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Topic: Casimir invariant

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	Re: SU(3) Casimirs
		It is only the "relative color" that matters, and all of the physical data about the relative color is encoded in the Casimir invariants of the combined configuration of all quarks under consideration.
		The interaction of N quarks can only depend on the colors through the "relative colors", which exactly means that it can only depend on the Casimir invariants of the configuration, and it is not surprising that the Cubic invariant becomes important for 3-quark interactions.
		Note that all Casimirs and their functions that I mentioned are always "constant" for the whole irreducible representation - they always act as a multiple of the identity operator, and the numerical constant in front of this identity number is the only nontrivial information about the Casimir.
	www.lns.cornell.edu /spr/2003-09/msg0054495.html (616 words)

	Casimir invariant - Encyclopedia Glossary Meaning Explanation Casimir invariant (Site not responding. Last check: 2007-10-08)
		Casimir invariant - Encyclopedia Glossary Meaning Explanation Casimir invariant.
		In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra.
		In other words, it is a member of the algebra that commutes with all the generators in the Lie algebra.
	www.encyclopedia-glossary.com /en/Casimir-invariant.html (216 words)

	Time Travel [Internet Encyclopedia of Philosophy] (Site not responding. Last check: 2007-10-08)
		The second principle is the invariance of the speed of light.
		This second principle profoundly affected the model of the cosmos: the constancy of c was inconsistent with Newtonian physics.
		The invariance of the speed of light according to Special Relativity replaces the invariance of time and distance in the Newtonian universe.
	www.iep.utm.edu /t/timetravel.htm (7833 words)

	Comments: 11-11-2003 (Site not responding. Last check: 2007-10-08)
		The gaige invariant field component in the Wu Yang Dirac electromagnetic phase is B(3), the Evans Vigier field.
		This field is gauge invariant and physical, the potential is not gauge invariant, but the contour integral over the potential is gauge invariant and thus physical.
		As soon as one intorduces this phase into electrodynamics, one introduces B(3), which is now known to be the spin Casimir invariant of the Einstein group, and so as fundamental as spin of a particle or mass of a particle.
	www.aias.us /Comments/cmnts111103b.html (515 words)

	DBLP: Casimir A. Kulikowski
		Akshay Vashist, Casimir A. Kulikowski, Ilya B. Muchnik: Ortholog Clustering on a Multipartite Graph.
		Ognyan Stoyanov, Alexander Kister, Israel Gelfand, Casimir A. Kulikowski, Cyrus Chothia: Geometric Invariant Core for the CL and CH1 Domains of Immunoglobulin Molecules.
		Poe Xing, Casimir A. Kulikowski, Ilya B. Muchnik, Inna Dubchak, Denise M. Wolf, Sylvia Spengler, Manfred Zorn: Analysis of Ribosomal RNA Sequences by Combinatorial Clustering.
	www.informatik.uni-trier.de /~ley/db/indices/a-tree/k/Kulikowski:Casimir_A=.html (609 words)

	Citations: On Unitary Representations of the Inhomogeneous Lorentz Group - Wigner (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		The time like representations of these representations are labeled by the Casimir invariants that may be interpreted in the rest frame as the fundamental concepts of spin and mass.
		is rank 5 and consequently there are 5 Casimir invariants that may be used to label the representation.
		Therewith mass is the Casimir invariant of the translations within the Poincar e group and therewith the abstract concept of inertial mass.
	citeseer.ist.psu.edu /context/109201/0 (1200 words)

	APPENDIX C (Site not responding. Last check: 2007-10-08)
		The normalized invariant Haar measure on the group manifold is d(mu) = (1/16 pi^2) sin(theta) d(phi_1) d(theta) d(phi_2) with coordinate ranges 0 <= phi_1 < 4 pi, 0 <= theta <= pi, 0 <= phi_2 < 2 pi
		The coordinates are those of a solid sphere in R^3 of radius (4 pi), to which the group manifold is homeomorphic.
		Both SU(2) and SU(1, 1) are Lie groups, of rank 1, and so possess one invariant (Casimir) operator whose eigenvalues label the IRREPS.
	graham.main.nc.us /~bhammel/FCCR/apdxC.html (798 words)

	Sets to Quarks - HyperDiamond Feynman Checkerboard
		The Casimirs are invariant under the Weyl group which is the reflection group related to the root vectors that I use in the decomposition 28 = 15 + 1 + 8 + 3 + 1.
		As a rank-2 space, it has 2 Casimirs, which should be related to 4-dim internal symmetry space and the 12-dim Standard Model.
		In calculations, it is sometimes convenient to use the volumes of compact manifolds that represent spacetime, internal symmetry space, and fermion representation space.
	www.valdostamuseum.org /hamsmith/Sets2Quarks4.html (3385 words)

	[No title]
		The invariant separated variables are independent on values of integrals of motion and Casimir functions.
		and belong to the invariant intersection of all the subfoliations of
		In this paper we show that for many known integrable systems this invariance property allows us to calculate their separated variables explicitly.
	www.turpion.org /php/paper.phtml?journal_id=rd&paper_id=179 (99 words)

	Committee on Scientific Ethics
		As a consequence, all theories with non-null curvature do not possess invariant units of space, time, energy, etc. Their applicability to measurements is then a matter of religion, not science.
		The first is that the selection assures the achievement of an invariant description, precisely because the unit is the most fundamental invariant of any theory, whether conventional or generalized..
		The second is that the generalization of the unit essentially implies the preservation of all original axioms, thus guaranteeing the preservation of quantum axioms and physical laws (Heisenberg’s uncertainty principle, Pauli’s exclusion principle, causality, probability, etc.), as presented in detail in the enclosed papers.
	www.scientificethics.org /ir00007.htm (6029 words)

	NEW STRUCTURE MODELS OF HADRONS, NUCLEI AND MOLECULES PERMITTED BY HADRONIC MECHANICS, THEIR EXPERIMENTAL VERIFICATIONS ...
		As recalled in Section 1, quantum mechanics predicts that the most stable state between two particles with the same spin is the singlet, for which the total spin is zero, thus predicting that the ground state of the deuteron should have spin zero, contrary to the experimental value of spin 1.
		When hadronic mechanics is used, the exact and invariant representation of the spin 1 of model (2.1) is achieved in a way similar to that of the Rutherford-Santilli neutron (Part V).
		Recall the the first exact and invariant representation of the magnetic moment of the deuteron was reached in Section III.4 under the conventional proton-neutron interpretation while the proton and the neutron are isoparticles.
	www.neutronstructure.org /part6.htm (8374 words)

	Notions of string-localization \| The String Coffee Table
		I didn’t even realize that this point could be controversial until D. Bahns told me that she believed that the construction of DDF invariants required fixing conformal gauge.
		Probably this wrong belief is due to the fact that usually the DDF invariants are written down in a CFT context, which is derived from a Polyakov action with conformal gauge fixed.
		I furthermore claim that the Pohlmeyer invariants are a proper subset of all DDF invariants, that the (or at least one) quantization of the latter is obvious, and that it trivially induces a consistent quantization of the Pohlmeyer invariants.
	golem.ph.utexas.edu /string/archives/000338.html (7695 words)

	Hamiltonian Extremal and Symplectic Physics (Site not responding. Last check: 2007-10-08)
		Cartan developed the idea, given a 1-form of Action describing a phsycial system, that if the Pfaff dimension was 2n+1 (state space) then there existed a unique vector field that had a null eigen value relative to the matrix of two forms, F=dA.
		Such processes defined by this unique vector field are said to be extremals (the variation of the action integral is zero along such directions.) The extremal vector fields have a Hamiltonian representation, for which the closed integral of Action (flux) is an evolutionary invariant and the open integrals of F are evolutionary invariants.
		An even more general situation that still preserves the open integrals of F=dA as an evolutionary invariant is given by those situations where the Work 1-form is closed, dW=0.
	www.uh.edu /~rkiehn/ed3/ed3fre12.htm (220 words)

	Representation theory of the Galilean group - Wikpedia (Site not responding. Last check: 2007-10-08)
		The central charge M is of course a Casimir invariant.
		Using Schur's lemma, an irreducible unitary representation would have both of these Casimir invariants as multiples of the identity.
		Remember we are talking about unitary representations here, which means these values have to be real.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Representation_theory_of_the_Galilean_group (620 words)

	SU(3) Casimirs (Site not responding. Last check: 2007-10-08)
		and (one of) the Casimir for the Poincaré Group is mass.
		what do you call the Casimir associated with the first (quadratic) Casimir invariant of color SU(3), and what is the cubic associated with.
		i did some googling for "cubic casimir" and found a paper that described three quark interactions with the cubic casimir, if i understood the paper, which i probably did not.
	www.lns.cornell.edu /spr/2003-09/msg0054386.html (316 words)

	Monitoring:aias.us - PESWiki
		O(3) electrodynamics is a theory of General Relativity and is a consequence of the fact that 3D space has an O(3) symmetry.
		The Evans Vigier field B(3), the second or fundamental electromagnetic spin Casimir invariant of the Einstein group of general relativity (original paper by M. Evans, 1992 from Cornell University).
		The first correct description of the electromagnetic phase factor, a gauge invariant and generally covariant description that for the first time correctly explains all physical optics, interferometry, the Aharonov Bohm effect, and topological phase effects.
	peswiki.com /index.php/Monitoring:aias.us (2701 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The adjoint representation of $\bar{\mathcal{P}}$ is given by \begin{displaymath} (\textrm{Ad}\,g)^{A}_{\verb+ +B}= \left(\begin{array}{ccc} \Lambda^{a}_{\verb+ +b} & \theta^{c}\varepsilon_{c}^{\verb+ +a}\sqrt{-h} & 0\\ 0 & 1 & 0\\ B\theta^{c}\varepsilon_{cd}\Lambda^{d}_{\verb+ +b} & -\frac{B}{2\sqrt{-h}}\theta^{a}\theta_{a} & 1 \end{array}\right), \end{displaymath} and the invariant Casimir operator determines the metric $h_{AB}$ such that $\langle V,V\rangle=h^{AB}V_{A}V_{B}=V^{a}V_{a}-2(B/\sqrt{-h})V_{2}V_{3}$, for any vector $V=V^{A}\bar{T}_{A}$ in $\bar{\textrm{\i}}^{1}_{2}$, with $A,B\in\{0,1,2,3\}$.
		The space $L(\bar{\mathcal{P}},H,U)$ invariant under right translations on $\bar{\mathcal{P}}$ is formed by the complex functions satisfying the condition (see the Appendix) \begin{eqnarray}\label{eq:rightinvfunc} F(h(\theta^{+'},\alpha',\beta')\cdot g(\theta^{a},\alpha,\beta)) = e^{-(\alpha'/2)}\chi(\theta^{+'},\alpha',\beta')F(g(\theta^{a},\alpha,\beta)), & &\nonumber\\ F\left(g\left(\Lambda^{a}_{\verb+ +b}(\alpha')\theta^{b}+\theta^{+'},\alpha'+\alpha, \beta'+\beta+\frac{B}{2}\theta^{+'}e^{\alpha'}(\theta^{0}-\theta^{1})\right)\right) & & \nonumber\\ =e^{-(\alpha'/2)}\exp\left(i\left(-\alpha'\frac{ \zeta^{A}\zeta_{A}\sqrt{-h}}{2B\zeta_{3}}+\beta'\zeta_{3}\right)\right)F(g(\theta^{a},\alpha,\beta)).
		This addition neutralizes the Wess--Zumino term $L_{WZ}$, causing the new Lagrangian $\bar{L}=L_{B}-\dot{\chi}$ to be invariant under the transformations of $\bar{\mathcal{P}}$.
	www.ma.utexas.edu /mp_arc/html/papers/04-157 (5574 words)

	Hentosh (Site not responding. Last check: 2007-10-08)
		The existence of Hamiltonian representation for these flows, added by correponding evolutions of associated spectral problem eigenfunctions and adjoint eigenfunctions, in the case of super-integro-differential operators of one and two anticommuting variables is investigated by use of the invariant Casimir functionals' property under some Lie-Backlund transformation [4].
		It is shown that the additional symmetry are generated by the Poisson structure, being a tensor product of the canonical Lie-Poisson and some canonical finite-dimensional super-Poisson ones, and a functional, being a sum of Casimir one and the corresponding power of a spectral eigenvalue.
		The connection of additional symmetry hierarchies with (21+1)- and (22+1)-dimensional supersymmetric Davey-Stewartson equations of one and two anticommuting variables accordingly and their triple linear representations are established.
	www.imath.kiev.ua /~appmath/Abstracts2005/Hentosh.html (213 words)

	MagneGas (Site not responding. Last check: 2007-10-08)
		A necessary well known condition for a mass to be physical, that is, to exist in our space-time, is that of being the eigenvalue of the second-order Casimir invariant of the Poincare' symmetry.
		In fact, the space component n-k represents the semiaxes of the fireball, while the forth component n-4 is a measure of the density of the fireball, all expressed in a scale invariant form.
		I, the isopoincare' symmetry P(3.1) is the invariance* for arbitrary causal speeds C = c/n4, without any restriction on n4 other than that of being nowhere null, thus admitting in a natural way speeds bigger than c.
	www.usmagnegas.com /science/ir00019b.htm (13101 words)

	Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach
		In particular, it is shown how control approaches already presented in the literature can be unified, and a new control methodology is presented and discussed.
		This control methodology relies on the generalization to infinite dimensions of the concept of structural invariant (Casimir function) and on the extension to distributed systems of the so-called control by interconnection methodology.
		In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced.
	epubs.siam.org /sam-bin/dbq/article/42953 (222 words)

	ATOMKI, Annual Report 2000 (Site not responding. Last check: 2007-10-08)
		The energy spectrum of these potentials was found to be purely real in both numerical and and analytical studies.
		In our study [1] we focused on the potential group approach, which allows simultaneous description of the bound and scattering states of certain potentials in terms of unitary irreducible representations of a non-compact potential group.
		There is an SO(2,1) potential group associated with the Hermitian version of this potential: its generators are first-order differential operators and its Casimir invariant is related to the Hamiltonian as H=−C
	www.atomki.hu /ar2000/1_general/g02.html (317 words)

	/export/home/WebPages/dasgroup/public_html/PUPress/hpfeiffer.html
		The definition is global - cohomologies, the Casimirs are local, would love to understand the connection.)
		Meyberg actually uses the structure of the decomposition of adjoint
		of a primitive quartic Casimir IS directly related to the Deligne
	cns.physics.gatech.edu /GroupTheory/PUPress/hpfeiffer.html (1596 words)

	CONCEPTUAL, THEORETICAL, AND EXPERIMENTAL FOUNDATIONS
		In fact, isoquarks have interactions that are linear and nonlinear (in the wavefunctions), local-differential and nonlocal-integral (e.g., integral over the volume of wave overlapping) and potential as well as nonpotential.
		The conventional interactions are represented with the conventional Hamiltonian, while the nonlinear, nonlocal and nonpotential interactions are represented with the isounit E that, being the fundamental invariant of the SU*(3) symmetry, has an unrestricted functional dependence, E = E(t, r, psi, dr/dt, dpsi/dr, etc.).
		Contact the IBR main office at ibr@gte.net for referral to experts.
	www.i-b-r.org /ir00019b.htm (13081 words)

	Paper Submission (Site not responding. Last check: 2007-10-08)
		Modeling and control of flexible structures, Stokes--Dirac structures, Infinite dimensional port Hamiltonian systems, Control by damping injection, Casimir functions, Control by interconnection
		The purpose of this paper is to show how the Timoshenko beam can be fruitfully approached within the framework of distributed port Hamiltonian systems (dpH systems) so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system.
		In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e.
	ce152.el.utwente.nl /papers3.php?paper=125 (190 words)

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