126 research outputs found
Harmonic Analysis on the quantum Lorentz group
This work begins with a review of complexification and realification of Hopf
algebras. We emphasize the notion of multiplier Hopf algebras for the
description of different classes of functions (compact supported, bounded,
unbounded) on complex quantum groups and the construction of the associated
left and right Haar measure. Using a continuation of symbols of
with complex spins, we give a new description of the unitary representations of
SL_q (2,\CC)_{\RR} and find explicit expressions for the characters of SL_q
(2,\CC)_{\RR}. The major theorem of this article is the Plancherel theorem for
the Quantum Lorentz Group.Comment: 60 pages, tared gzipped Postscript file, major revision of the
previous version, the Plancherel theorem is established in the more general
sense and we delay the study of Fusion theory to the next part of this pape
Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures
We study the quantization of Chern-Simons theory with group coupled to
dynamical sources. We first study the dynamics of Chern-Simons sources in the
Hamiltonian framework. The gauge group of this system is reduced to the Cartan
subgroup of We show that the Dirac bracket between the basic dynamical
variables can be expressed in term of dynamical matrix of rational type.
We then couple minimally these sources to Chern-Simons theory with the use of
a regularisation at the location of the sources. In this case, the gauge
symmetries of this theory split in two classes, the bulk gauge transformation
associated to the group and world lines gauge transformations associated to
the Cartan subgroup of . We give a complete hamiltonian analysis of this
system and analyze in detail the Poisson algebras of functions invariant under
the action of bulk gauge transformations. This algebra is larger than the
algebra of Dirac observables because it contains in particular functions which
are not invariant under reparametrization of the world line of the sources. We
show that the elements of this Poisson algebra have Poisson brackets expressed
in term of dynamical matrix of trigonometric type. This algebra is a
dynamical generalization of Fock-Rosly structure. We analyze the quantization
of these structures and describe different star structures on these algebras,
with a special care to the case where and
having in mind to apply these results to the
study of the quantization of massive spinning point particles coupled to
gravity with a cosmological constant in 2+1 dimensions.Comment: 32 pages and 1 eps figur
Universal Solutions of Quantum Dynamical Yang-Baxter Equations
We construct a universal trigonometric solution of the Gervais-Neveu-Felder
equation in the case of finite dimensional simple Lie algebras and finite
dimensional contragredient simple Lie superalgebras.Comment: 12 pages, LaTeX2e with packages vmargin, wasysym, amsmath, amssym
Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras
For a finite dimensional simple Lie algebra g, the standard universal
solution R(x) in of the Quantum Dynamical Yang--Baxter
Equation can be built from the standard R--matrix and from the solution F(x) in
of the Quantum Dynamical coCycle Equation as
It has been conjectured that, in the case
where g=sl(n+1) n greater than 1 only, there could exist an element M(x) in
such that in which is
the universal cocycle associated to the Cremmer--Gervais's solution. The aim of
this article is to prove this conjecture and to study the properties of the
solutions of the Quantum Dynamical coBoundary Equation. In particular, by
introducing new basic algebraic objects which are the building blocks of the
Gauss decomposition of M(x), we construct M(x) in as an explicit
infinite product which converges in every finite dimensional representation. We
emphasize the relations between these basic objects and some Non Standard Loop
algebras and exhibit relations with the dynamical quantum Weyl group.Comment: 46 page
Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
We analyze the hamiltonian quantization of Chern-Simons theory associated to
the universal covering of the Lorentz group SO(3,1). The algebra of observables
is generated by finite dimensional spin networks drawn on a punctured
topological surface. Our main result is a construction of a unitary
representation of this algebra. For this purpose, we use the formalism of
combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra
of polynomial functions on the space of flat SL(2,C)-connections on a
topological surface with punctures. This algebra admits a unitary
representation acting on an Hilbert space which consists in wave packets of
spin-networks associated to principal unitary representations of the quantum
Lorentz group. This representation is constructed using only Clebsch-Gordan
decomposition of a tensor product of a finite dimensional representation with a
principal unitary representation. The proof of unitarity of this representation
is non trivial and is a consequence of properties of intertwiners which are
studied in depth. We analyze the relationship between the insertion of a
puncture colored with a principal representation and the presence of a
world-line of a massive spinning particle in de Sitter space.Comment: 78 pages. Packages include
Unfashionable observations about three dimensional gravity
It is commonly accepted that the study of 2+1 dimensional quantum gravity
could teach us something about the 3+1 dimensional case. The non-perturbative
methods developed in this case share, as basic ingredient, a reformulation of
gravity as a gauge field theory. However, these methods suffer many problems.
Firstly, this perspective abandon the non-degeneracy of the metric and
causality as fundamental principles, hoping to recover them in a certain
low-energy limit. Then, it is not clear how these combinatorial techniques
could be used in the case where matter fields are added, which are however the
essential ingredients in order to produce non trivial observables in a
generally covariant approach. Endly, considering the status of the observer in
these approaches, it is not clear at all if they really could produce a
completely covariant description of quantum gravity. We propose to re-analyse
carefully these points. This study leads us to a really covariant description
of a set of self-gravitating point masses in a closed universe. This approach
is based on a set of observables associated to the measurements accessible to a
participant-observer, they manage to capture the whole dynamic in Chern-Simons
gravity as well as in true gravity. The Dirac algebra of these observables can
be explicitely computed, and exhibits interesting algebraic features related to
Poisson-Lie groupoids theory.Comment: 50 pages, written in LaTex, 3 pictures in encapsulated postscrip
Plebanski Theory and Covariant Canonical Formulation
We establish an equivalence between the Hamiltonian formulation of the
Plebanski action for general relativity and the covariant canonical formulation
of the Hilbert-Palatini action. This is done by comparing the symplectic
structures of the two theories through the computation of Dirac brackets. We
also construct a shifted connection with simplified Dirac brackets, playing an
important role in the covariant loop quantization program, in the Plebanski
framework. Implications for spin foam models are also discussed.Comment: 18 page
Comment on `Equilibrium crystal shape of the Potts model at the first-order transition point'
We comment on the article by Fujimoto (1997 J. Phys. A: Math. Gen., Vol. 30,
3779), where the exact equilibrium crystal shape (ECS) in the critical Q-state
Potts model on the square lattice was calculated, and its equivalence with ECS
in the Ising model was established. We confirm these results, giving their
alternative derivation applying the transformation properties of the
one-particle dispersion relation in the six-vertex model. It is shown, that
this dispersion relation is identical with that in the Ising model on the
square lattice.Comment: 4 pages, 1 figure, LaTeX2
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