Discrete Quantum Numbers Research Articles

Black Hole Microstate Counting and its Macroscopic Counterpart

We survey recent results on the exact dyon spectrum in a class of N = 4 supersymmetric string theories, and discuss how the results can be understood from the macroscopic viewpoint using AdS 2 / CFT 1 correspondence. The comparison between the microscopic and the macroscopic results includes power suppressed corrections to the entropy, the sign of the index, logarithmic corrections and also the twisted index measuring the distribution of discrete quantum numbers among the microstates. (Based on lectures given by A.S. at the 12th Marcel Grossmann Meeting On General Relativity, 12–18 Jul 2009, Paris, France; CERN Winter School on Supergravity, Strings, and Gauge Theory, 25–29 January 2010; String Theory: Formal Developments And Applications, 21 Jun – 3 Jul 2010, Cargese, France, and notes taken by I.M. at the Cargese school.)

Dimensions, nodes and phases in quantum numbers

Students of quantum mechanics encounter discrete quantum numbers in a somewhat incoherent and bewildering number of ways. For each physical system studied, quantum numbers seem to be introduced in their own specific way, some enumerating from 1 and others from 0, without a common uniting thread. This essay presents a point of view that builds on dimensions, boundary conditions and various inputs that, while known, are often not brought together to present a simple, consistent picture. At the same time, some surprisingly sophisticated connections are also made.

Black hole microstate counting and its macroscopic counterpart**Based on lectures given by AS at the 12th Marcel Grossmann Meeting on General Relativity, 12–18 July 2009, Paris, France; CERN Winter School on Supergravity, Strings, and Gauge Theory, 25–29 January 2010; String Theory: Formal Developments and Applications, 21 June–3 July 2010, Cargese, France and notes taken by IM at the Cargese school.

We survey recent results on the exact dyon spectrum in a class of supersymmetric string theories, and discuss how the results can be understood from the macroscopic viewpoint using AdS2/CFT1 correspondence. The comparison between the microscopic and the macroscopic results includes power suppressed corrections to the entropy, the sign of the index, logarithmic corrections and also the twisted index measuring the distribution of discrete quantum numbers among the microstates.

The magic nature of 132Sn explored through the single-particle states of 133Sn

Atomic nuclei have a shell structure in which nuclei with 'magic numbers' of neutrons and protons are analogous to the noble gases in atomic physics. Only ten nuclei with the standard magic numbers of both neutrons and protons have so far been observed. The nuclear shell model is founded on the precept that neutrons and protons can move as independent particles in orbitals with discrete quantum numbers, subject to a mean field generated by all the other nucleons. Knowledge of the properties of single-particle states outside nuclear shell closures in exotic nuclei is important for a fundamental understanding of nuclear structure and nucleosynthesis (for example the r-process, which is responsible for the production of about half of the heavy elements). However, as a result of their short lifetimes, there is a paucity of knowledge about the nature of single-particle states outside exotic doubly magic nuclei. Here we measure the single-particle character of the levels in (133)Sn that lie outside the double shell closure present at the short-lived nucleus (132)Sn. We use an inverse kinematics technique that involves the transfer of a single nucleon to the nucleus. The purity of the measured single-particle states clearly illustrates the magic nature of (132)Sn.

The low-energy spectrum of (2, 0) theory onT5× ℝ

We consider the ADE-series of (2, 0) supersymmetric quantum theories on T(5) x R, where the first factor is a flat spatial five-torus, and the second factor denotes time. The quantum states of such a theory Phi are characterized by a discrete quantum number f is an element of H(3)(T(5), C), where the finite abelian group C is the center subgroup of the corresponding simply connected simply laced Lie group G. At energies that are low compared to the inverse size of the T5, the spectrum consists of a set of continua of states, each of which is characterized by the value of f and some number 5r of additional continuous parameters. By exploiting the interpretation of this theory as the ultraviolet completion of maximally supersymmetric Yang-Mills theory on T(4) x S(1) x R with gauge group G(adj) = G/C and coupling constant g given by the square root of the radius of the S(1) factor, one may compute the number N(f)(r) (Phi) of such continua. We perform these calculations in detail for the A- and D-series. While the Yang-Mills theory formalism is manifestly invariant under the SL(4)(Z) mapping class group of T(4), the results are actually found to be invariant under the SL(5)(Z) mapping class group of T(5), which provides a strong consistency check.

Liouville equations for neutrino distribution matrices

The classical notion of a single-particle scalar distribution function or phase space density can be generalized to a matrix in order to accommodate superpositions of states of discrete quantum numbers, such as neutrino mass/flavor. Such a ``neutrino distribution matrix'' is thus an appropriate construct to describe a neutrino gas that may vary in space as well as time and in which flavor mixing competes with collisions. The Liouville equations obeyed by relativistic neutrino distribution matrices, including the spatial derivative and vacuum flavor mixing terms, can be explicitly but elegantly derived in two new ways: from a covariant version of the familiar simple model of flavor mixing, and from the Klein-Gordon equations satisfied by a quantum ``density function'' (mean value of paired quantum field operators). Associated with the latter derivation is a case study in how the joint position/momentum dependence of a classical gas (albeit with Fermi statistics) emerges from a formalism built on quantum fields.

Simple de Sitter solutions

We present a framework for de Sitter model building in type IIA string theory, illustrated with specific examples. We find metastable de Sitter (dS) minima of the potential for moduli obtained from a compactification on a product of two nil three-manifolds (which have negative scalar curvature) combined with orientifolds, branes, fractional Chern-Simons forms, and fluxes. As a discrete quantum number is taken large, the curvature, field strengths, inverse volume, and four-dimensional string coupling become parametrically small, and the de Sitter Hubble scale can be tuned parametrically smaller than the scales of the moduli, Kaluza Klein (KK), and winding mode masses. A subtle point in the construction is that although the curvature remains consistently weak, the circle fibers of the nilmanifolds become very small in this limit (though this is avoided in illustrative solutions at modest values of the parameters). In the simplest version of the construction, the heaviest moduli masses are parametrically of the same order as the lightest KK and winding masses. However, we provide a method for separating these marginally overlapping scales, and more generally the underlying supersymmetry of the model protects against large corrections to the low-energy moduli potential.

D matter

We study the properties and phenomenology of particle-like states originating from D-branes whose spatial dimensions are all compactified. They are non-perturbative states in string theory and we refer to them as D-matter. In contrast to other non-perturbative objects such as 't Hooft-Polyakov monopoles, D-matter states could have perturbative couplings among themselves and with ordinary matter. The lightest D-particle (LDP) could be stable because it is the lightest state carrying certain (integer or discrete) quantum numbers. Depending on the string scale, they could be cold dark matter candidates with properties similar to that of wimps or wimpzillas. The spectrum of excited states of D-matter exhibits an interesting pattern which could be distinguished from that of Kaluza-Klein modes, winding states, and string resonances. We speculate about possible signatures of D-matter from ultra-high energy cosmic rays and colliders.

Crossover from Vortex States to the Fulde-Ferrell- Larkin-Ovchinnikov State in Two-Dimensionals- andd-Wave Superconductors

The effects of a tilted magnetic field on two-dimensional type-II superconductors of s - and d -wave pairing are studied. This includes a study of the crossover from vortex states to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state when the direction of the magnetic field is changed from perpendicular to the conducting plane to a parallel orientation. The FFLO state is obtained in the limit of a parallel magnetic field, and vortex phases with a large quantum number n >0 (a higher Landau level) appear for tilted magnetic fields. At fixed tilt angle, the critical field exhibits a characteristic gtemperature dependence due to the change of the discrete quantum number. The relevance of the results for the observation and the identification of the FFLO state in the organic and high- T c superconductors is discussed.

Exchange-enhanced spin splitting in a two-dimensional electron system with lateral modulation.

The effective spin splitting of the Landau bands produced by an external unidirectional periodic-modulation potential is studied in the thermodynamic Hartree-Fock (HF) approximation. The HF energies, which depend on three quantum numbers, the center coordinates of the wave functions, and the discrete Landau and spin quantum numbers, and also the thermodynamic density of states (TDOS), which is experimentally accessible by magnetocapacitance measurements, are calculated numerically for representative values of wavelength and strength of the modulation potential. For GaAs at low temperatures, the exchange enhancement can yield an effective spin gap comparable with the cyclotron energy. The spin gaps occur in strips, in which the center coordinates correspond to HF energies near the Fermi energy. The width of these strips depends on the steepness of the modulation potential and, in an intricate manner, on temperature and screening effects. Screening effects may lead to broad peaks in the TDOS, while van Hove singularities due to the one-dimensional Landau bands lead to sharp peaks in the TDOS. The conditions, under which such structures should be resolved in experiments, are discussed.

Fast and stable algorithm for the analytical computation of two‐center Coulomb and overlap integrals over Slater‐type orbitals

AbstractProceeding from analytical expressions for two‐center kernel functions that we derived recently, we present new analytical formulas for the two‐center Coulomb and overlap integrals over Slater‐type orbitals. These formulas are of an exceptionally simple analytical structure and high numerical efficiency. An especially important point is that for the most frequently needed ranges of discrete quantum numbers, the formulas are completely stable in the cases of nearly equal scaling parameters or vanishing interatomic distances, except for one particular case of the Coulomb integral. No special asymptotic formulas are needed any more to compute the two‐center integrals over Slater‐type orbitals in these case. Furthermore, a largely recursive formulation makes the integral evaluation very economical and fast. In particular, we assess the numerical performance of a new kind of angular momentum recurrences that we have proposed in a previous article [W. Hierse and P.M. Oppeneer, J. Chem. Phys. 99, 1278 (1993)]. © 1994 John Wiley & Sons, Inc.

Ward identities and sum rules for composite systems described in the dispersion relation technique: The deuteron as a composite two-nucleon system

Here we develop an approach to the description of composite systems, using dispersion integrals over the composite-particle mass. A corresponding diagram technique, based on the dispersion relation N/ D method, is constructed. Lorentz covariant amplitudes for a two-particle composite system interacting with external fields are built up in the framework of this technique. Ambiguities connected with the subtraction terms in double dispersion relations are removed due to the requirements of gauge invariance and analyticity. Sum rules for energy-momentum and discrete quantum numbers are shown to be valid for the elastic form factors and deep inelastic structure functions of the composite system. We carry out realistic description of the structure of the deuteron within the framework of the developed technique. We take into account relativistic effects, using nucleon degrees of freedom only. The pn phase-shift data (δ S, δ D and ε) for J P = 1 + at kinetic energies up to 0.6 GeV were used for a determination of the deuteron vertices (which are the relativistic analogues of the deuteron wave function). We obtained a good description of the following deuteron properties: (i) the form factor combinations A( Q 2) and B( Q 2) up to Q ⩽ 1 GeV/ c; (ii) the deuteron binding energy, magnetic and quadrupole moments. The performed investigation can be considered as a reference frame for extracting non-nucleonic degrees of freedom in the deuteron.

Tests of Conservation Laws

For quite a while it has been realized that some discrete quantum numbers are conserved in some interactions but not in others. The most conspicuous cases are parity P, charge conjugation C, and the product CP which are conserved in strong and electromagnetic interactions but not in weak interactions. The question arises whether for some of the other conserved quantities, which are conserved in strong, electromagnetic and weak interactions, there is an interaction intermediate in strength between weak and gravitational which violates these quantum numbers, e.g., baryon number B and lepton number L. The possibility exists that these conservation laws, if they are broken at all, are only broken by the gravitational force which would make the mass of an intermediate boson which induces the breakdown equal to the Planck mass.

Quantum Numbers for Bloch Electrons in a Magnetic Field

AbstractLandau functions characterize the motion of an electron in a plane perpendicular to a magnetic field. They have the degeneracy parameter ky. This parameter is resolved into four quantum numbers by a periodic crystalline potential. Two are phase angles; two are restricted to a finite set of values. Only one of the latter is a degeneracy parameter. A constructive proof is given for this proposition; it is valid under wide conditions, the only restriction being the famous ‚rationality condition’︁. The two discrete quantum numbers have as their range the numerator and denominator of the representative fraction. Group theory yields three of the numbers; the fourth labels the subbands, that is the equivalent representations within the spectrum. In the Peierls‐Onsager low field approach, the same four quantum numbers arise from one (two‐dimensional) Bloch band.

Quark-parton models and the neutron charge radius

We point out that in parton models of nucleons, including standard constituent-quark models, the nonvanishing of the neutron charge radius implies an essential mixing of the space part of the nucleon wave function with those parts involving isospin and other discrete quantum numbers. We state these conditions precisely, and give a new value for this charge radius, derived from precise electron-deuteron scattering data.

Sum Rules for Inclusive Cross Sections

In generalization of known results, we present rigorous sum rules for inclusive cross sections based upon conservation of momentum or of any additive discrete quantum number. Included are formulas which normalize inclusive cross sections in terms of ${\ensuremath{\sigma}}_{\mathrm{total}}$ independently of multiplicities. The sum rules could be useful in constraining theoretical models which construct inclusive cross sections without detailed reference to exclusive processes. Some applications are described and a generalization to the tensors governing weak and electromagnetic inclusive reactions is given.