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Topic: Lorentz group

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Lie group

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Invariant (mathematics)

	Lorentz group - Wikipedia, the free encyclopedia
		Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime.
		The Lorentz group is a 6-dimensional noncompact Lie group which is not connected, and whose connected components are not simply connected.
		The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction).
	en.wikipedia.org /wiki/Lorentz_group (3220 words)

	Lorentz transformation - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		The Lorentz transformation is a group transformation that is used to transform the space and time coordinates (or in general any four-vector) of one inertial reference frame, $S, into those of another one, S', with S' traveling at a relative speed of {v} to S along the x-axis.$
		Lorentz believed the luminiferous aether hypothesis; it was Albert Einstein who developed the theory of relativity to provide a proper foundation for its application.
		The Lorentz transformations were first published in 1904, but their formalism was at the time imperfect.
	www.newlenox.us /project/wikipedia/index.php/Lorentz_transformation_equations (792 words)

	Lorentz group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		The Lorentz group is the ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group of all (Click link for more info and facts about Lorentz transformation) Lorentz transformations of (Click link for more info and facts about Minkowski spacetime) Minkowski spacetime.
		Mathematically, the Lorentz group is the (Click link for more info and facts about generalized orthogonal group) generalized orthogonal group O(1, 3), (or O(3, 1) depending on the (Click link for more info and facts about sign convention) sign convention).
		The restricted Lorentz group is generated by ordinary (Click link for more info and facts about spatial rotations) spatial rotations and (Click link for more info and facts about Lorentz boost) Lorentz boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction).
	www.absoluteastronomy.com /encyclopedia/L/Lo/Lorentz_group.htm (844 words)

	Lorentz group (Site not responding. Last check: 2007-10-20)
		The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime.
		It is the subgroup of the Poincaré group consisting of all isometries that leave the origin fixed.
		The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts, which can be thought of as rotations in a plane that includes a time-like direction.
	www.sciencedaily.com /encyclopedia/lorentz_group (547 words)

	LORENTZ GROUP FACTS AND INFORMATION
		The Lorentz group is a subgroup of the PoincarÃ©_group, the group of all isometries of Minkowski spacetime.
		The Lorentz group is a 6-dimensional noncompact Lie_group which is not connected, and whose connected components are not simply_connected.
		The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz_boosts (which can be thought of as hyperbolic rotations in a plane that includes a time-like direction).
	www.witwib.com /Lorentz_group (3757 words)

	Special relativity - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-20)
		Lorentz suggested an aether theory in which objects and observers travelling with respect to a stationary aether underwent a physical shortening (Lorentz-Fitzgerald contraction) and a change in temporal rate (time dilation).
		While Lorentz suggested the Lorentz transformation equations, Einstein's contribution was, inter alia, to derive these equations from a more fundamental principle without assuming the presence of an aether.
		It is also worth noting that Maxwell's equations, combined with the Lorentz force law, can also be used to mathematically demonstrate several consequences of special relativity such as Lorentz contraction and time dilation, at least for rulers and clocks which operate via electromagnetic forces.
	encyclopedia.worldsearch.com /special_relativity.htm (5220 words)

	Henri Poincaré, great mathematician
		Lorentz group the "Poincaré group", a terminology favoured by Wightman, in honour of his contributions.
		The group thus generated in now known as the conformal group, This, of course, is not an invariance group for particles of non-zero mass.
		Pais says that Lorentz and Poincaré did not regard Einstein's paper as the final word, since it just postulates what we would want to be true (the constancy of the speed of light) but did not construct a theory in which it was true.
	www.mth.kcl.ac.uk /~streater/poincare.html (630 words)

	Special relativity (Site not responding. Last check: 2007-10-20)
		The theory, known as Lorentz Ether Theory (LET) was criticized, even by Lorentz himself, because of its ad hoc nature.
		While Lorentz suggested the Lorentz transformation equations, Einstein's contribution was, inter alia, to derive these equations from a more fundamental theory, which theory did not require the presence of an aether.
		Under Special Relativity, the seemingly complex transformations of Lorentz and Fitgerald derived cleanly from simple geometry and the Pythagorean theorem.
	www.sciencedaily.com /encyclopedia/special_relativity (2286 words)

	Lorentz group
		Lorentz group is a subgroup of the Poincaré group.
		The Lorentz group is generated by rotations and Lorentz boosts.
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Lorentz_group.html (52 words)

	Phase Space Picture
		The Lorentz group is known to be a difficult subject to mathematicians, because it is a non-compact group.
		The groups Sp(2) and Sp(4) are locally isomorphic to the (2 + 1)-dimensional and (3 + 2)-dimensional Lorentz groups.
		Since we are combining the Wigner function with group theory, we have reprinted in the Appendix Wigner's 1932 paper on the Wigner function as well as his 1939 paper on the representations of the inhomogeneous Lorentz group.
	www2.physics.umd.edu /~yskim/home/book91.html (2134 words)

	Group Theory & Rubik's Cube
		Group theory is the study of the algebra of transformations and symmetry.
		Given a group G with a subgroup H={h1,h2,...}, the "left coset" of H corresponding to an element x of G is defined as the set { x h1, x h2, x h3,...
		A representation of a group G is a set of matrices M which are homomorphic to the group.
	akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html (3602 words)

	CMS group: Lorentz angle of silicon detectors
		The Lorentz angle has to be determined for silicon detectors in high energy physics.
		The Lorentz angle is the angle under which charge carrier are deflected in a magnetic field perpendicular to the electric field.
		In Karlsruhe a comprehensive study of the Lorentz angle of non-irradiated and irradiated silicon detectors was performed.
	www-ekp.physik.uni-karlsruhe.de /cms/lorentz/index.html (486 words)

	Symmetry and Symmetry Breaking
		The use of the mathematics of group theory to study physical theories was central to the work, early in the twentieth century in Göttingen, of the group whose central figures were F. Klein (who earlier collaborated with Lie) and D.
		The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in the concept of symmetry in physics that took place in the twentieth century.
		It is therefore possible to describe symmetry breaking in terms of relations between transformations groups, in particular between a group (the unbroken symmetry group) and its subgroup(s).
	plato.stanford.edu /entries/symmetry-breaking (9818 words)

	[No title]
		(*****************************************************************************) ( :Title: Lorentz Group ) ( :Author: Jeff Olson ) ( :Summary: Extends the definitions in ClassicalGroups to include the Lorentz group.
		LorentzGroup[omega, zeta]." LorentzGroupQ::usage = "LorentzGroupQ[lambda] determines whether lambda is an element of the Lorentz group." LorentzInverse::usage = "LorentzInverse[lambda] gives the inverse transformation for lambda.
		LorentzInverse is more efficient than Inverse for Lorentz transformations." LorentzRotation::usage = "LorentzRotation gives an element of the rotation subgroup of the proper, orthochronus Lorentz group.
	www.ph.utexas.edu /~jdolson/math/LorentzGroup.m (136 words)

	Biblioteca Virtual Leite Lopes (Site not responding. Last check: 2007-10-20)
		The study of the finite-dimenslonal representations (irreducible) of the proper Lorentz group yields the result that the field variables can only be scalars, spinors, four-vectors, and tensors and spinors of higher rank.
		We remark that while the transformations U(L) which constitute an infinite-dimensional representation of the Lorentz group can be taken as unitary, the transformations such as L, D, which transform the (finite-dimensional) 4 - vector and spinor space into themselves respectively, can not be unitary.
		The latter are seen to be the infinitesimal operators which determine the infinite-dimensional representations (unitary) of the inhomogeneous Lorentz group.
	www4.prossiga.br /lopes/prodcien/inversionoperations/inver1-1.html (489 words)

	Can You See the Lorentz-Fitzgerald Contraction?
		The appearance of "half angles" p/2 is characteristic of the two-fold "spinorial covering" of the Lorentz group by SL For instance, matrices of the second form above yield a "one parameter subgroup" of SL (C) as we let p vary, which "covers" the one parameter subgroup SO(2) of PSL
		(C), the Lorentz group, and the Moebius group are all isomorphic as abstract groups.
		Lorentz transformations may be classified into four types according to their geometric effect on the night sky:
	math.ucr.edu /home/baez/physics/Relativity/SR/penrose.html (1653 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Jones-matrix formalism as a representation of the Lorentz group
		It is shown that the two-by-two Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group.
		The attenuation and phase-shift filters are represented, respectively, by the three-parameter rotation subgroup and the three-parameter Lorentz group for two spatial dimensions and one time dimension.
	imaqs.uh.edu /kasa/paper/1997/sci/han.d.1997.2.html (106 words)

	The Lorentz Group Uning Rotations and Dilations
		The approach is too general, and must be restricted to graft the results to the Lorentz group.
		This is the general form of the Lorentz transformation presented by Möller.
		Real quaternions are used in a rotation and a dilation to perform the work of the Lorentz group.
	world.std.com /~sweetser/quaternions/relativity/rotationdilation/rotationdilation.html (685 words)

	Quantum field theory - Wikibooks
		3.1 Relativity principle and the group of coordinate transformations.
		Any quantity which transforms like the space-time coordinates under Lorentz transformation is defined as a four-vector.
		Relativity principle and the group of coordinate transformations.
	en.wikibooks.org /wiki/Quantum_field_theory (487 words)

	ABSTRACTS OF PAPERS ERICH W. ELLERS
		ERICH W. Hermitian presentations of Chevalley groups I. We give a presentation for a Chevalley group arising from a Hermitian Lie algebra whose roots have all the same length.
		The Lorentz group Omega(V) is bireflectional and all involutions in Omega(V) are conjugate.
		Let G be a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells of G with conjugacy classes C of G, in particular, we consider the question, when is such an intersection not empty.
	www.math.toronto.edu /ellers/abstracts.html (817 words)

	[No title]
		This doesn't happen in the relativistic case because elements of D (spinors) are not left unchanged by the action of the Lorentz group.
		When I said "unitary action of the Lorentz group," I meant the (cover of the) of the Poincare group, which is the the semi-direct product of the Lorentz group with spacetime translations.
		A Lorentz transformation takes this to 4-momentum (p^0, p^1, p^2, p^3) subject to the constraint (p^0)^2 - [(p^1)^2 + (p^2)^2 + (p^3)^2] = m^2, i.e., the 4-momentum of the electron must stay on the mass hyperboloid.
	www.math.niu.edu /~rusin/known-math/01_incoming/QFT (2052 words)

	Seeing all Six Dimensions of the Lorentz Group of Special Relativity, in the Planetarium Sky (Site not responding. Last check: 2007-10-20)
		The transformation is the exponential of an infinitesimal Lorentz transformation, times a steadily increasing parameter; so the figure is actually a stage for a movie of a 1-parameter group of transformations.
		Four of the six dimensions of the Lorentz group are used up in telling where in the sky to place the two poles, as each place is worth two parameters.
		So you are indeed ``Seeing all Six Dimensions of the Lorentz Group of Special Relativity, in the Planetarium Sky'', four dimensions directly in the figure, and two more in your mind's eye by imagining a constant flow on the Mercator map suggested by the figure's latitudes and longitudes.
	www.uwm.edu /~eli/thisfigure.html (713 words)

	Groups in GR - Physics Help and Math Help - Physics Forums
		Is the guage group for gravity defined as the group of all possible Weyl tensors on a general 4D Riemann manifold?
		The equivalent group is the group of continuous transformations that locally look like lorentz transformations(essentially a lorentz transformation for each point in space-time such that the mapping from space-time to lorentz transformations is continuous).
		I've been told elsewhere that the gauge group for gravity is the general diffeomorphism group (i.e.
	www.physicsforums.com /showthread.php?t=12518 (710 words)

	In the Mind's Eye
		Now there are a number of mathematical ways to represent the Lorentz group.
		Those representations, that cannot be decomposed into subgroups that also represent the Lorentz group, are called irreducible representations of the Lorentz group.
		What it means is that the irreducible representations of the basic symmetries of our universe give rise to the possibilities of all the sub-atomic particles, and thus to all the matter, in the universe.
	www.dogchurch.com /tract/lorentzgroup.html (675 words)

	GROUP THEORY AND GENERAL RELATIVITY
		This is the only book on the subject of group theory and Einstein's theory of gravitation.
		The first six are devoted to rotation and Lorentz groups, and their representations.
		The entire book is self-contained in both group theory and general relativity theory, and no prior knowledge of either is assumed.
	www.worldscibooks.com /physics/p199.html (338 words)

	Quantum Gravity Concept Map - Symmetries
		When the angular momentum of any of these femionic quanta is parallel to their momentum they are "left-handed" and transform as an "isospin doublet" (the up and down quarks form a doublet, as do the electron and neutrino).
		The use of gauge bosons as intermediaries in nonlocal interactions serves to restore the local nature of the theory.
		The parameter space of the "unitary" groups (U(1), SU(2) and SU(3)) are "isomorphic to" (have a one-to-one correspondence with) the circle, the sphere (a surface!) and the "three sphere" (not a ball!).
	www.rwc.uc.edu /koehler/qg/sym.html (593 words)

	Lorentz Group Lie Algebra Map of Ultra-Relativistic Radiating Electron (Site not responding. Last check: 2007-10-20)
		Well-known high accuracy integrator scheme, the symplectic integrator, cannot guarantee its calculation accuracy for the radiating electron orbit because the symplectic integrator is valid for the Hamilton systems and the radiating electron motions are not classified into the Hamilton systems.
		In this paper, it is shown that one can obtain highly accurate numerical solutions to the radiating ultra-relativistic electron motion, based on the Lorentz group Lie algebra^1 concept and the concrete numerical calculation scheme is constructed.
		After the theoretical discussions of the Lorentz group Lie algebra and some practical numerical calculation technics, the accuracy of the presented method is checked to use some numerical examples.
	flux.aps.org /meetings/YR97/BAPSPAC97/abs/S510086.html (219 words)

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