Spherically symmetric equations in gauge theories for an arbitrary semisimple compact lie group

An explicit construction of spherically symmetric equations (not only static and/or self-dual) in gauge theories for the minimal embedding of SU(2) in an arbitrary semisimple compact Lie group G is given. The final equations are written in a form containing only gauge invariant quantities in R₂. The whole group structure is concentrated in the only matrix, which is directly related to the Cartan matrix of G. In particular, the developed technique allows to generalize the Witten duality equation [1] and to obtain the spectrum of pointlike solutions in G.     

Equations of motion for N = 1 massless super-Poincaré representations in D = 3,4 space-time

We derive the complete set of equations of motion for an arbitrary N = 1 massless super-Poincaré representation in D = 3 and D = 4 space-time. These equations are derived from the constraints imposed by the on-shell conformal invariance of a massless super-Poincaré representation.     

On Gapped Phases with a Continuous Symmetry and Boundary Operators

We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different methods, for G=SU(2) in one, and G?SU(d), in the other we construct explicitly an ‘excess spin’ operator that implements rotations of half of the infinite chain on the GNS Hilbert space of the ground state of the full chain. Since this operator is constructed as the limit of a sequence of observables, the representation itself is, in principle, experimentally observable. We claim that the corresponding unitary representation of G is closely related to the representation found at the boundary of half-infinite chains. We conclude with determining the precise relation between the two representations for the class of frustration-free models with matrix product ground states.     

Comments on a Minimal Quasiparticle Approach for the QGP and Its Large-N c Limits

A minimal quasiparticle approach for describing QGP at temperatures much higher than the critical one is discussed. It involves an ideal-gas framework in which quark and gluon masses depend on temperature. This model is able to reproduce the recent equations of state computed in lattice QCD for temperatures typically higher than 2 T _c , in a range in which it is reasonable to neglect interactions between quasiparticles. In addition, the equations of state for a generic gauge theory with gauge groups SU(N _c ) and quarks in an arbitrary representation are studied. The gauge independence in the pure glue sector and the large-N _c equivalence between the gauge groups SU(N _c ) and SO(2N _c ) in a full plasma is finally shown for normalized thermodynamic quantities.     

Implications of a natural fermion mass hierarchy

The hierarchical struture of the fundamental fermion mass spectra is required to arise in a non-accidental way from a unified model G_family with a horizontal symmetry factor group G_generation. A quark or lepton must then not be in the same representation of G_family as its anti-particle. Models for G_family of the type SU(4)_C × SU(2)_L × SU(2)_R are favoured over SU(5) or SO(10).     

Analytic structure and explicit solution of an important implicit equation

The equationz=2G(z)?expG(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation forG(z) explicitly. It turns out, after study of the complex structure of thez andG planes, that an explicit integral representation ofG(z) can be given, which may be directly used for numerical calculations of high precision.     

Finite modular groups and lepton mixing

We study lepton mixing patterns which are derived from finite modular groups ΓN, requiring subgroups Gν and Ge to be preserved in the neutrino and charged lepton sectors, respectively. We show that only six groups ΓN with N=3, 4, 5, 7, 8, 16 are relevant. A comprehensive analysis is presented for Ge arbitrary and Gν=Z₂×Z₂, as demanded if neutrinos are Majorana particles. We discuss interesting patterns arising from both groups Ge and Gν being arbitrary. Several of the most promising patterns are specific deviations from tri-bimaximal mixing, all predicting θ₁₃ non-zero as favoured by the latest experimental data. We also comment on prospects to extend this idea to the quark sector.     

New relativistic wave equations associated with indecomposable representations of the Poincaré group

A general formalism for constructing wave equations associated with an induced representation of a topological group G is developed. Next, this formalism is applied in constructing new relativistic wave equations associated with indecomposable representations of the Poincaré group. The properties of new equations for spin-1/2 and spin-1 are discussed in some detail.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

Brueckner G matrix for a planar slab of nuclear matter

The equation for the Brueckner G matrix is investigated for planar-slab geometry. A method for calculating the G matrix for a planar slab of nuclear matter is developed for a separable form of NN interaction. Actually, the separable version of the Paris NN potential is used. The singlet ¹ S ₀ and the triplet ³ S ₁–³ D ₁ channel are considered. The present analysis relies on the mixed momentum-coordinate representation, where use is made of the momentum representation in the slab plane and of the coordinate representation in the orthogonal direction. The full two-particle Hilbert space is broken down into the model subspace, where the two-particle propagator is considered exactly, and the complementary subspace, where the local-potential approximation is used, which was proposed previously for calculating the effective pairing potential. Specific calculations are performed for the case where the model subspace is constructed on the basis of negative-energy single-particle states. The G matrix is parametrically dependent on the total two-particle energy E and the total momentum P _⊥ in the slab plane. Since the G matrix is assumed to be further used to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at the value E=2μ, where μ is the chemical potential of the system under investigation. The calculations are performed predominantly for P _⊥=0. The role of nonzero values of P _⊥ is assessed. The resulting G matrix is found to depend greatly on μ in the surface region.     

Stability of higher-dimensional Einstein-Yang-Mills theories on symmetric spaces

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, G_YM. The theory is compactified on a d-dimensional symmetric coset space $\text{G}\text{H}$ with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if G_YM = H, $\text{G}\text{H}$ is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CP_n, where the stability criterion appears less restrictive.     

Generalizing some properties of short range classical solutions to Yang-Mills theory

We extend a theorem which states that for classical solutions of Yang-Mills theory, the field G_μν has to decrease at least as fast as the source S_μ at spatial infinity, provided G_μν decreases exponentially [G_μν ～ exp(?Mr)]. This generalization encompasses all decreases G_μν ～ exp(?Mr^η) with η > 0, r→∞. This is done by assuming an integral representation for A_μ, the vector potential, and imposing some regularity conditions on A_μ, valid as r→∞.     

Application of the method of integral equations for calculating the vertex constants for an α-particle

The vertex constants G²_αTN and G²_αdd for the virtual decays α → t(τ) + p(n) and α → d + d are calculated by solving the Faddeev-Yakubovsky equations for four nucleons. A spin-dependent separable potential with the Hulthén form factor is used as the NN interaction. The resonant Hilbert-Schmidt expansion is applied to solve the integral equations. The values obtained G²_αTN = 17.9 ± 1.7 fm and G²_αdd = 18.1 ± 1.3 fm are compared with the phenomenological values extracted from data on nuclear reactions.     

G2-Monopoles with Singularities (Examples)

G₂-Monopoles are solutions to gauge theoretical equations on G₂-manifolds. If the G₂-manifolds under consideration are compact, then any irreducible G₂-monopole must have singularities. It is then important to understand which kind of singularities G₂-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.     

Some general properties of representations of local currents

Some properties of representations of local current operators are studied. The currents are assumed to be conserved and to have charge densities transforming like the regular representation of any internal symmetry group G containing the isospin SU₂. The representation space is the “physical” Hilbert space, having a positive definite metric and carrying time-like positive-energy representations of the Poincaré group. The main results are that in every irreducible representation space, (A) arbitrarily large irreducible representations of G must occur, and (B) the mass spectrum is unbounded and continuous from some point onwards if it is not strictly degenerate. These results have strong implications for current algebra saturation schemes, both at finite and infinite momentum.     

Crossing resonance of wave fields in a medium with an inhomogeneous coupling parameter

The dynamic susceptibilities (Green’s functions) of the system of two coupled wave fields of different physical natures in a medium with an arbitrary relation between the mean value ? and rms fluctuation Δ? of the coupling parameter have been examined. The self-consistent approximation involving all diagrams with noncrossing correlation lines has been developed for the case where the initial Green’s function of the homogeneous medium describes the system of coupled wave fields. The analysis has been performed for spin and elastic waves. Expressions have been obtained for the diagonal elements G _mm and G _uu of the matrix Green’s function, which describe spin and elastic waves in the case of magnetic and elastic excitations, and for the off-diagonal elements G _mu and G _um , which describe these waves in the case of cross excitation. Change in the forms of these elements has been numerically studied for the case of one-dimensional inhomogeneities with an increase in Δ? and with a decrease in ? under the condition that the sum of the squares of these quantities is conserved: two peaks in the frequency dependences of imaginary parts of G _mm and G _uu are broadened and then joined into one broad peak; a fine structure appears in the form of narrow resonance at the vertex of the Green’s function of one wave field and narrow antiresonance at the vertex of the Green’s function of the other field; peaks of the fine structure are broadened and then disappear with an increase in the correlation wavenumber of the inhomogeneities of the coupling parameter; and the amplitudes of the off-diagonal elements vanish in the limit ? → 0.     

On the renormalization of the quantum field theory of point-like monopoles and charges

The quantum field theory of point-like monopoles and charges is first formulated on a euclidean lattice for a convenient regularization. The regularization preserves the peculiar features of the theory, namely those related to the invariance and to the quantization condition. The partition function is expressed as a path integral over the particle's closed paths and the action is constructed in terms of arbitrary surfaces having those paths as boundaries. The possible divergences of the continuum limit are discussed, in particular the vacuum polarization ones. It is found that, although both the electric charge Q and the magnetic charge G are renormalized as Q = Z_QQ_R and G = Z_GG_R, the quantization condition is preserved by the renormalization i.e. Z_QZ_G = 1 so that QG = Q_RG_R = 2πn. Due to the dual symmetry of the theory, then, for Q = G we get Z_Q = Z_G = 1.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory II. Direct analytic representations of tensorial field equations

By using a characterization of the concept of analytic representation and a variational approach to self-adjointness introduced in a preceding paper, we prove a theorem, according to which a necessary and sufficient condition for a class $\text{C}$², regular, tensorial, quasi-linear system of field equations to admit an ordered direct analytic representation in terms of the Lagrange equations in a region R of its variables is that the system is self-adjoint in R. We point out as a first corollary that if the ordering requirement is removed from the definition of analytic representation, then the condition of self-adjointness of the field equations is only sufficient for the existence of a Lagrangian density. We then provide as a second corollary a methodology for the computation of the Lagrangian density for the representation of self-adjoint quasi-linear tensorial field equations. This methodology is also particularized for ordinary semilinear systems of tensorial field equations through a third corollary. The above results are interpreted from the viewpoint of interactions. We first recover, through a fourth corollary, the conventional structure of the total Lagrangian density L_Tot = Σ_{1 a}ⁿL_Free^(a) + L_Int for the semilinear form of the field equations, and then introduce through a fifth corollary a generalized structure of the type L_Tot = Σ_{1 a}ⁿL_{Int, I}^(a)L_Free^(a) + L_Int.II for t representations of the field equations in the quasi-linear form. Therefore, our analysis seems to indicate that a general form of representing interacting fields is characterized by (n+1)-interaction terms in the Lagrangian: n multiplicative terms and one additive term to the Lagrangian for the free fields.     

Perturbed angular distributions in the presence of isotropic paramagnetic relaxation

The effect of paramagnetic relaxation on perturbed angular distributions is treated for nuclei interacting with their electronic shells via isotropic hyperfine interaction. The conditions are given under which Blume's analytical stochastic-model result for the nuclear perturbation factorsG _k (t) can be derived quantum mechanically. Systems with arbitrary nuclear spin, but electronic spinS=1/2 may be calculated without resorting to the assumption necessary forS>1/2. Explicit closed expressions forG _k (t) can be found for this particular case.