Quantum Fractals : From Heisenberg's Uncertainty to Barnsley's Fractality
Arkadiusz Jadczyk
Language: English
Pages: 360
ISBN: 9814569860
Format: PDF / Kindle (mobi) / ePub
Starting with numerical algorithms resulting in new kinds of amazing fractal patterns on the sphere, this book describes the theory underlying these phenomena and indicates possible future applications. The book also explores the following questions:
- What are fractals?
- How do fractal patterns emerge from quantum observations and relativistic light aberration effects?
- What are the open problems with iterated function systems based on Mobius
- transformations?
- Can quantum fractals be experimentally detected?
- What are quantum jumps?
- Is quantum theory complete and/or universal?
- Is the standard interpretation of Heisenberg's uncertainty relations accurate?
- What is Event Enhanced Quantum Theory and how does it differs from spontaneous localization theories?
- What are the possible applications of quantum fractals?
Readership: Advanced undergraduate students and professionals in quantum chaos, as well as philosophers of science.
. . . . . . . . . . . 2.1.1 Cantor set through “Chaos Game” . . . . . . . . Iterated function systems . . . . . . . . . . . . . . . . . . 2.2.1 Definition of IFS . . . . . . . . . . . . . . . . . . . 2.2.2 Frobenius-Perron operator . . . . . . . . . . . . . Cantor set through matrix eigenvector . . . . . . . . . . . Quantum iterated function systems . . . . . . . . . . . . . Example: The “impossible” quantum fractal . . . . . . . . 2.5.1 24 symmetries — the octahedral group . . . . . . 2.5.2
equation X 2 + Y 2 + Z 2 + W 2 = 1, and parametrized by three angles φ, θ, ψ, onto two-dimensional sphere S 2 described by the equation n2x + n2y + n2z = 1 and parametrized by the angles φ, θ. This map S 3 → S 2 is known as Hopf fibration. The fibers of the Hopf fibration are the circles parametrized by the ψ angle while φ and θ are constant. We will now describe the physical meaning of the angle φ. Let n be a spin direction, and let P (n) be the orthogonal projection on the corresponding spin
(2.165) pg. 70/2 May 29, 2014 10:6 World Scientific Book - 9in x 6in What are Quantum Fractals? QuantumFractals3 71 Let us denote by a dot ˙ the derivative with respect to t. Taking the derivative of both sides of the equation det(A(t)) = a(t)d(t) − b(t)c(t) = 1 at t = 0, and taking into account the fact that A(0) = I, i.e. a(0) = d(0) = ˙ 1, c(0) = b(0) = 0, we obtain a(0) ˙ + d(0) = 0. Therefore the tangent vector ˙ a(0) ˙ b(0) (2.166) ˙ c(0) ˙ d(0) to a path in SL(2, C) is a matrix of
some easy properties of Pythagorean quadruples are derived as follows. Proposition 2.15. If (x, y, z, t) is a primitive Pythagorean quadruple, then two of the three numbers x, y, z are even, while the third one is odd. t is always odd. Proof. We start with recalling some elementary properties of numbers. While they are evident for mathematicians, it may not be so for people outside this community. Every even number can be written as 2k, k being an integer. Therefore sums, products (therefore
(or “pure quantum states”) are purely subjective, they represent our knowledge. As Henry Stapp puts it in his book “Mind, Matter and Quantum Mechanics” [Stapp (1993)]: “The Copenhagen interpretation is often criticized on the grounds that it is subjective, i.e., that it deals with the observer’s knowledge of things, rather than those things themselves. This charge arises mainly from Heisenberg’s frequent use of the words ‘knowledge’ and ‘observer’. Since quantum theory is fundamentally a