Lattice formulation of the CPn−1 non-linear σ models

We set up a general lattice version of non-linear σ models defined on homogeneous spaces. We then apply this to the CP^n?1 models which are the correct extension of the SU(2) σ model to SU(N). We exhibit their “confinement” property: the elementary multiplets Z_α used to describe the system do not appear as physical particles but only as bound states $\text{Z}{}_{\mathrm{\alpha }}\text{λ}{}_{\mathrm{\alpha \beta }}{}^{\mathrm{a}}\text{Z}{}_{\mathrm{\beta }}$. The method enable us to examine the “θ vacua” in the strong coupling limit by using a “dilute loop” approximation. We discuss the effect of the low activation energy for instantons which means that on a lattice, topological number is not conserved.     

On inflation in the heterotic superstring model

The possibility is considered of achieving inflation in the field-theory limit of the E₈ X E₈ superstring model, which is an N = 1 supergravity theory possessing a “no-scale” SU(n, 1)/SU(n) × U(1) structure. It is shown that neither type I inflation (due to higher-derivative terms O(R²)), nor inflation due to a SUSY-breaking gaugino-condensation potential, is possible, essentially because of the absence of free dimensionless parameters. Kaluza-Klein type inflation is ruled out because the internal space is Ricci flat. The occurence of type II inflation (due to some gauge singlet “inflation” field φ) depends upon the form of the superpotential F and of the Kähler potential G, but this also seems not to be possible, unless the >SU(n, 1) symmetry can be broken in a particular way. Hence, some new type of compactification scheme may be called for, or a different type of inflation.     

Phases of resonant amplitudes: πN → ϱN

Using SU(6)_W symmetry in its l-broken form we find that the newly reported relative signs of πN → ?N resonant amplitudes seem to indicate a universal “SU (6)_W-like” preference for all observed multiplets. Since the corresponding πN → πΔ phases favour an “anti-SU(6)_W” solution for the 70, L^P = 1^? multiplet the l-broken SU (6)_W model now faces a serious contradiction.     

A generalized Hartree-Fock-Bogoliubov approach for the description of band structures in nuclei: (I). Formulation of the theory and its exact solution in the framework of the SU(3) model

The SU(3) model is studied in the framework of the generalized Hartree-Fock-Bogoliubov approach, proposed in the papers of Kerman and Klein, and Beliaev and Zelevinski. In this model the mentioned approach is equivalent to the exact solution of the problem. The generalized density matrix R defined in the product space of single-particle and nuclear states is calculated analytically. Its eigenvalues, the occupation numbers in the average field, differ essentially from one. Thus the normalization condition $\text{R}{}^2\text{=}\text{R}$ used in the HFB theory is not valid in the generalized approach.     

Bound state approach for strange dibaryons in the skyrme model

The bound state approach to strange dibaryons in the Skyrme model is extended to baryon number n > 1. Kaon bound states are obtained in a (variational) axially symmetric SU (2) skyrmion background field. Collective quantization of isospin and spatial zero modes leads to dibaryon quantum states. These are classified in flavor multiplets.     

Breaking of the SO(n) symmetry with vector and adjoint higgses

The spontaneous symmetry breaking of SO(n) is investigated by studying the most general quartic SO(n)-invariant Higgs potential with two multiplets of scalars belonging to a vector and to an adjoint representation. In the most general cases largest residual symmetry is found to be SU(l), SU(l-1), SO(l), SO(l-1) or SO(n-2) wherel is the rank of SO(n). In particular, the breaking of SO(n) into SU(n ₁)×SU(n ₂) is found to occur only in special cases.     

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J(J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for ²⁰Ne and ²⁴Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the ²⁴Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.     

How chiral are type-II superstrings?

We study the type-II superstrings in four dimensions by studying vacua where massless chiral multiplets transform as complex representations of the non-abelian gauge group. We show that the gauge group can only be SU(3) and that such fields transform as 3 of SU(3). However, attempts to obtain the theory with N=1 supergravity fail. It turns out that the “different” constructions via asymmetric orbifolds give the same massless spectrum with necessarily N=2 supergravity.     

The complete N = 3 matter coupled supergravity

The complete N = 3 matter coupling to supergravity is obtained in a geometrical framework. This coupling always exists if the 3n complex scalars of the n vector multiplets are co-ordinates of the Kähler-grassmannian manifold SU(3, n)/SU(3) × SU(n) × U(1). Subgroups of SO(3, n) ⊂ SU(3, n) of dimension 3 + n can be gauged and give rise to a non-trivial scalar potential. The techniques used in this paper allow for the calculation of scalar potentials of extended supergravities in any dimension without explicit construction of the lagrangian. This opens the possibility of discussing patterns of partial supersymmetry and gauge symmetry breaking on a purely group-theoretical ground.     

Quantum Discreteness is an Illusion

I review arguments demonstrating how the concept of “particle” numbers arises in the form of equidistant energy eigenvalues of coupled harmonic oscillators representing free fields. Their quantum numbers (numbers of nodes of the wave functions) can be interpreted as occupation numbers for objects with a formal mass (defined by the field equation) and spatial wave number (“momentum”) characterizing classical field modes. A superposition of different oscillator eigenstates, all consisting of n modes having one node, while all others have none, defines a non-degenerate “n-particle wave function”. Other discrete properties and phenomena (such as particle positions and “events”) can be understood by means of the fast but continuous process of decoherence: the irreversible dislocalization of superpositions. Any wave-particle dualism thus becomes obsolete. The observation of individual outcomes of this decoherence process in measurements requires either a subsequent collapse of the wave function or a “branching observer” in accordance with the Schrödinger equation—both possibilities applying clearly after the decoherence process. Any probability interpretation of the wave function in terms of local elements of reality, such as particles or other classical concepts, would open a Pandora’s box of paradoxes, as is illustrated by various misnomers that have become popular in quantum theory.     

Left-right symmetry breaking scale and supersymmetry in unified theories

We propose a class of supersymmetric grand unified models where parity and SU(2)_R breaking scales are widely separated and compatible with a low-lying mass for the right-handed gauge boson W_R. The intermediate symmetry SU(4)_c×SU(2)_L×SU(2)_R and Higgs content are uniquely fixed if m_WR < 10⁹ GeV. The unification scale lies within an order of magnitude below the Planck mass.     

A new representation for the density matrix: (II). Equations of motion

The equations of motion for the new SU(n) parameters representing the density matrix for arbitrary spin are derived. It is shown that for a class of hamiltonians which are diagonal, the equations are exactly solvable by using the device of combining the SU(n) parameters in pairs using the Pauli spin matrix σ_y. It is also shown that using the correspondence relations between the SU(n) parameters and the spherical tensor moments, it is possible to picture the time evolution of the tensor parameters using the explicit solutions for the SU(n) parameters. This procedure has been illustrated by discussing in detail the problem of spin-1 and $\text{spin}\text{-}\text{3}\text{2}$ systems interacting with an external magnetic field and subjected to quadrupole interactions.     

Non-trivial saddle points and band structure of bound states of the two-dimensional O(N) vector model

We discuss O(N) invariant scalar field theories in 0 + 1 space-time dimensions (quantum mechanics) and in 1 + 1 space-time dimensions (field theory). Combining ordinary “Large N” saddle point techniques and simple properties of the diagonal resolvent of one-dimensional Schrödinger operators we find non-trivial (non-constant) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. In the “Large N” limit these saddle points are exact for the quantum mechanical case, but only approximate in the two-dimensional theory. In the latter case they are the leading contributions to the time evolution kernel at short times, or equivalently, the leading contribution to the high temperature expansion of partition function stemming from space dependent static configurations in case of the Euclidean theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra.     

Structure of supergravity

It is shown how to construct actions and kinetic multiplets out of scalar multiplets of supergravity. The multiplets containing the scalar and the conformal tensor are obtained and yield the gauge actions of Einstein and Weyl supergravity. The complete supersymmetric R² and Bel-Robinson invariants are also obtained as well as general n-loop counterterms.     

A test of the collective correlations calculation of the dipole states of28Si

We solve a one band hopping model in the presence of a homogeneous static electric fieldF. In this case the exact eigenvalues do form a ladder:E _m =m e Fa, the corresponding eigenfunctions are localized within the “tilted” band.     

Scaling limit of the one-dimensional attractive Hubbard model: the non-half-filled band case

The scaling limit of the less than half-filled attractive Hubbard chain is studied. This is a continuum limit in which the particle number per lattice site, n, is kept finite (0 < n < 1) while adjusting the interaction and bandwidth in such a way that there is a finite mass gap. We construct this limit both for the spectrum and the secular equations describing the excitations. We find that similarly to the half-filled case, the limiting model has a massive and a massless sector. The structure of the massive sector is closely analogous to that of the half-filled band and consequently to the chiral invariant SU(2) Gross-Neveu (CGN) model. The structure of the massless sector differs from that of the half-filled band case: the excitations are of particle and hole type, however they are not uniquely defined. The energy and the momentum of this sector exhibits a tower structure corresponding to a conformal field theory with c = 1 and SU(2) × SU(2) symmetry. The energy-momentum spectrum and the zero temperature free energy of the states with finite density coincides with that of the half-filled case supporting the identification of the limiting model with the SU(2) symmetric CGN theory.     

Essential restriction on the symmetry of a unified theory for the case of massive gluons

In unified gauge theories with massive “color” gluons, the physical requirement of maintaining “color” SU(3) as a global classification symmetry is shown to lead to the following restrictions: (i) the local unifying symmetry group must be of the form G_flavor?G_color; (ii) quarks are to be integer charged; (iii) the number of flavors is an integral multiple of the number of “colors”.     

Mesons in the large-N collective field method

The large-N limit of SU(N) matrix quantum mechanics has been studied recently as a model for large-N Yang-Mills theory. Here we solve this model with fundamental representation fermions (“quarks”) added. The “meson” spectrum is given by an integral equation and exhibits asymptotically linear “Regge trajectories” with the same spacing as that of the “glueballs”.