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Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics

Abraham A. Ungar

Foundations of Physics 27 (6):881-951 (1997)

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The Bifurcation Approach to Hyperbolic Geometry.Abraham A. Ungar - 2000 - Foundations of Physics 30 (8):1257-1282.details The Thomas precession of relativity physics gives rise to important isometries in hyperbolic geometry that expose analogies with Euclidean geometry. These, in turn, suggest our bifurcation approach to hyperbolic geometry, according to which Euclidean geometry bifurcates into two mutually dual branches of hyperbolic geometry in its transition to non-Euclidean geometry. One of the two resulting branches turns out to be the standard hyperbolic geometry of Bolyai and Lobachevsky. The corresponding bifurcation of Newtonian mechanics in the transition to Einsteinian mechanics indicates (...) Download Export citation Bookmark 1 citation
The Bloch Gyrovector.Jing-Ling Chen & Abraham A. Ungar - 2002 - Foundations of Physics 32 (4):531-565.details Hyperbolic vectors are called gyrovectors. We show that the Bloch vector of quantum mechanics is a gyrovector. The Bures fidelity between two states of a qubit is generated by two Bloch vectors. Treating these as gyrovectors rather than vectors results in our novel expression for the Bures fidelity, expressed in terms of its two generating Bloch gyrovectors. Taming the Thomas precession of Einstein's special theory of relativity led to the advent of the theory of gyrogroups and gyrovector spaces. Gyrovector spaces, (...) Download Export citation Bookmark
The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits.Abraham A. Ungar - 2002 - Foundations of Physics 32 (11):1671-1699.details Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects that form (...) Download Export citation Bookmark
From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem. [REVIEW]Abraham A. Ungar - 1998 - Foundations of Physics 28 (8):1283-1321.details A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic velocities (...) Download Export citation Bookmark 4 citations
From the Group SL(2, C) to Gyrogroups and Gyrovector Spaces and Hyperbolic Geometry.Jingling Chen & Abraham A. Ungar - 2001 - Foundations of Physics 31 (11):1611-1639.details We show that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. The superiority of the use of the gyrogroup formalism over the use of the SL(2, C) formalism for dealing with the Lorentz group in some cases is indicated by (i) the validity of gyrogroups and gyrovector spaces in higher dimensions, by (ii) the analogies that they share with groups and (...) Download Export citation Bookmark 1 citation

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