2,556 research outputs found
Strong contraction of the representations of the three dimensional Lie algebras
For any Inonu-Wigner contraction of a three dimensional Lie algebra we
construct the corresponding contractions of representations. Our method is
quite canonical in the sense that in all cases we deal with realizations of the
representations on some spaces of functions; we contract the differential
operators on those spaces along with the representation spaces themselves by
taking certain pointwise limit of functions. We call such contractions strong
contractions. We show that this pointwise limit gives rise to a direct limit
space. Many of these contractions are new and in other examples we give a
different proof
On the continuous spectral component of the Floquet operator for a periodically kicked quantum system
By a straightforward generalisation, we extend the work of Combescure from
rank-1 to rank-N perturbations. The requirement for the Floquet operator to be
pure point is established and compared to that in Combescure. The result
matches that in McCaw. The method here is an alternative to that work. We show
that if the condition for the Floquet operator to be pure point is relaxed,
then in the case of the delta-kicked Harmonic oscillator, a singularly
continuous component of the Floquet operator spectrum exists. We also provide
an in depth discussion of the conjecture presented in Combescure of the case
where the unperturbed Hamiltonian is more general. We link the physics
conjecture directly to a number-theoretic conjecture of Vinogradov and show
that a solution of Vinogradov's conjecture solves the physics conjecture. The
result is extended to the rank-N case. The relationship between our work and
the work of Bourget on the physics conjecture is discussed.Comment: 25 pages, published in Journal of Mathematical Physic
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
Quantum Gravity and the Algebra of Tangles
In Rovelli and Smolin's loop representation of nonperturbative quantum
gravity in 4 dimensions, there is a space of solutions to the Hamiltonian
constraint having as a basis isotopy classes of links in R^3. The physically
correct inner product on this space of states is not yet known, or in other
words, the *-algebra structure of the algebra of observables has not been
determined. In order to approach this problem, we consider a larger space H of
solutions of the Hamiltonian constraint, which has as a basis isotopy classes
of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on
H. The ``empty state'', corresponding to the class of the empty tangle, is
conjectured to be a cyclic vector for T. We construct simpler representations
of T as quotients of H by the skein relations for the HOMFLY polynomial, and
calculate a *-algebra structure for T using these representations. We use this
to determine the inner product of certain states of quantum gravity associated
to the Jones polynomial (or more precisely, Kauffman bracket).Comment: 16 pages (with major corrections
Particle Topology, Braids, and Braided Belts
Recent work suggests that topological features of certain quantum gravity
theories can be interpreted as particles, matching the known fermions and
bosons of the first generation in the Standard Model. This is achieved by
identifying topological structures with elements of the framed Artin braid
group on three strands, and demonstrating a correspondence between the
invariants used to characterise these braids (a braid is a set of
non-intersecting curves, that connect one set of points with another set of
points), and quantities like electric charge, colour charge, and so on. In
this paper we show how to manipulate a modified form of framed braids to yield
an invariant standard form for sets of isomorphic braids, characterised by a
vector of real numbers. This will serve as a basis for more complete
discussions of quantum numbers in future work.Comment: 21 pages, 16 figure
Stable Determination of the Electromagnetic Coefficients by Boundary Measurements
The goal of this paper is to prove a stable determination of the coefficients
for the time-harmonic Maxwell equations, in a Lipschitz domain, by boundary
measurements
Quantum Versus Mean Field Behavior of Normal Modes of a Bose-Einstein Condensate in a Magnetic Trap
Quantum evolution of a collective mode of a Bose-Einstein condensate
containing a finite number N of particles shows the phenomena of collapses and
revivals. The characteristic collapse time depends on the scattering length,
the initial amplitude of the mode and N. The corresponding time values have
been derived analytically under certain approximation and numerically for the
parabolic atomic trap. The revival of the mode at time of several seconds, as a
direct evidence of the effect, can occur, if the normal component is
significantly suppressed.
We also discuss alternative means to verify the proposed mechanism.Comment: minor corrections are introduced into the tex
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