936 research outputs found
The Reversed q-Exponential Functional Relation
After obtaining some useful identities, we prove an additional functional
relation for exponentials with reversed order of multiplication, as well as
the well known direct one in a completely rigorous manner.Comment: 6 pages, LaTeX, no figure
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
Finite Euler Hierarchies And Integrable Universal Equations
Recent work on Euler hierarchies of field theory Lagrangians iteratively
constructed {}from their successive equations of motion is briefly reviewed. On
the one hand, a certain triality structure is described, relating arbitrary
field theories, {\it classical\ts} topological field theories -- whose
classical solutions span topological classes of manifolds -- and
reparametrisation invariant theories -- generalising ordinary string and
membrane theories. On the other hand, {\it finite} Euler hierarchies are
constructed for all three classes of theories. These hierarchies terminate with
{\it universal\ts} equations of motion, probably defining new integrable
systems as they admit an infinity of Lagrangians. Speculations as to the
possible relevance of these theories to quantum gravity are also suggested.Comment: (replaces previous unprintable version corrupted mailer) 13 p.,
(Plain TeX), DTP-92/3
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
Why Matrix theory works for oddly shaped membranes
We give a simple proof of why there is a Matrix theory approximation for a
membrane shaped like an arbitrary Riemann surface. As corollaries, we show that
noncompact membranes cannot be approximated by matrices and that the Poisson
algebra on any compact phase space is U(infinity). The matrix approximation
does not appear to work properly in theories such as IIB string theory or
bosonic membrane theory where there is no conserved 3-form charge to which the
membranes couple.Comment: 8 pages, 4 figures, revtex; references adde
Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space
Wigner's quasi-probability distribution function in phase-space is a special
(Weyl) representation of the density matrix. It has been useful in describing
quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum
computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It
is also of importance in signal processing.
Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal,
has only emerged in the last quarter-century: It furnishes a third,
alternative, formulation of Quantum Mechanics, independent of the conventional
Hilbert Space, or Path Integral formulations. In this logically complete and
self-standing formulation, one need not choose sides--coordinate or momentum
space. It works in full phase-space, accommodating the uncertainty principle.
This is an introductory overview of the formulation with simple illustrations.Comment: LaTeX, 22 pages, 2 figure
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
Quaternionic Spin
We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic
2-component spinors, leading to a formalism which treats both massive and
massless particles on an equal footing. The resulting unified description has
the correct particle spectrum to be a generation of leptons, with the correct
number of spin/helicity states. Furthermore, precisely three such generations
naturally combine into an octonionic description of the 10-dimensional massless
Dirac equation, as discussed in previous work.Comment: LaTeX2e, 15 pages, 1 PS figure; to appear in Clifford '99 proceeding
Self-gravitating Yang Monopoles in all Dimensions
The (2k+2)-dimensional Einstein-Yang-Mills equations for gauge group SO(2k)
(or SU(2) for k=2 and SU(3) for k=3) are shown to admit a family of
spherically-symmetric magnetic monopole solutions, for both zero and non-zero
cosmological constant Lambda, characterized by a mass m and a magnetic-type
charge. The k=1 case is the Reissner-Nordstrom black hole. The k=2 case yields
a family of self-gravitating Yang monopoles. The asymptotic spacetime is
Minkowski for Lambda=0 and anti-de Sitter for Lambda<0, but the total energy is
infinite for k>1. In all cases, there is an event horizon when m>m_c, for some
critical mass , which is negative for k>1. The horizon is degenerate when
m=m_c, and the near-horizon solution is then an adS_2 x S^{2k} vacuum.Comment: 16 pp. Extensive revision to include case of non-zero cosmological
constant and implications for adS/CFT. Numerous additional reference
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