173 research outputs found
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
Efficient Algorithm for Asymptotics-Based Configuration-Interaction Methods and Electronic Structure of Transition Metal Atoms
Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B.
D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods
for atoms which reproduce, at fixed finite subspace dimension, the exact
Schr\"odinger eigenstates in the limit of fixed electron number and large
nuclear charge. Here we develop, implement, and apply to 3d transition metal
atoms an efficient and accurate algorithm for asymptotics-based CI.
Efficiency gains come from exact (symbolic) decomposition of the CI space
into irreducible symmetry subspaces at essentially linear computational cost in
the number of radial subshells with fixed angular momentum, use of reduced
density matrices in order to avoid having to store wavefunctions, and use of
Slater-type orbitals (STO's). The required Coulomb integrals for STO's are
evaluated in closed form, with the help of Hankel matrices, Fourier analysis,
and residue calculus.
Applications to 3d transition metal atoms are in good agreement with
experimental data. In particular we reproduce the anomalous magnetic moment and
orbital filling of Chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.Comment: 14 pages, 1 figur
Irreversibility and symmetry principles in quantum information
Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the
concept of quantum operations. If there is an
underlying symmetry principle, additional structures emerge.
The central question we address is, What are the consequences of global or local gauge symmetry
on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation.
Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.Open Acces
Exotic Baryons and Monopole Excitations in a Chiral Soliton Model
We compute the spectra of exotic pentaquarks and monopole excitations of the
low--lying and baryons in a chiral soliton model. Once the
low--lying baryon properties are fit, the other states are predicted without
any more adjustable parameters. This approach naturally leads to a scenario in
which the mass spectrum of the next to lowest--lying states is
fairly well approximated by the ideal mixing pattern of the
representation of flavor SU(3). We compare
our results to predictions obtained in other pictures for pentaquarks and
speculate about the spin--parity assignment for and Comment: 16 pages, 2 figures, 6 table
Computing Multiplicities of Lie Group Representations
For fixed compact connected Lie groups H \subseteq G, we provide a polynomial
time algorithm to compute the multiplicity of a given irreducible
representation of H in the restriction of an irreducible representation of G.
Our algorithm is based on a finite difference formula which makes the
multiplicities amenable to Barvinok's algorithm for counting integral points in
polytopes.
The Kronecker coefficients of the symmetric group, which can be seen to be a
special case of such multiplicities, play an important role in the geometric
complexity theory approach to the P vs. NP problem. Whereas their computation
is known to be #P-hard for Young diagrams with an arbitrary number of rows, our
algorithm computes them in polynomial time if the number of rows is bounded. We
complement our work by showing that information on the asymptotic growth rates
of multiplicities in the coordinate rings of orbit closures does not directly
lead to new complexity-theoretic obstructions beyond what can be obtained from
the moment polytopes of the orbit closures. Non-asymptotic information on the
multiplicities, such as provided by our algorithm, may therefore be essential
in order to find obstructions in geometric complexity theory.Comment: 10 page
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Relativistic Studies of the Charmonium and Bottomonium Systems Using the Sucher Equation
In this dissertation, bound states of quarks and anti-quarks (mesons) are studied with a relativistic equation known as the Sucher equation. Prior to the work in this dissertation, the Sucher equation had never been used for meson mass spectra. Furthermore, a full angular momentum analysis of the Sucher equation has never been studied. The Sucher equation is a relativistic equation with positive energy projectors imposed on the interaction. Since spin is inherent to the equation, the Sucher equation is equivalent to a relativistic Schrödinger equation with a spin-dependent effective potential. Through a complete general angular momentum analysis of the equation, we found that different angular momenta can couple through the effective potential without explicitly using tensor interaction. Next we expanded the wave functions in a complete set of basis functions and converted the Sucher equation into a matrix eigenvalue equation. As a practical application, we fit to the low lying states of the bottomonium and charmonium systems with the minimal number of input parameters, and we were able to predict the remaining spectra. We find that the the Sucher equation may be used for charmonium and bottomonium spectra. However, the spin dependent interactions inherent to the Sucher equation do not produce adequate energy level splitting between singlet and triplet states
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