3,098 research outputs found
Some Simple (Integrable) Models of Fractional Statistics
In the first part, we introduce the notion of fractional statistics in the
sense of Haldane. We illustrate it on simple models related to anyon physics
and to integrable models solvable by the Bethe ansatz. In the second part, we
describe the properties of the long-range interacting spin chains. We describe
its infinite dimensional symmetry, and we explain how the fractional statistics
of its elementary excitations is an echo of this symmetry. In the third part,
we review recent results on the Yangian representation theory which emerged
from the study of the integrable long-range interacting models.Comment: 18 pages, Latex. Two lectures presented at 1994 Les Houches Summer
School ``Fluctuating Geometries in Statistical Mechanics and Field Theory''
(also available at http://xxx.lanl.gov/lh94/
Time-dependent q-deformed coherent states for generalized uncertainty relations
We investigate properties of generalized time-dependent q-deformed coherent states for a noncommutative harmonic oscillator. The states are shown to satisfy a generalized version of Heisenberg's uncertainty relations. For the initial value in time the states are demonstrated to be squeezed, i.e. the inequalities are saturated, whereas when time evolves the uncertainty product oscillates away from this value albeit still respecting the relations. For the canonical variables on a noncommutative space we verify explicitly that Ehrenfest's theorem hold at all times. We conjecture that the model exhibits revival times to infinite order. Explicit sample computations for the fractional revival times and superrevival times are presented
Comments on the Properties of Mittag-Leffler Function
The properties of Mittag-Leffler function is reviewed within the framework of
an umbral formalism. We take advantage from the formal equivalence with the
exponential function to define the relevant semigroup properties. We analyse
the relevant role in the solution of Schr\"odinger type and heat-type
fractional partial differential equations and explore the problem of
operatorial ordering finding appropriate rules when non-commuting operators are
involved. We discuss the coherent states associated with the fractional
Sch\"odinger equation, analyze the relevant Poisson type probability amplitude
and compare with analogous results already obtained in the literature.Comment: 16 pages, 9 figure
K-matrices for 2D conformal field theories
In this paper we examine fermionic type characters (Universal Chiral
Partition Functions) for general 2D conformal field theories with a bilinear
form given by a matrix of the form K \oplus K^{-1}. We provide various
techniques for determining these K-matrices, and apply these to a variety of
examples including (higher level) WZW and coset conformal field theories.
Applications of our results to fractional quantum Hall systems and (level
restricted) Kostka polynomials are discussed.Comment: 59 pages, 2 figures, v2: note added, minor changes, references added,
v3: typos correcte
Spectral Asymptotics of Eigen-value Problems with Non-linear Dependence on the Spectral Parameter
We study asymptotic distribution of eigen-values of a quadratic
operator polynomial of the following form ,
where is a second order differential positive elliptic operator
with quadratic dependence on the spectral parameter . We derive
asymptotics of the spectral density in this problem and show how to compute
coefficients of its asymptotic expansion from coefficients of the asymptotic
expansion of the trace of the heat kernel of . The leading term in
the spectral asymptotics is the same as for a Laplacian in a cavity. The
results have a number of physical applications. We illustrate them by examples
of field equations in external stationary gravitational and gauge backgrounds.Comment: latex, 20 page
Bounds for low-energy spectral properties of center-of-mass conserving positive two-body interactions
We study the low-energy spectral properties of positive center-of-mass
conserving two-body Hamiltonians as they arise in models of fractional quantum
Hall states. Starting from the observation that positive many-body Hamiltonians
must have ground-state energies that increase monotonously in particle number,
we explore what general additional constraints can be obtained for two-body
interactions with "center-of-mass conservation" symmetry, both in the presence
and absence of particle-hole symmetry. We find general bounds that constrain
the evolution of the ground-state energy with particle number, and in
particular, constrain the chemical potential at . Special attention is
given to Hamiltonians with zero modes, in which case similar bounds on the
first excited state are also obtained, using a duality property. In this case,
in particular, an upper bound on the charge gap is also obtained. We further
comment on center of mass and relative decomposition in disk geometry within
the framework of second quantization.Comment: 8 pages, published versio
Constrained KP Hierarchies: Additional Symmetries, Darboux-B\"{a}cklund Solutions and Relations to Multi-Matrix Models
This paper provides a systematic description of the interplay between a
specific class of reductions denoted as \cKPrm () of the primary
continuum integrable system -- the Kadomtsev-Petviashvili ({\sf KP}) hierarchy
and discrete multi-matrix models. The relevant integrable \cKPrm structure is a
generalization of the familiar -reduction of the full {\sf KP} hierarchy to
the generalized KdV hierarchy . The important feature
of \cKPrm hierarchies is the presence of a discrete symmetry structure
generated by successive Darboux-B\"{a}cklund (DB) transformations. This
symmetry allows for expressing the relevant tau-functions as Wronskians within
a formalism which realizes the tau-functions as DB orbits of simple initial
solutions. In particular, it is shown that any DB orbit of a
defines a generalized 2-dimensional Toda lattice structure. Furthermore, we
consider the class of truncated {\sf KP} hierarchies ({\sl i.e.}, those defined
via Wilson-Sato dressing operator with a finite truncated pseudo-differential
series) and establish explicitly their close relationship with DB orbits of
\cKPrm hierarchies. This construction is relevant for finding partition
functions of the discrete multi-matrix models.
The next important step involves the reformulation of the familiar
non-isospectral additional symmetries of the full {\sf KP} hierarchy so that
their action on \cKPrm hierarchies becomes consistent with the constraints of
the reduction. Moreover, we show that the correct modified additional
symmetries are compatible with the discrete DB symmetry on the \cKPrm DB
orbits.
The above technical arsenal is subsequently applied to obtain completeComment: LaTeX, 63 pg
On Multi-step BCFW Recursion Relations
In this paper, we extensively investigate the new algorithm known as the
multi-step BCFW recursion relations. Many interesting mathematical properties
are found and understanding these aspects, one can find a systematic way to
complete the calculation of amplitude after finite, definite steps and get the
correct answer, without recourse to any specific knowledge from field theories,
besides mass dimension and helicities. This process consists of the pole
concentration and inconsistency elimination. Terms that survive inconsistency
elimination cannot be determined by the new algorithm. They include polynomials
and their generalizations, which turn out to be useful objects to be explored.
Afterwards, we apply it to the Standard Model plus gravity to illustrate its
power and limitation. Ensuring its workability, we also tentatively discuss how
to improve its efficiency by reducing the steps.Comment: 38 pages, 13 figures, 3 appendice
Dynamical Correlation Functions and Finite-size Scaling in Ruijsenaars-Schneider Model
The trigonometric Ruijsenaars-Schneider model is diagonalized by means of the
Macdonald symmetric functions. We evaluate the dynamical density-density
correlation function and the one-particle retarded Green function as well as
their thermodynamic limit. Based on these results and finite-size scaling
analysis, we show that the low-energy behavior of the model is described by the
Gaussian conformal field theory under a new fractional selection rule for
the quantum numbers labeling the critical exponents.Comment: 27 pages, PS file, to be published in Nucl.Phys.
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