Comment on a recent calculation of the Ham's reduction factors by the use of the exponential transformation method

The reduction factors p and q for an E electronic state with linear Jahn-Teller coupling to an E vibrational mode have been calculated numerically. The results together with other evidence suggest that the results obtained by Sigmund et al. using the transformation method are incorrect in the intermediate and strong coupling regimes.     

Ham reduction factors for the general Г8-(e+t) Jahn-Teller Hamiltonian

The quenching factors (Ham reduction factors) of the Г₈-(e+t) Jahn-Teller system with arbitrary coupling parameters D_? and D_τ are presented. The calculations are based on a group-theoretical approach combined with the application of a canonical transformation which leads to 2 different diagonalization procedures for the weak and intermediate as well as for the intermediate and strong coupling case. The results are accurate in the whole coupling range.     

On the von Staudt-Clausen theorem for q-Euler numbers

The q-Euler numbers and polynomials were recently constructed [T. Kim, “The Modified q-Euler Numbers and Polynomials,” Adv. Stud. Contemp. Math., 16, 161–170 (2008)]. These q-Euler numbers and polynomials have interesting properties. In this paper, we prove a theorem of the von Staudt-Clausen type for q-Euler numbers; namely, we prove that the q-Euler numbers are p-adic integers. Finally, we prove Kummer-type congruences for the q-Euler numbers.     

Causal A-statistics I. General properties

It is shown that the second quantization axioms can, in principle, be satisfied by creation and annihilation operators generating the Lie algebra of the unimodular group. The Fock spaces W(p,q) are labelled with two arbitrary non-negative numbers p and q. The Pauli principles are formulated. In the Fock space W(p,q) there can not be more than p+q particles in a single state. The charge of an arbitrary ensemble of particles can not exceed p and be less than q.     

3nj coefficients of u q(2) and multiple basic hypergeometric series

Triple-sum formulas for 9j coefficients and multiple-sum expressions [with five or four separate sums of the _p+1 F _p(1) or _p+1 φ _p type, p=2, 3, 4] for the 12j coefficients of both kinds (with or without braiding) of the SU(2) group and the quantum algebra u _q(2) are derived, eliminating sums over the j type parameters [q generalizations of the very well poised (Dougall’s type) hypergeometric ₄ F ₃(?1), ₅ F ₄(1), and ₆ F ₅(?1) series] from their expansions in terms of q-6j coefficients. The rearrangements of the derived formulas for generic and stretched q-9j coefficients (related to the q versions of some Kampé de Fériet series) are discussed, as well as the different versions of stretched and doubly stretched q-12j coefficients.     

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).     

Q-dependent stoner factors and reformulated spin-fluctuation enhancements: V and Pd

Spin-polarized self-consistent LMTO band calculations are performed for supercells containing induced, “frozen” spin-waves. Momentum dependent Stoner factors S(q) are obtained as the local ratio between induced and applied spin splitting. The results are used to calculate the spin-fluctuation enhancement λ_sp on a similar basis as a calculation of the electron phonon coupling. The methods are tested on vanadium and palladium. For the latter a strong reduction of S(q) is found for large q. Obtained values for λ_sp are of reasonable magnitude.     

Large-pT production of single and double photons in proton-proton and pion-proton collisions

Quantum chromodynamic (QCD) predictions are made for the large transverse momentum production of single and double photons in proton-proton, proton-antiproton, and pion-proton collisions. In π^?p collisions at center-of-mass energy W = 27.4 GeV and p_T = 4.0 GeV, it is estimated that about 0.3% of the 90° single photon triggers will be balanced on the “away-side” by a single photon with roughly the same transverse momentum. In π⁺p collisions this fraction drops to about 0.09%. These fractions increase with p_T. In addition to the pure QED annihilation term $\text{q}\text{q}\text{→γγ}$, it is found that the QCD-induced subprocess gg→γγ provides an important source of double photons. Photon bremsstrahlung contributions are also examined. Experimental study of the systematics of single and double photon production in hadron-hadron collisions will provide information on the size of the strong interaction coupling constant, α_s(Q), and on the charges of the quarks. Knowledge of the gluon distributions within hadrons and of the effective transverse momentum of partons in hadrons can also be gained.     

Generalized susceptibilities and phonon anomalies in Pd and Pt metals

The generalized susceptibility, χ(q), in Pd and Pt for q along the [100], [110], [111], and [120] directions was determined from their APW and RAPW energy band structures, respectively, using the analytic tetrahedron linear energy scheme of Rath and Freeman. The band structures were previously found to yield Fermi surface radii, temperature dependencies of the static magnetic susceptibility, χ(T), resistivity, and a spin lattice relaxation, T₁T, in very good agreement with experiment. In the χ(q) calculations, we used 2048 tetrahedra in 1/48th irreducible BZ and the energy eigenvalues for bands 4, 5, and 6 which cross the Fermi energy as fitted to a Fourier series representation. The intraband parts of χ(q) at q = 0 for both metals are found to agree with the density of states at the Fermi energy to without 0.5%. Our results show that the dominant contribution to χ_intra arises from the dominant band 5 whose “jungle-gym” FS has strong nesting features; the main peak for Pd occurs at the same q value (= 0.65π/a) for q along the [0q0], [q, q, 0], and [q, q, q] directions. The locus of this main peak is a square in the (0, 0, 1) plane. The maximum of χ_intra for q along the [110] and [111] directions are 23% and 13%, respectively, higher than the value of χ(q) at q = 0. For q along the [010] and [120] directions, the peak is, however, lower than the value of χ_intra at q = 0. Hence, while phonon anomalies are predicted for the [110] and [111] directions, no anomaly is predicted for either the [100] or [120] direction. The predicted q value for the [110] anomaly, $\text{q}\text{= 0.65π/a}$ is close to the experimental value of ～0.7 π/a. Although there may be a hint of an anomaly at 0.56 [111] in the measurements, a more detailed investigation of this region is called for. For platinum, χ_intra for q along the [010], [110] and [111] directions has main peaks which occur at q = 0.68 π/a, 0.75 π/a, and 0.85 π/a, respectively. Here too, this main peak comes from the nesting of the jungle-gym Fermi surface which is not, however, as flat as that of palladium. Anomalies are predicted (although weaker in Pt than in Pd) along [110] and [111] but not along [100] and [120]. The [110] anomaly is close to the measured q value (～0.7–0.8 π/a). Also in agreement with experiment, we predict a weaker [110] anomaly for Pt than for Pd. In both Pd and Pt, weaker anomalies are predicted for the [111] direction than for the [110] direction.     

The perturbative approach to the critical behaviour of two-matrix models in the limit N → ∞

We construct representations of the Heisenberg algebra by pushing the perturbation expansion to high orders. If the multiplication operators B_1,2 tend to differential operators of order l_2,1, respectively, the singularity is characterized by (l₁, l₂). Let l₁ ⩾ l₂. Then the two cases, (A) “l₂ does not divide l₁” and (B) “l₂ divides l₁”, need a different treatment. The universality classes are labelled [p, q] where [p, q] = [l₁,l₂] in case (A) and [p, q]=[l₁ + 1,l₂] in case (B).     

Quark dynamics and pion-nucleon coupling

In the framework of nonperturbative QCD phenomenology we discuss: (1) The elementary process for the creation of color-singlet q$\text{q}$-pairs inside a hadron. (2) The interaction of the q$\text{q}$-pair with the surrounding quark-gluon medium. An important consequence of these discussions is that meson emission takes place preferentially, if the primary q$\text{q}$-pair is created in the surface region of the hadron. For the case of pseudoscalar coupling we employ PCAC to obtain the coupling of the q$\text{q}$-pair to the pion. The resulting form and coupling strength of the πNN vertex is consistent with the phenomenological OPEP.     

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

In the present paper, we study a new kind of p-adic measures for q?+?1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.     

Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

In the previous papers of this series we discussed the representation in quantum mechanics of canonical transformations leading to action and angle variables, for Hamiltonians with bounded or unbounded orbits, i.e., whose spectra is either discrete, continuous, or mixed. In the present paper the results are extended to Hamiltonians with periodic potentials which have a band spectra. Again the canonical transformations are non-linear and non-bijective and the classical analysis shows that the angle variable φ (always in the interval 0≤φ≤2π) and action J can be defined for energies both below and above the maximum height of the potential. In all of the original phase space the variables (q, p) are then periodic functions of φ. Inversely, because of the invariance of the Hamiltonian under translations q → q + a, the (φ, J) are also periodic functions of q. thus to recover bijectiveness we require an infinite sheet structure in both the (q, p) and (φ, J) phase spaces. In turn the sheet structure can be replaced by appropriate ambiguity groups and spins, with the help of which we propose an explicit expression for the representation in quantum mechanics of the canonical transformation, and recover the latter when we pass to the classical limit with the help of the WKB approximation. The present analysis corroborates the previous surmise that the nature of the spectra of a quantum mechanical Hamiltonian, i.e., continuous, discrete, mixed, or of bands, is related to the ambiguity group and spin of the problem. As the latter originates in classical mechanics when we discuss the canonical transformations from (q, p) to (φ, J), we conclude that some quantum features, such as the nature of spectra of operators, are already implicit in the classical picture.     

Supersymmetry in the AdS/CFT correspondence

We study how local symmetry transformations of (p,q) anti de Sitter supergravities in three dimensions act on fields on the two-dimensional boundary. The boundary transformation laws are shown to be the same as those of two-dimensional (p,q) conformal supergravities for p,q≤2. Weyl and super Weyl transformations are generated from three-dimensional general coordinate and super transformations.     

Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy

We study the asymptotic behavior of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions, we prove that the vector (p, q) of moments and coordinates of the oscillators in the network satisfies (p, q)(t) → (0, q _c ) as t → ∞, where q _c is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a nontrivial limiting regime was established.The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.     

Spin-glass-like transition in the majority-vote model with anticonformists

Majority-vote model on scale-free networks and random graphs is investigated in which a randomly chosen fraction p of agents (called anticonformists) follows an antiferromagnetic update rule, i.e., they assume, with probability governed by a parameter q (0 < q < 1∕2), the opinion opposite to that of the majority of their neighbors, while the remaining 1 ? p fraction of agents (conformists) follows the usual ferromagnetic update rule assuming, with probability governed by the same parameter q, the opinion in accordance with that of the majority of their neighbors. For p = 1 it is shown by Monte Carlo simulations and using the Binder cumulants method that for decreasing q the model undergoes second-order phase transition from a disordered (paramagnetic) state to a spin-glass-like state, characterized by a non-zero value of the spin-glass order parameter measuring the overlap of agents’ opinions in two replicas of the system, and simultaneously by the magnetization close to zero. In the case of the model on scale-free networks the critical value of the parameter q weakly depends on the details of the degree distribution. As p is decreased, the critical value of q falls quickly to zero and only the disordered phase is observed. On the other hand, for p close to zero for decreasing q the usual ferromagnetic transition is observed.     

Momentum distributions in the nucleus

The nuclear form factor F(q) and one particle momentum distribution p(q) can be shown to have a power law decrease for large momenta. For the form factor F(q) we show that it is q/A that must be large for this asymptotic behavior to be important. For only q large the form factor, in a simple model, is shown to decrease exponentially in q. A similar behavior for p(q) is proposed.     

Exact results for the diluted Ising model with competing interactions

We consider the random-bond Ising model with the exchange integrals J > 0, ?J and 0 with the respective probabilities p, q and r, where p + q + r = 1. We give the exact value of the averaged internal energy and an exact upper bound to the averaged specific heat at temperature T determined by $\text{k}{}_{\mathrm{B}}\text{T =}\text{2J}\text{In[}\text{p}\text{(1 ? p ? r)}\text{]}\text{,}$ where k_B is the Boltsmann constant. We show that all the averaged correlation functions of even spins are non-negative at this temperature.     

Liouville dynamics for optimal stochastic phase-space representations of quantum mechanics

It is shown that the space P(Γ_s) of Γ_s-distribution functions ?(q, p) (Husimi transforms) can be described without reference to any conventional representation of the density operator ?. A Liouville-type differential equation governing the free time-evolution of ?_t(q, p) is derived and solved explicitly; the time dependence of this solution supports the thesis that ?(q, p) is a bona fide probability density observable with optimally accurate apparatus for the simultaneous measurement of position and momentum. Liouville-type equations are derived also for the case when local interactions described by analytic potentials are present. Probability currents corresponding to ?(q, p) are defined and it is shown that they obey a continuity equation at space-time points. Reduced Γ_s-distribution functions are defined and it is shown that they obey a BBGKY hierarchy of equations. A Brownian-motion experimental test of the underlying theory of measurement is suggested.     

Sensitivity of polarization observables in elastic ed scattering to the neutron form factors

The deuteron elastic form factors are calculated within the Bethe-Salpeter approach with separable interaction. The charge, quadrupole, and magnetic form factors [F _C(q ²), F _Q(q ²), and F _M(q ²), respectively]; the structure functions A(q ²) and B(q ²); and the tensor polarization components T ₂₀(q ², T ₂₁(q ²), and T ₂₂(q ²) are investigated up to ?q ²=50 fm^?2. The role of relativistic effects is discussed, and a comparison with nonrelativistic calculations is performed. The effect of the neutron form factors on the deuteron form factors and especially on tensor polarization components is discussed too.