Quantum integrable systems related to lie algebras

Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g²v(q) of the following 5 types: v_I(q) = q^?2, v_II(q) = sinh^?2q, v_III(q) = sin^?2q, $\text{v}{}_{\mathrm{<mtext>IV</mtext>}}\text{(q) =}\text{P}\text{(q)}$, v_V(q) = q^?2 + ω²q². Here $\text{P}\text{(q)}$ is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp(q_jq_{j+ 1}) is moreover considered.This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.     

An analytic calculation of the weak field limit of the static color dielectric constant

We analyze the Dyson equation/Ward identity system for the axial gauge n · A = 0 gluon propagator Δ_μν(q)whenn · q = 0. The solution behaves like (q^?4 + (q²)^ν?1) for small q₂, and we are able to calculate the power ν analytically. It turns out to be 0.1737. This analytic calculation verifies our earlier numerical solutions to these equations. For static problems, n · q = 0 is the temporal gauge, and in this gauge the gluon propagator is directly related to the color dielectric constant. We can thus calculate the dielectric constant in the infrared limit.     

Temperature quenching of luminescence for linear and derivative nuclear operators

An earlier study of the thermal quenching of luminescence using the single-configurational-coordinate model is extended from Condon-approximation overlap integrals 〈u_n|v_m〉² to the linear and derivative integrals 〈u_n|z_v|v_m〉² and 〈u_n|?/?z_v|v_m〉². For non-radiative transitions, the thermally weighted nuclear factor in the transition rate is, for the linear and derivative integrals, the corresponding factor for 〈u_n|v_m〉² integrals multiplied by 2E_Xv/?ω_v and $\text{2[E}{}_{\mathrm{Xv}}\text{- E}{}_{\mathrm{p}}{}_{\mathrm{U}}\text{(T)]/}\text{h}\text{?}\text{ω}{}_{\mathrm{v}}$, respectively. E_Xv is the energy of the crossover above the initial- v-parabola minimum, and E_pU(T) is the single activation energy fitted to the nuclear factor's temperature dependence for 〈u_n|v_m〉² integrals. These multiplying factors are exact for equal parabola force constants and good approximations for unequal force constants. These multiplying factors will be difficult to distinguish experimentally. The more important considerations for fitting the model to thermal-quenching data are the parabola placement and the Condon-approximation integrals described previously.     

On spectral representations for the functions Wi(v, q2) (i = 1, 2) describing electroproduction process

It is shown that the Deser, Gilbert, Sudarshan representation (DGSR) does not follow from microcausality and spectrality only. Examples of the functions W_i(v, q²) satisfying the microcausality and spectrality conditions are given which cannot be written as the DGSR with spectral function h(a, α) that is a temperature distribution. Instead of the DGSR the spectral representation for W_i(v, q²) has been proved (eq. (3)) which follows only from microcausality and spectrality.     

Analytical properties of the forward elastic scattering amplitudes following from microcausality

The forward elastic scattering amplitudes T_i(v,q²), (i = 1,2), of the virtual photon with the mass q² are considered in variables σ and ? where σ and ? are related to q² and v by the formulae q² = exp (2σ) and v = exp (σ) ch?. It has been proved that microcausality requirement implies the analyticity of amplitudes in the tube region [Im σ + π/2] + |Im?| < π/2 as a function of both complex variables σ and ?. Formulae are obtained expressing amplitudes T_l(v,q²) at arbitrary v and q² through functions W_l^(?)(v, q²) describing the electroproduction process ?2v < q² < 0.     

The nucleon axial form factor and the pion-nucleon vertex function

The nucleon axial current and related form factors are investigated in a model of relativistic quarks confined by a scalar potential of the formM(r)=c r ⁿ , with special emphasis on center-of-mass corrections and pionic effects. Pionic contributions to the axial form factorG _A (q ²) from af _π?_μφ term with constantf _π are demonstrated to vanish. The pion-nucleon form factorG _πNN (q ²) is derived and turns out to be longer ranged thanG _A (q ²). The induced pseudo-scalar form factorG _p (q ²) is shown to be connected toG _πNN (q ²) by the standard PCAC relation, contributions from the quark core toG _p (q ²) being negligibly small.     

Quantum oscillations in screening in a two-dimensional electron gas

The static dielectric constant of a two-dimensional electron gas is studied as a function of the strength of a dc magnetic field applied normal to the plane of the electron gas. At high temperatures $\text{(kT ?}\text{h}\text{?}\text{ω}{}_{\mathrm{c}}\text{)}$ the static dielectric function is independent of magnetic field, and for long wavelengths is given by ? ? ?₀ + 2n_vme²/q, where ?₀ is the background dielectric constant and n_v is the valley degeneracy. At low-temperatures, quantum oscillations become important and dramatically modify the screening.     

Spatial-temporal dispersion of the kinetic coefficients near the Anderson transition

A generalization of the Vollhardt-Wölfle localization theory is proposed to make it possible to study the spatial-temporal dispersion of the kinetic coefficients of a d-dimensional disordered system in the low-frequency, long-wavelength range (ω?_F and q?k _F ). It is shown that the critical behavior of the generalized diffusion coefficient D(q,ω) near the Anderson transition agrees with the general Berezinskii-Gor’kov localization criterion. More precisely, on the metallic side of the transition the static diffusion coefficient D(q,0) vanishes at a mobility threshold λ _c common for all q: D(q, 0)∝t=(λ _c ?λ)/λ _c →0, where λ=1/(2π?_F τ) is a dimensionless coupling constant. On the insulator side, q≠0 D(q,ω)∝?iω as ω→0 for all finite q. Within these limits, the scale of the spatial dispersion of D(q,ω) decreases in proportion to t in the metallic phase and in proportion to ωξ ², where ξ is the localization length, in the insulator phase until it reaches its lower limit ～λ _F. The suppression of the spatial dispersion of D(q,ω) near the Anderson transition up to the atomic scale confirms the asymptotic validity of the Vollhardt-Wölfle approximation: D(q,ω)?D(ω) as |t|→0 and ω→0. By contrast, the scale of the spatial dispersion of the electrical conductivity in the insulator phase is of order of the localization length and diverges in proportion to |t|^?v as |t|→0.     

A generalized interference model with indefinitely rising Regge trajectories

A class of analytic crossing-symmetric amplitudes is constructed which is compatible with any Regge trajectory bounded by ¦α(s)¦q, q<1/2. There are examples with an infinite number of Regge poles and without cuts. The model shows that the restrictions on the Regge trajectory and the residue function by analyticity and crossing symmetry are extremely weak.     

A new method to determine the shear coefficient of Timoshenko beam theory

The frequencies ω of flexural vibrations in a uniform beam of arbitrary cross-section and length L are analysed by expanding the exact elastodynamics equations in powers of the wavenumber q=mπ/L, where m is the mode number: ω²=A₄q⁴+A₆q⁶+?. The coefficients A₄ and A₆ are obtained without further assumptions; the former captures Euler-Bernoulli theory while the latter, when compared with Timoshenko beam theory rendered into the same form, unambiguously yields the shear coefficient κ for any cross-section. The result agrees with the consensus best values in the literature, and provides a derivation of κ that does not rely on physical assumptions.     

Nuclear magnetic relaxation of spin<Emphasis Type="Italic">I</Emphasis>=1/2 scalar coupled to a quadrupolar nucleus in the presence of cross-correlation effects

The rigorous treatment of relaxation for the dipolar-multipolarAX spin system (I=1/2,S>1/2) in the presence of the dipolarI-S coupling, anisotropy chemical shift and quadrupolar interaction ofS spin is proposed. The calculations of the spin evolution under the relaxation Hamiltonian are based on the second-order time-dependent perturbation theory and are carried out in the operator representation. For this task the double commutator identities of the type [[I _±S _z ⁿ ,A _q ^μp ]A _?q ^μp ] and [[I _zS _z ⁿ ,A _q ^μp ]A _?q ^μp. ] are derived. The fist-order differential equations for the evolution of longitudinal two-spin orderI _zS _z ⁿ , z=magnetization ofS spinS _z ⁿ and coherences <I _±S _z ⁿ > in the spin systemIS with scalar coupling between spin 1/2 and quadrupolar spinS>1/2 were obtained. These equations are used to get equations for the evolutions of each component of the multiplet structure of spinI. The imaginary part of the cross-correlation spectral density function and indirect spin-spin coupling Hamiltonian are taken into account. Equations for the longitudinal components of theI spin spectrum in the presence of cross-correlation effects were obtained also. Longitudinal and transverse relaxation times and cross-relaxation times in the presence of cross-correlation D-CSA, Q-CSA, Q-D were analyzed.     

Moments of the spectra for rotational transitions induced by collisions or by external perturbations

Considered first is the cross sectiond σ(J′←J)/dω for the rotational transitionJ′←J of a linear rotator in collisions. The set of cross sections for a givenJ and for allJ′ defines a discrete spectral distribution as a function of the transition energy ΔE=E(J′)?E(J). In both the classical and the quantum mechanical sudden approximations the moments \(S(\mu ;J) = \sum\limits_{J'} {(\Delta {\rm E}} )^\mu d\sigma (J' \leftarrow J)/d\omega \) of ordersμ=2n andμ=2n+1 (n=0,1,...) are polynomials of degreen in the rotational energyE(J). The special cases forn=0 indicate thatS(0;J) andS(1;J) are independent ofJ. This is a theorem proved elsewhere. Explicit expressions forS(μ; J) of low orders are presented. They are derived on the assumption of equal relative velocitiesv andv′ before and after the collision. Correction onS(μ;J) for the difference betweenv andv′ is made to the lowest order in terms ofS(μ+1;J) forv=v′. The quantum mechanical results for linear rotators are extended to spherical-top and symmetric-top rotators. These results apply not only to the collision cross section but also to a physical quantity expressible in the form ¦〈Γ′¦F¦Γ〉¦² of a transition probability withany F that is independent of the initial and final rotational states ¦Γ〉 and ¦Γ′〉.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. II

We generalize to any order q, the methods developed in a companion paper for q = 2,3 for finding bi-solitons, solutions of the class of non-integrable non-linear equations L_qK = K²; L_q = ? + Σ_i+j≤qa_ij?_xⁱ?_lⁱ, ? ≠ 0 in 1 + 1 dimensions. We call bi-solitons K(ω₁,ω₂) of the exponential type variables ω_i = exp(γ_ix + ρ_it), i = 1,2 and deal only with the so-called “non trivial” solutions which may be written as a finite sum K = Σ^lmax₀ω¹₂F_i(Z)_, F₁ rational function of Z = ω₁Z = ω₁ + ω₁. To any such polynomial K, we associate a linear transformation such that L_qK has only the power ω¹₂ of K² and we find that there are particular polynomialswhere the above restriction provide a factorization of the linear operator L_q in the product of smaller order differential operators. After this linear phase, we show in a second step that these forms yield solutions for the full non linear equation which can be derived in an intrinsic manner. Examples in the monomial and binomial cases are given.     

A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.     

Quantum hall effect of two-dimensional interacting boson systems

While non-interacting two-dimensional (2-D) boson systems cannot have the quantum Hall effect (QHE), we show that 2-D interacting boson systems can have the QHE if there is a downward cusp in the ground state energy. A necessary condition to have the QHE at filling factor v = p/q with σ_H = (p/q)e²/h is that p and q must satisfy e^ipqπ= 1, so the possible candidates are v = 2n₁/(2n₂ + 1) and (2n₁ + 1)/2n₂ where n₁ and n₂ are integers.     

Space-time dependence of the gluon condensate correlation function and quarkonium spectra

We consider the influence of the gluon condensate in QCD on the energy levels of quarkonia, taking into account the x-dependence of 〈ω|:G_μv^a(x) G^aμv(0):|ω〉. The modification compared to earlier approaches which approxim ated the above vacuum expectation value by a constant is quite sizeable; for the b?b system we find that the effect can be essentially described in terms of a local potential.     

Dispersive viscosity coefficient of nuclear matter

By application of Landau's kinetic theory of a Fermi liquid in non-equilibrium, we have deduced a dispersive, momentum and frequency dependent shear viscosity coefficient η(q, ω) for symmetric nuclear matter. The resulting formula for η is found to be complicated for arbitrary q, ω; however, simple interpretation is possible in the limit of small momentum q → 0. In the static limit, ω → 0, η reduces to the well known kinetic formula for the hydrodynamic viscosity coefficient η_o, while in the high frequency (zero sound) limit, ωτ ? 1, η(ω) is found proportional to (?iω)^? indicating elastic behaviour of nuclear matter. As is discussed shortly, application of these results to finite nuclei unfortunately does not seem justified.     

超导临界温度理论(Ⅰ)

本文求出了Eliashberg方程在T=T_c时的解,得到了下面的临界温度级数表示式:T_c=α₀(μ^*)(λ〈ω²〉)^1/2{1+1/λα₁(μ^*)〈ω⁴>/〈ω²>²+1/λ²(α₂₁(μ^*)〈ω⁶>/〈ω²>³+α₂₂(μ^*)〈ω⁴>²/〈ω²>⁴) +1/λ³(α₃₁(μ^*)〈ω⁸>/〈ω²>⁴+α₃₂(μ^*)(〈ω⁴>〈ω⁶>)/〈ω²>⁵)+α₃₃(μ^*)〈ω⁴>³/〈ω²>⁶+…},其中α₀(μ^*),α₁(μ^*)等仅是μ^*的函数。新的T_c公式表明了,T_c不仅依赖于λ、μ^*和〈ω²〉,而且依赖于有效声子谱α²F(ω)的各级矩〈ω²ⁿ〉。