I have finished a paper dealing with "not so sacred geometry" - quantum fractals in four dimensions. As it is mostly math, though with illustrations at the end, probably it is too technical to be of any use except of few experts :). The new paper, with the title "Quantum fractals on n-spheres" can be downloaded as pdf from
http://quantumfuture.net/quantum_future/papers/quantum_fractals_on_n-spheres.pdf
Here I am only quoting the Abstract and part of the Introduction from this paper, as it can be read independently of the math:
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Quantum fractals on n-spheres
(submitted for publication in "Advances in Applied Clifford Algebra")
Arkadiusz Jadczyk
Abstract
Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Moebius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by ''density matrices" in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we discuss the Hamilton's ''icossian calculus" and its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell.
Introduction
Footnote: Nowadays the defenders of the ''antiquated scheme" of q.m. go as far as to assign ''crackpot index'' to those who question this scheme. So, for instance, 10 points (on the scale of 1--50), are assigned for each claim that quantum mechanics is fundamentally misguided, and another 10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain ''why" they occur, or fails to provide a ''mechanism" [2].
The present paper follows the line of ideas developed in a series of papers that has led to the Event Enhanced Quantum Theory (EEQT), as summarized in [3], and recently extended in [4], but we now go beyond that framework. While, following von Neumann, we keep the algebraic structure as one of the most important for the mathematical formalism of q.m., and we propose to dispose of the concept of ''observables" and of ''expectation values" at the fundamental level. We also dispose of the concept of ''time", understood as a ''continuous parameter", external to the theory. Our philosophy, concerning ''time" is that of the German social philosopher Ernest Bloch:
which the ''future" is only ''probable", rather than determined, is necessarily an open system. The mathematical formalism of the
standard quantum theory is based on complex Hilbert spaces and Jordan algebras of self--adjoint operators. It involves interpretational axioms for expectation values and eigenvalues of self--adjoint operators as ''possible results of measurements", yet it does not provide a framework for defining the measurements [5,6]. In view of these considerations, Gell-Mann would certainly score a high crackpot index [2] for this statement:
[7] with the following quote from Niels Bohr's friend Piet Hein:
Footnote: C.f. also \[9], where a similar idea, based on a KMS equilibrium state is discussed in a broader, philosophical framework
But their philosophy applies, at most, to equilibrium states, while ''quantum foams" before the Planck era are certainly far from equilibrium. David Hestenes [10,11] proposed to understand the role of the complex numbers in quantum theory in terms of the Clifford algebra. This is also our view. L. Nottale, in his theory of ''scale relativity" [12] proposed an alternative idea, where the complex structure arises from a stochastic differential equation in a fractal space--time. We think that our approach may serve as a connecting bridge between fractality, the nontrivial topology of dodecahedral models of space--time, as discussed by J--P.~Luminet et al. [13] (cf. also [14].), and the late thoughts of A. Einstein [15], who wrote:
[................]
Sec. Examples contains the results of the numerical simulations of IFS of Moebius transformations that lead to ''quantum fractals". We study quantum fractals on the circle (one-dimensional sphere and pentagon), two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell, and 120 cell). The last section contains the summary and conclusions and also points out some open problems.
In the appendices we discuss the Hamilton's ''icossian calculus" (in particular we quote in extenso the original Hamilton's paper published in 1856), and its application to quaternionic realization of the binary icosahedral group that is at the basis of 600 cell and its dual, the 120 cell.
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And here is one of the illustrations from the paper
http://quantumfuture.net/quantum_future/papers/quantum_fractals_on_n-spheres.pdf
Here I am only quoting the Abstract and part of the Introduction from this paper, as it can be read independently of the math:
***************************************
Quantum fractals on n-spheres
(submitted for publication in "Advances in Applied Clifford Algebra")
Arkadiusz Jadczyk
Abstract
Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Moebius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by ''density matrices" in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we discuss the Hamilton's ''icossian calculus" and its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell.
Introduction
So wrote Ervin Schrodinger fifty years ago [1]. Today the standard scheme of q.m. is as antiquated as it ever was, and provides no answer to the most fundamental questions such as ''what is time?", and how to describe events that happen in a single physical system, such as our Universe.
Footnote: Nowadays the defenders of the ''antiquated scheme" of q.m. go as far as to assign ''crackpot index'' to those who question this scheme. So, for instance, 10 points (on the scale of 1--50), are assigned for each claim that quantum mechanics is fundamentally misguided, and another 10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain ''why" they occur, or fails to provide a ''mechanism" [2].
The present paper follows the line of ideas developed in a series of papers that has led to the Event Enhanced Quantum Theory (EEQT), as summarized in [3], and recently extended in [4], but we now go beyond that framework. While, following von Neumann, we keep the algebraic structure as one of the most important for the mathematical formalism of q.m., and we propose to dispose of the concept of ''observables" and of ''expectation values" at the fundamental level. We also dispose of the concept of ''time", understood as a ''continuous parameter", external to the theory. Our philosophy, concerning ''time" is that of the German social philosopher Ernest Bloch:
So, time is only then, when something happens, and only there where something happens. Therefore the primary concept is that of an event, and of the process - that is a sequence of events. Time, as a continuous, global variable, comes in only in the limit of a large number of events. The primary process is that of ''quantum jumps". It is an irreversible process in an open system, and every system in
which the ''future" is only ''probable", rather than determined, is necessarily an open system. The mathematical formalism of the
standard quantum theory is based on complex Hilbert spaces and Jordan algebras of self--adjoint operators. It involves interpretational axioms for expectation values and eigenvalues of self--adjoint operators as ''possible results of measurements", yet it does not provide a framework for defining the measurements [5,6]. In view of these considerations, Gell-Mann would certainly score a high crackpot index [2] for this statement:
The same can be said about the last paragraph of Schrodingers paper [1], where he wrote
The need for an open--minded approach is well noted by John A. Wheeler, who ends his book ''Geons, Black Holes & Quantum Foam"
[7] with the following quote from Niels Bohr's friend Piet Hein:
Alain Connes and Carlo Rovelli [8] proposed to explain the classical time parameter as arising from the modular automorphism group of a KMS state on a von Neumann algebra over the field of complex numbers C
Footnote: C.f. also \[9], where a similar idea, based on a KMS equilibrium state is discussed in a broader, philosophical framework
But their philosophy applies, at most, to equilibrium states, while ''quantum foams" before the Planck era are certainly far from equilibrium. David Hestenes [10,11] proposed to understand the role of the complex numbers in quantum theory in terms of the Clifford algebra. This is also our view. L. Nottale, in his theory of ''scale relativity" [12] proposed an alternative idea, where the complex structure arises from a stochastic differential equation in a fractal space--time. We think that our approach may serve as a connecting bridge between fractality, the nontrivial topology of dodecahedral models of space--time, as discussed by J--P.~Luminet et al. [13] (cf. also [14].), and the late thoughts of A. Einstein [15], who wrote:
The present paper is a technical one. It fills the empty space with discrete structures, and it deals with the discrete random aspects of quantum jumps generated by the algebraic structure of real Clifford algebras of Euclidean spaces, and of their conformal extensions. The jumps are generated by Moebius transformations and lead to iterated function systems with place dependent probabilities, thus to fractal patterns on n--spheres. Our ideas are close to those of W. E. Baylis, who also noticed [16] the similarities between the Clifford algebra scheme and the formal algebraic structure of q.m.
[................]
Sec. Examples contains the results of the numerical simulations of IFS of Moebius transformations that lead to ''quantum fractals". We study quantum fractals on the circle (one-dimensional sphere and pentagon), two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell, and 120 cell). The last section contains the summary and conclusions and also points out some open problems.
In the appendices we discuss the Hamilton's ''icossian calculus" (in particular we quote in extenso the original Hamilton's paper published in 1856), and its application to quaternionic realization of the binary icosahedral group that is at the basis of 600 cell and its dual, the 120 cell.
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And here is one of the illustrations from the paper
