dorsal/arxiv
View SchemaQuantum fractals on n-spheres. Clifford Algebra approach
| Authors | Arkadiusz Jadczyk |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0608117 |
| URL | https://arxiv.org/abs/quant-ph/0608117 |
| DOI | 10.1007/s00006-006-0020-9 |
| Journal | Adv.Appl.CliffordAlgebras17:201-240,2007 |
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 (Mobius 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". As a numerical illustration we study quantum fractals on the circle, two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell,and 120 cell). The invariant measure on the attractor is approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n+1) invariant measure on S^n under SO(1,n+1) transformations and discuss the Hamilton's "icossian calculus" as well as 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. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra Cl(n+1) as generalized Lorentz boosts, and their action as Moebius transformation on n-sphere, and a decomposition of any element of Spin^+(1,n+1) into a boost and a rotation, including the explicit formula for the pullback of the O(n+1) invariant Riemannian metric with respect to the associated Mobius transformation.
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"abstract": "Using the Clifford algebra formalism we extend the quantum jumps algorithm of\nthe Event Enhanced Quantum Theory (EEQT) to convex state figures other than\nthose stemming from convex hulls of complex projective spaces that form the\nbasis for the standard quantum theory. We study quantum jumps on n-dimensional\nspheres, jumps that are induced by symmetric configurations of non-commuting\nstate monitoring detectors. The detectors cause quantum jumps via geometrically\ninduced conformal maps (Mobius transformations) and realize iterated function\nsystems (IFS) with fractal attractors located on n-dimensional spheres. We also\nextend the formalism to mixed states, represented by \"density matrices\". As a\nnumerical illustration we study quantum fractals on the circle, two--sphere\n(octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell,\n600 cell,and 120 cell). The invariant measure on the attractor is approximated\nby the powers of the Markov operator. In the appendices we calculate the\nRadon-Nikodym derivative of the SO(n+1) invariant measure on S^n under\nSO(1,n+1) transformations and discuss the Hamilton\u0027s \"icossian calculus\" as\nwell as its application to quaternionic realization of the binary icosahedral\ngroup that is at the basis of the 600 cell and its dual, the 120 cell. As a\nby-product of this work we obtain several Clifford algebraic results, such as a\ncharacterization of positive elements in a Clifford algebra Cl(n+1) as\ngeneralized Lorentz boosts, and their action as Moebius transformation on\nn-sphere, and a decomposition of any element of Spin^+(1,n+1) into a boost and\na rotation, including the explicit formula for the pullback of the O(n+1)\ninvariant Riemannian metric with respect to the associated Mobius\ntransformation.",
"arxiv_id": "quant-ph/0608117",
"authors": [
"Arkadiusz Jadczyk"
],
"categories": [
"quant-ph",
"nlin.CD"
],
"doi": "10.1007/s00006-006-0020-9",
"journal_ref": "Adv.Appl.CliffordAlgebras17:201-240,2007",
"title": "Quantum fractals on n-spheres. Clifford Algebra approach",
"url": "https://arxiv.org/abs/quant-ph/0608117"
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