Fuzzy σ algebras of physics

Following the idea of Zadeh, the concept of a statistical (or fuzzy) algebra is introduced. For two extreme cases of classical and quantum statistical algebras the representation theorems are proved. The basic feature distinguishing these two cases is the possibility of producing nontrivial superpositions of pure quantum states, which is absent in the classical case.A part of this work has been written during the author's stay at the Mathematics Department, University of Toronto (Canada). The financial support from the NSERC research grant No. A5206 is gratefully acknowledged.On leave of absence from the Institute of Theoretical Physics, University of Gdask, 80-952 Gdask, Poland.     

Inequivalent quantizations for non-linear σ model

We compute the homotopy groups ₀ and ₁ of the classical configuration space of anO(3) invariant field theory on ×, where is a compact two dimensional manifold for arbitrary genusg and- denotes the time coordinate. We also present the finite dimensional, unitary, irreducible, inequivalent representations of the appropriate fundamental groups and comment on some of their implications.     

Integrable non-linear σ models with fermions

The two-dimensional non-linear model on a Riemannian symmetric spaceM=G/H is coupled to fermions with quartic self-interactions. The resulting hybrid model is presented in a gauge-dependent formulation, with a bosonic field taking values inG and a fermionic field transforming under a given representation of the gauge groupH. General criteria for classical integrability are presented: they essentially fix the Lagrangian of the model but leave the fermion representation completely arbitrary. It is shown that by a special choice for the fermion representation (derived from the adjoint representation ofG by an appropriate reduction) one arrives naturally at the supersymmetric non-linear model onM=G/H. The issue of quantum integrability is also discussed, though with less stringent results.Work partially supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil, and KFA Jülich, Federal Republic of GermanyOn leave of absence from Fakultät für Physik der Universität Freiburg, Federal Republic of Germany     

Conformally invariant σ models in 2n dimensions

The conformally invariant σ-model of Felsager and Leinaas in four dimensions (n=2) is generalised to 2n dimensions, and the unit-charge instantons are presented. This hierarchy of σ-models starts with the O(3) σ-model (n=1). The relationship of these models with the generalised YM systems for even n=2p is pointed out.     

Current algebra Ward identities in the renormalized σ model

The many-current Ward identities corresponding to the Gell-Mann current algebra are discussed in the renormalized model. The Ward identities are verified in the case of the SU(2)×SU(2) chiral symmetry. In the SU(3)×SU(3) case the uniqueness of the Adler-Bardeen anomaly is proved using the Wess-Zumino consistency conditions.On leave of absence from the University of Genova (Italy).Chercheur Associé au C.N.R.S. (C.P.T./Marseille).     

Studies on σ Resonance in a New Unitarization Approach

The pole position of the σ meson is determined using the recently proposed new unitarization method combined with the two-loop SU(2)×SU(2) chiral perturbation theory results. The scattering length parameters and the effective range parameters predicted by our method are in good agreement with experimental results. It is found that M_σ=542±26±39 MeV, Γ_σ=546±122±25 MeV.     

Renormalisation group and two-point functions in bosonic σ models

The renormalisation group equations for two-point functions involving the energy-momentum tensor in two-dimensional non-linear σ models derived previously using dimensional regularisation are solved to give the leading singular term in the operator product expansion. The results are in accord with the general framework suggested by Zamolodchikov, although the equation describing the renormalisation flow of the Virasoro central charge takes a modified form.     

Vextex Corrections and one σ Exchange Potential at Finite Temperature

By using the imaginary-time Green's function method in finite temperature field theory,we calculate the effects of the three-line vertex function on finite temperature one σ-meson exchange potential.It is found that the finite temperature coupling constant of the sigma-nucleon interaction decreases as the temperature increases,it approaches zero when temperature arrives at 210MeV.This result is quite similar to that the temperature dependence of coupling constant given by Nambu-Jona-Lasinia model at the quark level.     

Classical and quantum algebras of non-local charges in σ models

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear models. In the case of the classical chiral models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.Laboratoire associé au CNRS No. LA 280     

Cancellations of infrared divergences in the two-dimensional non-linear σ models

In the two-dimensionalO(N) nonlinear models, the expectation value of anyO(N) invariant observable is shown to have an infrared finite weak coupling perturbative expansion, although it is computed in the wrong spontaneously broken symmetry phase. This result is proved by extracting all infrared divergences of any bare Feynman amplitude atD=2– dimension. The divergences cancel at any order only for invariant observables. The renormalization atD=2 preserves the infrared finiteness of the theory.     

Monopoles,non-linear σ models,and two-fold loop spaces

In this paper we study the topology of , the moduli spaces ofSU(2) monopoles associated with the Yang-Mills-Higgs and Bogomol'nyi equations, and (m) _k , non-linear models from quantum field theory. Beautiful work of Donaldson [18, 19], Hitchin [24, 25] and Taubes [37, 39, 40] shows that gauge equivalence classes of monopoles correspond to based rational self-maps of the Riemann sphere. Similarly, the non-linear models we consider here are based harmonic maps from the Riemann sphere to complex projectivem space. In seminal work, Segal [35] studied (m) _k , the space of based rational maps from the Riemann sphere to complex projectivem space of a fixed degreek. Any element of (m) _k is clearly an element of _k ² CP(m), the space of all based continuous maps from the Riemann sphere to complex projectivem space of a fixed degreek, and this assignment gives rise to the natural inclusion of (m) _k in _k ² CP(m). Segal showed that these natural inclusions are homotopy equivalences through dimensionk(2m – 1). As the topology of the two-fold loop space ² CP(m) is well understood, Segal's result gives a very efficient way to explicitly determine the low dimensional topology of (m) _k . Thus iterated loop spaces have much to say about the topology of monopoles and non-linear models.Partially supported by NSF grant DMS-8508950Partially supported by NSF grant DMS-8701539     

Quantal symmetries in the non-linear σ model with Maxwell and non-Abelian Chern-Simons terms

The quantal symmetry property of the CP1 nonlinear （y model with Maxwell non-Abelian Chern- Simons terms in （2＋1） dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev-Senjanovic （FS） path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.     

Quantal symmetries in the non-linear σ model with Maxwell and non-Abelian Chern-Simons terms

The quantal symmetry property of the CP1 nonlinear σ model with Maxwell non-Abelian ChernSimons terms in(2+1) dimension is studied.In the Coulomb gauge,the system is quantized by using the Faddeev-Senjanovic(FS) path-integral formalism.Based on the quantaum Noether theorem,the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.     

Nonlocal field correlation functions on a lattice in the HP1 σ model

Connected two-point field-strength correlation functions are measured on a lattice in the quaternionic projective σ model within pure SU(2) theory. The correlation lengths extracted from exponential fits for these correlation functions, λ₁⁻¹ = 1.40(3) GeV and λ⁻¹ = 1.51(3) GeV, are found to be in good agreement with the results of other known calculations. The dependence of bilocal functions on the connector shape is also studied.     

The Analytical Approach for the Two-dimensional O(3) Non-linear σ Model

In the lattice gauge theory,by using the variational cumulant expansion,we calculate the internal energy and specific heat for the two-dimensional O(3) non-linear σ model.A comparison with Monte Carlo data is also presented.     

Decay of a scalar σ meson near the critical end point in the PNJL model

Properties of a scalar a meson are investigated in the two-flavor Nambu-Jona-Lasinio model with the Polyakov loop (PNJL). Model analysis of the phase diagram of strong interacting matter is performed. The temperature dependence of the σ → ππ decay width is studied at the zero chemical potential and near the critical end point. The calculated strong coupling constant g _σππ and the decay width are compared with available experimental data and other model results. Nonthermal enhancement of the total decay width is noted for the σ meson near the critical end point when the condition m _σ ≥ 2m _π is broken.     

Anilinium Ions: Relations Between pK, σ and Intensity of the UV Transition

Spectroscopists have often assumed, based only on the UV spectrum of the anilinium ion (φNH₃ ⁺), that ammonium groups cannot bring a “mesomeric” effect upon π systems. This assumption^(1–3) which could be justified for the anilinium ion itself, whose OO band of the secondary transition (towards 260 nm) is very weak, has been unduly extended to all the ammonium groups. Actually, the interaction between a π system and (NX_aX_bX_c)⁺ groups has been studied by several authors^(4–8) and we have shown that contrary to what had been assumed by spectroscopists, such an interaction induces strong effects on the UV spectrum of the benzene chromophore^(9–11). Using the MNDO and the CNDO methods, we have studied the quantum factors which play a part in the long range interaction, between: an ω group X on the one hand, N on the other hand, and then φ, in φNH₂ ⁺CH₂X, φNH⁺ (CH₂X_a) CH₂X_b and φNH⁺ (C₂H₄)₂X.     

Graded orthosymplectic geometry and OSp (4;1)-invariant fermionic σ models

A geometry based on the fundamental 4+N-dimensional linear representation of the extended super-de Sitter group OSp(4;N) is considered. For N=1 two OSp(4;1)-invariant fermionic models are discussed.On leave of absence from Institute of Theoretical Physics, University of Wrocaw, Wrocaw, Poland.     

On the coincidence of an Alday-Maldacena regularized σ model and Nambu-Goto areas of minimal surfaces

For the σ model and Nambu-Goto actions, values of the Alday-Maldacena regularized actions are calculated on the solutions of the equations of motion with constant unregularized Lagrangian. It turns out that these values coincide up to a factor independent of boundary conditions.