and the state .
These are all identifiable (ie delineated) states . Their boundaries can be described as discontinuities . Therefore we can use the above methods to get a better description than possible with continuous mathematics .
A good indication that we are on the right track is the use of so-called Recurrence Relations in the Wolfram article . These are simply our old friend Fibonacci (ie an iterative growth statement of a finite process . )
The form is g(n+1) = g1(n) +g2(n-1) . A term is dependant on the previous two terms . )
Note that the Fibonacci Golden Ratio squared is approximately equal to Khinchin's constant ( 1.618^2 ~ 2.685 ) . Some relationship would be expected .
If we can develop an Invariant Quantal Tensor , we will be able to assign (ie calculate) a number giving a one-dimensional value to the whole spectrum of physical reality , from sub-atomic particles to humans to companies to national states , rating each .
I cannot see this being very popular .
"All persons with a number less than 100 , please report to the Soylent Green Department."
Nearly as bad as Health Insurance .