The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation

We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T ²π, whereT=h·A·?, with?(x)=x ^(q?1/q,h(x)=(x∥x∥ _q ) ^q ,A=(a(r, s)∶r, s∈S), and $$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$ We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n) ^p ∥Ax(n) ^p ∥₁ withp>0 has limit pointsξ of period?2. We show thatA positive definite implies limit points are fixed points that satisfy the equationAξ ^p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.     

Equation of state and quantal crossover of the XY model in a longitudinal magnetic field

The equation of state of the XY model in a longitudinal (Γ) and transverse field (B) is investigated in the vicinity of a multicritical point (Γ_c, T =0) using the renormalization group method developed recently by the author and K. Walasek. For Γ = Γ_c a quantal crossover behaviour of the form H = M_{R³ψ (z)} is obtained, where H ～ B, M_R denotes the transverse magnetization, z ～ TM_R^?/βq, while ?,β_q are crossover and multicritical exponents, respectively.     

Laplace operator and hodge decomposition for quantum groups and quantum spaces

LetΓ=Γ _±,z be one of theN ²-dimensional bicovariant first order differential calculi for the quantum groups GL _q (N), SL _q (N), SO _q (N), or Sp _q (N), whereq is a transcendental complex number andz is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’s external algebraΓ ^{^} coincides with the de Rham cohomologies of its leftinvariant, its right-invariant and its biinvariant subcomplexes. In the cases GL _q (N) and SL _q (N) the cohomology ring is isomorphic to the biinvariant external algebraΓ _inv ^{^} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. It is also applicable for quantum Euclidean spheres. The eigenvalues of the Laplace-Beltrami operator in cases of the general linear quantum group, the orthogonal quantum group, and the quantum Euclidean spheres are given.     

On the Deser,Gilbert, Sudarshan representation for functions Wi(ν, q2)(i = 1, 2) describing the electroproduction process

The Deser, Gilbert, Sudarshan representation (D.G.S.R.) for the functions W_i(ν, q²) (i = 1,2) is considered as equations determining spectral functions h_i(a, α) via the values W_i(ν, q²) in the physical region of the electroproduction channel. It is shown that if W_i(ν, q²) obey the microcausality and spectrality conditions, then the equations for h_i(a, α) have solutions in the class of Schwartz temperated distributions and thereby the D.G.S.R. is proved. Formulae are obtained expressing spectral functions in the D.G.S.R. through the values of functions W_i(ν, q²) in the physical region of the electroproduction channel.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

Groundstate threshold in disordered anisotropic +/−J Ising models

In disordered anisotropic square +/− J Ising models SQ(p, q) at groundstates we investigate the pairs (p_c, q_c) of critical concentrations of antiferromagnetic bonds with concentrations p,q, respectively in orthogonal coordinate directions. We are led to p_c(q) ≈ π(q) with π(q) from the so-called adjoined problem. This approach is well supported by simulations for different values of q on the basis of minimal matchings of frustrated plaquettes. In particular, p_c(0) ≈ 0.21 from simulations and π(0) = 0.2113248 …, with the conjecture that p_c(0) = π(0). The concept of the adjoined problem is extended to d-dimensional (hyper-) cubic lattices. We hereby obtain for p_c,d especially in the sotropic case: p_c,3 ≈ 0.154, p_c,4 ≈ 0.170, p_c,5 ≈ 0.178, p_c,6 ≈ 0.182. Moreover, in analogy to SQ(p,q) we used the approach also for honeycomb Ising models HC(p,q,r) with no antiferromagnetic bonds in the third plaquette direction (r = 0).     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Higher order QCD corrections to an exclusive two-photon process

We calculate the next-to-leading order QCD corrections in the $\text{MS}$ scheme to the coefficient functions in an operator product expansion of the amplitude T(q², p²) for the process $\text{γ}{}^1\text{(q) + γ}{}^1\text{(p) →}\text{helicity-zero}$, flavour non-singlet meson in which ?q² is large and ?p² ? 0. For an asymptotic wave function the complete O(α_s) correction for a pseudoscalar meson is about 16% for p² = 0 and α_s = 0.3; most of this correction can be removed by using a modified evolution equation for the wave function, leaving a correction of about 7%. For large p² the complete O(α_s) correction for a pseudoscalar meson is about 10%. We discuss how our results can be combined with similar calculations for the pion form factor F_π(Q²) to give a prediction: F_π(Q²) = Aα_s(Q²)T_π²(q², 0)(1 + Bα_s) that is independent of the as yet unknown two-loop anomalous dimension matrix.     

The sub-ODE method for finding exact travelling wave solutions of generalized nonlinear Camassa-Holm, and generalized nonlinear Schrödinger equations

With the aid of the ordinary differential equation (ODE) involving an arbitrary positive power of dependent variable proposed by Li and Wang and an indirect F-function method very close to the F-expansion method, we solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p) and the generalized nonlinear Schrödinger equation with nonlinear dispersion GNLS(l,n,p,q). Taking advantage of the new subsidiary ODE, this F-function method is used to map the solutions of C(l,n,p) and GNLS(l,n,p,q) equations to those of that nonlinear ODE. As result, we can successfully obtain in a unified way, many exact solutions.     

Simple tests of quark-nucleon reciprocity

The inclusive cross-section for the production of a single hadron in deep inelastic electroproduction is studied in a dual resonance model. The Bjorken scaling behaviour in the virtual photon fragmentation region for finite x (≡ 2p_L^c.m./√s) is (1/σ^{T,L))d3σT,L/E?}d³p ～ (1/q²) F(x,p_⊥²/q²) and thus the transverse momentum grows like q², whereas in the parton model (1/σ^T,L)d³σ^T,L/E^?1d³p ～ F(x,p_⊥²). A related effect is the absence of two-jet structure in e⁺e^? annihilation. We believe that dual model results may give a more reliable indication of the deep inelastic behaviour for composite hadrons than the parton model.     

Partial cross sections for Rb− photodetachment in the region of the Rb(5p 2 P 1/2 3/2) thresholds and their analysis by multichannel quantum defect theory

A crossed ion-laser beam apparatus has been used to measure accurate relative total and partial cross sections for photodetachment from Rb^? ions with high photon energy resolution (0.1–0.6 cm^?1) in the region of the Rb(5p ² P _1/2,3/2) thresholds (photon energy range 16,350–16,820 cm^?1). Satisfactory fits to these data by multichannel quantum defect theory (MQDT) have been obtained, resulting in a reliable set of seven MQDT parameters. The electron angular distribution parameter for the Rb(5_s) channel was found to beβ(5s)=2, independent of photon energy. MQDT predicts a sharp, window-type variation of?(5s) around the minimum of the 5s-cross section below the Rb(5p ² P _1/2) threshold (where an accurate experimental measurement ofβ(5s) was not possible) and a similar behaviour ofβ(6s) in the case of Cs^?photodetachment below the Cs(6p ² P _1/2) threshold.     

Phenomenology of light- and strange-quark simultaneous production at high energies

This letter presents an extension of EPL116(2017)62001 to light- and strange-quark nonequilibrium chemical phase-space occupancy factors (γ_q,s). The resulting damped trigonometric functionalities relating γ_q,s to the nucleon-nucleon center-of-mass energies (\(\sqrt {{s_{NN}}} \)) looks very similar except different coefficients. The phenomenology of the resulting γ_q,s(\(\sqrt {{s_{NN}}} \)) describes a rapid decrease at \(\sqrt {{s_{NN}}} \) ? 7GeV followed by a faster increase up to ~20 GeV. Then, both γ_q,s become nonsensitive to \(\sqrt {{s_{NN}}} \). Although these differ from γ _s (\(\sqrt {{s_{NN}}} \))obtained at γ _q (\(\sqrt {{s_{NN}}} \))=1, various particle ratios including K⁺/π⁺, K^?/π^?, Λ/π^?, Λ?/π^?, Ξ⁺/π⁺, and Ω/π^?, can well be reproduced, as well. We conclude that γ_q,s(\(\sqrt {{s_{NN}}} \)) should be instead determined from fits of various particle yields and ratios but not merely from fits to the particle ratio K⁺/π⁺.     

Features of multiplicity distributions in a bootstrap model of inclusive spectra

Using a bootstrap model of inclusive spectra we derive an integral equation satisfied by the generating function for multiplicity distributions. The (semi-asymptotic) solution of this equations has the form Ψ(λ, s) = Ψ(λ, s₀)(s/s₀)^b(λ) where s is the usual energy variable and b(λ) satisfies an eigenvalue equation and is completely determined by the leading particle distribution. Closed formulae for the binomial moments and for correlation coefficients are also given, and in addition we discuss some general features of the bootstrap model. As a phenomenological application we discuss the rate of variation with energy of multiplicity moments. Our results are expected to be representative for multiperipheral-like models.     

The irreducible Raman scattering tensor operator for the 32 crystallographic point groups and its application to the resonance and electronic Raman effect

The irreducible components of the Raman scattering tensor operator $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ under the symmetry of a general point group are calculated. The unitary transformations U(Γ_γΓ_ksΓ_k′s′, ρσ) from the Cartesian $\text{α}\text{?}{}_{\mathrm{\rho \sigma }}$ and spherical $\text{α}\text{?}{}_{\mathrm{Q}}{}^{\mathrm{K}}$ components, respectively, to the irreducible components $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ for the 32 crystallographic point groups are collected in tables. As an example the unitary transformation U(Γ_γΓ_ksΓ_k′s′, ρσ) is used to discuss the behavior of the scattering tensor in a resonance Raman experiment. With the help of the general formalism the scattering tensor for electronic Raman transitions of transition metal ions is calculated. As an example the scattering tensors of electronic Raman transitions within the ⁵T₂ state of the high-spin trigonal distorted octahedral Fe²⁺ are calculated and the refinement of the selection rules is discussed.     

Free energies of the spherical model of a spin glass

Free energies g(m, m_s) and f(m, q) of the spherical spin glass (SG) model due to Kosterlitz et al. are calculated explicitly as functions of the uniform magnetization m, and SG order parameter m_s and the Edwards-Anderson order parameter q. It is shown that g(0, m_s) and f(0, q) below the transition temperature T_g are constant in the ranges 0 ≦ m_s ≦ m_s⁰ and 0 ≦ q ≦ q₀ respectively, where $\text{q}{}_0\text{= (1 -?}\text{T}\text{T}{}_{\mathrm{g}}\text{) = m}{}^2{}_{\mathrm{s}}{}^0$. The proper equilibrium values of m_s( = m_s⁰) and q( d=q₀) are then fixed from the inspection of their behaviors under infinitesimal uniform field proproportional to N^-a($\text{a ≧ 0}$).     

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.     

The energy-momentum tensor in scalar and gauge field theories

A previous study of the energy-momentum tensor in ?⁴ theory and spontaneously broken non-Abelian gauge field theories is extended here to show finiteness to all orders in perturbation theory. Divergences of Green's functions Γ_μν^(j) (q; p₁, …, p_j) involving the energy-momentum tensor θ_μν and j particle fields are removed by counterterms of the ordinary Lagrangian plus a renormalization of the coefficient of the Callan-Coleman-Jackiw improvement term in θ_μν. Physically the extra renormalization means that the mean square “mass radius” of elementary spin zero particles must be specified from experiment.     

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.     

Isometric immersions of pseudo-Riemannian space forms

In this paper we study local isometric immersions f:M_sⁿ(K)→N_s+q²ⁿ⁻¹(c) of a time-like n-submanifold M_sⁿ(K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N_s+q²ⁿ⁻¹(c) with constant sectional curvature c and index s+q, where q≥0, 1≤s≤n−1 and K≠c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M_sⁿ(K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N_n−1²ⁿ⁻¹(c) and N₁²ⁿ⁻¹(c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.     

Remarks on the Poisson equations of motion

In respect of Poisson's equations of motion, we give solutions of the form F_o(H)exp{iU(p, q)}, which have a physical significance and are of importance for the calculation of phase operators in quantum mechanics.