571 research outputs found
Unified Octonionic Representation of the 10-13 Dimensional Clifford Algebra
We give a one dimensional octonionic representation of the different Clifford
algebra Cliff(5,5)\sim Cliff(9,1), Cliff(6,6)\sim Cliff(10,2) and lastly
Cliff(7,6)\sim Cliff(10,3) which can be given by (8x8) real matrices taking
into account some suitable manipulation rules.Comment: RevTex file, 19 pages, to be published in Int. J. of Mod. Phys.
Properties of the Scalar Universal Equations
The variational properties of the scalar so--called ``Universal'' equations
are reviewed and generalised. In particular, we note that contrary to earlier
claims, each member of the Euler hierarchy may have an explicit field
dependence. The Euler hierarchy itself is given a new interpretation in terms
of the formal complex of variational calculus, and is shown to be related to
the algebra of distinguished symmetries of the first source form.Comment: 15 pages, LaTeX articl
The Moyal bracket and the dispersionless limit of the KP hierarchy
A new Lax equation is introduced for the KP hierarchy which avoids the use of
pseudo-differential operators, as used in the Sato approach. This Lax equation
is closer to that used in the study of the dispersionless KP hierarchy, and is
obtained by replacing the Poisson bracket with the Moyal bracket. The
dispersionless limit, underwhich the Moyal bracket collapses to the Poisson
bracket, is particularly simple.Comment: 9 pages, LaTe
Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras
In the present note we suggest an affinization of a theorem by Kostrikin
et.al. about the decomposition of some complex simple Lie algebras
into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out
that the untwisted affine Kac-Moody algebras of types ( prime,
), can be decomposed into
the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The
and cases are discussed in great detail. Some possible
applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure
Lagrangian and Hamiltonian Formalism on a Quantum Plane
We examine the problem of defining Lagrangian and Hamiltonian mechanics for a
particle moving on a quantum plane . For Lagrangian mechanics, we
first define a tangent quantum plane spanned by noncommuting
particle coordinates and velocities. Using techniques similar to those of Wess
and Zumino, we construct two different differential calculi on .
These two differential calculi can in principle give rise to two different
particle dynamics, starting from a single Lagrangian. For Hamiltonian
mechanics, we define a phase space spanned by noncommuting
particle coordinates and momenta. The commutation relations for the momenta can
be determined only after knowing their functional dependence on coordinates and
velocities.
Thus these commutation relations, as well as the differential calculus on
, depend on the initial choice of Lagrangian. We obtain the
deformed Hamilton's equations of motion and the deformed Poisson brackets, and
their definitions also depend on our initial choice of Lagrangian. We
illustrate these ideas for two sample Lagrangians. The first system we examine
corresponds to that of a nonrelativistic particle in a scalar potential. The
other Lagrangian we consider is first order in time derivative
Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
Among the simple finite dimensional Lie algebras, only sl(n) possesses two
automorphisms of finite order which have no common nonzero eigenvector with
eigenvalue one. It turns out that these automorphisms are inner and form a pair
of generators that allow one to generate all of sl(n) under bracketing. It
seems that Sylvester was the first to mention these generators, but he used
them as generators of the associative algebra of all n times n matrices Mat(n).
These generators appear in the description of elliptic solutions of the
classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't
Hooft's work on confinement operators in QCD, and various other instances. Here
I give an algorithm which both generates sl(n) and explicitly describes a set
of defining relations. For simple (up to center) Lie superalgebras, analogs of
Sylvester generators exist only for sl(n|n). The relations for this case are
also computed.Comment: 14 pages, 6 figure
Effective theoretical approach of Gauge-Higgs unification model and its phenomenological applications
We derive the low energy effective theory of Gauge-Higgs unification (GHU)
models in the usual four dimensional framework. We find that the theories are
described by only the zero-modes with a particular renormalization condition in
which essential informations about GHU models are included. We call this
condition ``Gauge-Higgs condition'' in this letter. In other wards, we can
describe the low energy theory as the SM with this condition if GHU is a model
as the UV completion of the Standard Model. This approach will be a powerful
tool to construct realistic models for GHU and to investigate their low energy
phenomena.Comment: 18 pages, 2 figures; Two paragraphs discussing the applicable scope
of this approach are adde
Symmetry breaking from Scherk-Schwarz compactification
We analyze the classical stable configurations of an extra-dimensional gauge
theory, in which the extra dimensions are compactified on a torus. Depending on
the particular choice of gauge group and the number of extra dimensions, the
classical vacua compatible with four-dimensional Poincar\'e invariance and zero
instanton number may have zero energy. For SU(N) on a two-dimensional torus, we
find and catalogue all possible degenerate zero-energy stable configurations in
terms of continuous or discrete parameters, for the case of trivial or
non-trivial 't Hooft non-abelian flux, respectively. We then describe the
residual symmetries of each vacua.Comment: 24 pages, 1 figure, Section 4 modifie
Explicit Lie-Poisson integration and the Euler equations
We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson
integrators, preserving all Casimirs, can be constructed. The integrators are
extremely simple. Examples are the rigid body, a moment truncation, and a new,
fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho
U*(1,1) Noncommutative Gauge Theory As The Foundation of 2T-Physics in Field Theory
A very simple field theory in noncommutative phase space X^{M},P^{M} in d+2
dimensions, with a gauge symmetry based on noncommutative u*(1,1), furnishes
the foundation for the field theoretic formulation of Two-Time Physics. This
leads to a remarkable unification of several gauge principles in d dimensions,
including Maxwell, Einstein and high spin gauge principles, packaged together
into one of the simplest fundamental gauge symmetries in noncommutative quantum
phase space in d+2 dimensions. A gauge invariant action is constructed and its
nonlinear equations of motion are analyzed. Besides elegantly reproducing the
first quantized worldline theory with all background fields, the field theory
prescribes unique interactions among the gauge fields. A matrix version of the
theory, with a large N limit, is also outlinedComment: 24 pages, LaTe
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