12 found
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  1.  23
    Formalism to Deal with Reichenbach's Special Theory of Relativity.Abraham A. Ungar - 1991 - Foundations of Physics 21 (6):691-726.
    The objective of this article is to provide a formalism to deal with the special theory of relativity (STR, in short) as riewed by Reichenbach, according to which STR involves an ineradicableconventionality of simultaneity. One of the two postulates of STR asserts that, in empty space, the one-way speed of light relative to inertial frames is constant. Experimental evidence, however, is related to the constancy of the round-trip speed of light and has no bearing on one-way speeds. Following Reichenbach's viewpoint, (...)
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  2.  19
    The Relativistic Velocity Composition Paradox and the Thomas Rotation.Abraham A. Ungar - 1989 - Foundations of Physics 19 (11):1385-1396.
    The relativistic velocity composition paradox of Mocanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non-communtativity of the relativistic velocity composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are “some difficulties within the framework of relativistic electrodynamics.” The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by composite velocities in special relativity, as opposed to the role they play (...)
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  3.  72
    From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem. [REVIEW]Abraham A. Ungar - 1998 - Foundations of Physics 28 (8):1283-1321.
    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 (...)
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  4.  49
    Midpoints in Gyrogroups.Abraham A. Ungar - 1996 - Foundations of Physics 26 (10):1277-1328.
    The obscured Thomas precessionof the special theory of relativity (STR) has been soared into prominence by exposing the mathematical structure, called a gyrogroup,to which it gives rise [A. A. Ungar, Amer. J. Phys.59,824 (1991)], and the role that it plays in the study of Lorentz groups [A. A. Ungar, Amer. J. Phys.60,815 (1992); A. A. Ungar, J. Math. Phys.35,1408 (1994); A. A. Ungar, J. Math. Phys.35,1881 (1994)]. Thomas gyrationresults from the abstraction of Thomas precession.As such, its study sheds light on (...)
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  5.  68
    The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits.Abraham A. Ungar - 2002 - Foundations of Physics 32 (11):1671-1699.
    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 (...)
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  6.  31
    Book Review: Theory of K-Loops. By Hubert Kiechle. Springer, Berlin, 2002, X + 186 Pp., $36.80 (Softcover). ISBN 3-540-43262-0. [REVIEW]Abraham A. Ungar - 2003 - Foundations of Physics 33 (4):677-678.
  7.  26
    The Bloch Gyrovector.Jing-Ling Chen & Abraham A. Ungar - 2002 - Foundations of Physics 32 (4):531-565.
    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, (...)
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  8.  66
    The Bifurcation Approach to Hyperbolic Geometry.Abraham A. Ungar - 2000 - Foundations of Physics 30 (8):1257-1282.
    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 (...)
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  9.  61
    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.
    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 (...)
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  10.  56
    The Relativistic Composite-Velocity Reciprocity Principle.Abraham A. Ungar - 2000 - Foundations of Physics 30 (2):331-342.
    Gyrogroup theory [A. A. Ungar, Found. Phys. 27, 881–951 ] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the relativistic composite-velocity reciprocity principle. The breakdown of commutativity in the Einstein velocity addition ⊕ of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly on (...)
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  11.  47
    Thomas Precession: Its Underlying Gyrogroup Axioms and Their Use in Hyperbolic Geometry and Relativistic Physics.Abraham A. Ungar - 1997 - Foundations of Physics 27 (6):881-951.
    Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 ] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor made (...)
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  12.  29
    Successive Lorentz Transformations of the Electromagnetic Field.Abraham A. Ungar - 1991 - Foundations of Physics 21 (5):569-589.
    A velocity-orientation formalism to deal with compositions of successive Lorentz transformations, emphasizing analogies shared by Lorentz and Galilean transformations, has recently been developed. The emphasis in the present article is on the convenience of using the velocity-orientation formalism by resolving a paradox in the study of successive Lorentz transformations of the electromagnetic field that was recently raised by Mocanu. The paradox encountered by Mocanu results from the omission of the Thomas rotation (or, precession) which is involved in the composition of (...)
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