105,809 research outputs found
Pseudo-differential equations, and the Bethe ansatz for the classical Lie algebras
No full textThe correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observe
Bethe Ansatz equations for the classical A^(1)_n affine Toda field theories
No full textWe establish a correspondence between classical A(1)n affine Toda field theories and An Bethe Ansatz systems. We show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model.1751-812
Z(_N)-symmetric field theories and the thermodynamic Bethe ansatz
No full textThis thesis is concerned with perturbed conformal field theory, the thermodynamic Bethe ansatz technique and applications to statistical mechanics. In particular, the phase space of two dimensional Z(_N)-symmetric statistical models is examined using these techniques. The aim of the first two chapters is to review some general material concerning statistical mechanics, perturbed conformal field theory, integrable two-dimensional quantum field theory and the thermodynamic Bethe ansatz (TBA) technique. In the third chapter Z(_N)-symmetric statistical theories are discussed and the known features of the phase space of such models are surveyed. The field content of the conformal models in this space (called parafermionic models) is investigated in order to determine which perturbations can be used to investigate the phase space. In the fourth and fifth chapters TBA equations are proposed to describe massless and massive renormalisation flows from the Z(_N)-symmetric conformal theories under self-dual Z(_N)-symmetric perturbations. According to the sign of the perturbation parameter the infrared limits are shown to be either conformal c = 1 or massive theories. The ground state energies of these models can be discovered in all perturbative regimes via the TBA method and the results agree with perturbation theory in ultraviolet and infrared limits. Results from detailed studies of the N = 5, 6..10 models are presented throughout. It is also deduced that the parafermionic models lie exactly at the bifurcation point of the first-order transition region into the Kosterlitz-Thouless region of the Z(_N)-symmetric phase space. The sixth and seventh chapters deal solely with massive perturbations. In chapter six, results from the TBA equations are used to deduce the mass spectrum and the vacuum structure of the underlying scattering theory. In chapter seven, proposals for the massive S-matrices are made. For N odd the mass spectra proposed by the TBA method and that predicted by the S-matrix approach (using the minimality principle) differ. It is suggested therefore, that the N odd S-matrices contain zeroes in the physical strip, violating the minimality principle
The Bethe-Ansatz for Gaudin Spin Chains
No full textWe investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(<C). The key problem is to find the spectrum and the corresponding eigenvectors of the commuting Hamiltonians. The standard method to solve this type of classical problem was introduced by H. Bethe and is known as the Bethe-Ansatz. Bethe's technique has proven to be very powerful in various areas of modem many-body theory and statistical mechanics. [19], [14], [4] Following Sklyanin's ideas in [19], we derive the Bethe-Ansatz equations for sl2(<C). Solving the Bethe-Ansatz equations is equivalent to finding polynomial solutions of the Lame differential equation, which has a meaning in electrostatics. We derive this equation for sl2(<C), and investigate its special cases. We discuss classical and more recent results on the Gaudin spin chain for sl2(<C) and provide numerical evidence for new observations in the real case of the Lame equation. Using roots of classical polynomials known as Jacobi polynomials, which are solutions to a special case of the Lame equation, we numerically approximate solutions to the Lame equation in more complicated settings. We discuss the Gaudin model associated to the Lie algebra sl3(C). Using the Bethe-Ansatz equations for sl3(C), we provide solutions in special cases.ThesisMaster of Science (MSc
Mass-Energy Relation for SN 1987A from Observations
No full textWe derive a relation between mass and kinetic energy for the hydrogen envelope of SN 1987A, using direct observations of the luminosities and photospheric velocities of the supernova. With a schematic but realistic treatment of the radiative transfer problem, which allows us to follow the position in mass of the photosphere as a function of time, we find that the observations determine uniquely the kinetic energy of the envelope once its mass is known. We do not use any input from the existing computer calculations of supernova explosions: therefore, our results are independent from any model assumed for the explosion mechanism and can actually constrain such models. Although the mass of the envelope has not so far been determined from observations, from different studies of the progenitor star there seems to be evidence for a quite large value, in the range Menv > 7 M ⊙. Corresponding to this, our approach gives for SN 1987 A a kinetic energy of the ejecta > 1 foe (=1051 ergs), the energy scaling roughly linearly with Menv
Perturbed conformal field theory, nonlinear integral equations and spectral problems
No full textThis thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results
On quantum phase crossovers in finite systems
No full textIn this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine the crossover couplings. By considering the behaviour of particular ground state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers
Perturbative and non-perturbative studies in low dimensional quantum field theory
No full textA relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches
Static and dynamic properties of the pion from continuum modelling of strong QCD
No full textWe present nonperturbative numerical solutions for the quark propagator Schwinger-Dyson
equation (SDE) and pseudoscalar meson Bethe-Salpeter equation (BSE) at and beyond the
rainbow-ladder truncation level of this system of equations. We solve this coupled system
of integral equations using a phenomenological model for the dressed gluon propagator in
Landau gauge as input. In the rainbow-ladder truncation scheme, we systematically calculate
static properties of the pion and kaon. After combining the rainbow-ladder truncation for the
SDE-BSE system with the impulse approximation for the pion-photon vertex, we present
numerical results for the pion form factor using the Ball-Chiu and bare vertices for the
nonperturbative quark-photon vertex. We find that the Ball-Chiu vertex satisfies
electromagnetic current conservation automatically, however, this vertex gives a charge
pion radius that is less than its experimental value, leaving room for further improvement.
We go beyond the rainbow-ladder truncation by including pion cloud effects into the quark
propagation, and then all the way up into the pion form factor. Here we find significant
changes for the mass and decay constant of the pion. For the pion form factor, on the other
hand, we find no qualitative changes in the region studied for both vertices.
Nevertheless, more work remains to be done at and beyond the rainbow-ladder truncation in
order to connect the pion form factor to the model-independent perturbative result
Big Effects from Small Changes: Possible Ways to Explore Nature's Chemical Diversity
No full textFungi or bacteria that produce secondary metabolites often have the potential to bring up various compounds from a single strain. The molecular basis for this well-known observation was confirmed in the last few years by several sequencing projects of different microorganisms. Besides well-known examples about induction of a selected biosynthesis (for example, by high- or low-phosphate cultivation media), no overview about the potential in this field for finding natural products was given. We have investigated the systematic alteration of easily accessible cultivation parameters (for example, media composition, aeration, culture vessel, addition of enzyme inhibitors) in order to increase the number of secondary metabolites available from one microbial source. We termed this way of revealing nature's chemical diversity the 'OSMAC (One Strain-Many Compounds) approach' and by using it we were able to isolate up to 20 different metabolites in yields up to 2.6 g L(-1) from a single organism. These compounds cover nearly all major natural product families, and in some cases the high production titer opens new possibilities for semisynthetic methods to enhance even more the chemical diversity of selected compounds. The OSMAC approach offers a good alternative to industrial high-throughput screening that focuses on the active principle in a distinct bioassay. In consequence, the detection of additional compounds that might be of interest as lead structures in further bioassays is impossible and clearly demonstrates the deficiency of the industrial procedure. Furthermore, our approach seems to be a useful tool to detect those metabolites that are postulated to be the final products of an amazing number of typical secondary metabolite gene clusters identified in several microorganisms. If one assumes a (more or less) defined reservoir of genetic possibilities for several biosynthetic pathways in one strain that is used for a highly flexible production of secondary metabolites depending on the environment, the OSMAC approach might give more insight into the role of secondary metabolism in the microbial community or during the evolution of life itself
- …
