Thermo field dynamics and quantum algebras

The algebraic structure of thermo field dynamics lies in the q-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of h_q(1) and of h_q(1|1), respectively.     

N-particle Bogoliubov vacuum state

We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We show that, in the particle representation, the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N-particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single-particle density matrix of the vacuum. We compare N-body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N → ∞ but a small fraction of depleted particles is kept constant.     

Symmetry properties and renormalization group in the stable superfluid phase of bosons at zero temperature

The appearence of divergences in perturbation theory for a system in a stable phase requires the existence of an underlying symmetry and related Ward identities to remove exactly these singularities in the physical response functions. In this paper we consider explicitly the Ward identities associated with the (gauge) symmetry of the interacting neutral Bose gas at zero temperature to solve the old-standing problem of infrared divergencies due to Bose- Einstein condensation. The exact infrared behavior of the system is achieved for any dimension d > 1. For 1 < d ? 3 the system is controlled by a non trivial line of fixed point distant from the Bogoliubov solution, while above d = 3 the Bogoliubov fixed point is found to be stable.     

Bounding the Bogoliubov coefficients

While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [M. Visser, Phys. Rev. A 59 (1999) 427-438, arXiv:quant-ph/9901030], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2×2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.     

Vector boson in the constant electromagnetic field

The propagator and the complete sets of in-and out-solutions of the wave equation, together with the Bogoliubov coefficients relating these solutions are obtained for the vector W-boson (with the gyromagnetic ratio g=2) in a constant electromagnetic field. When only the electric field is present, the Bogoliubov coefficients are independent of the boson polarization and are the same as for the scalar boson. For the collinear electric and magnetic fields, the Bogoliubov coefficients for states with the boson spin perpendicular to the field are again the same as in the scalar case. For the W ^? spin parallel (antiparallel) to the magnetic field, the Bogoliubov coefficients and the one-loop contributions to the imaginary part of the Lagrange function are obtained from the corresponding expressions for the scalar case by the substitution m ² → m ²+2eH (m ² → m ²-2eH). For the gyromagnetic ratio g=2, the vector boson interaction with the constant electromagnetic field is described by the functions that can be expected by comparing the scalar and Dirac particle wave functions in the constant electromagnetic field.     

Propagator for the Chebyshev q object

We propose an alternative role of the harmonic oscillator algebra. Observing that the q-deformed harmonic oscillator algebra defines the Chebyshev q object, we show that the q-free particle and the pulsed oscillator are special cases of the Chebyshev q object, characterized by a common deformation parameter q and reduced to a usual free particle as q tends to unity. For the deformed free particle, q is a real number, whereas for the pulsed oscillator it belongs to S ¹. Then, we derive the propagator for the Chebyshev q object, from which we obtain the propagators for the deformed free particle and the pulsed oscillator.     

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.     

New q-Deformed Oscillator Algebra and Thermodynamic Model

In this paper we propose the new q-oscillator algebra. We discuss the coherent state and the deformed su(2) algebra for this algebra when q is real. As is different from Arik–Coon algebra (J. Math. Phys. 17:524, 1976), this algebra is invariant under the hermitian conjugation for complex q. When q is a root of unity, we obtain the finite dimensional Fock space. Finally we discuss the thermodynamics of particle obeying this algebra when q is a root of unity.     

Perturbation expansion, Bogoliubov inequality and integral representations in nonextensive Tsallis statistics

By using integral representations the perturbation expansion and the Bogoliubov inequality in nonextensive Tsallis statistics are investigated in a unified way. This procedure extends the analysis performed recently by Lenzi et al. [Phys. Rev. Lett. 80, 218 (1998)] to the quantum (discrete spectra) case, for q<1. An example is presented in order to illustrate the method. Received 19 November 1998     

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.     

The infinite number of generalized dimensions of fractals and strange attractors

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_q, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_q. We prove that lim _q→0D_q = fractal dimension (D), lim_q→1D_q = information dimension (σ) and D_q=2 = correlation exponent (v). D_q with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_q for any q′ > q. For homogeneous fractals D_q = D_q. A particularly interesting dimension is D_q=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D_∞.     

Discontinuity of surface tensions in the q-state Potts model

For the ferromagnetic scalar q-state Potts model on a d-dimensional cubic lattice we prove the following results: (1) We derive a correlation inequality and then we prove that the surface tension between two ordered phases exists in dimension d ⩾ 2 whenever q ⩾ 2 and it is discontinuous at the transition point whenever q is large enough. (2) At the limit q↗ ∞ the surface tension between an ordered phase and the disordered one vanishes everywhere except at the transition point.     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

The structure of the multi-instanton gas

We parametrize the q-instanton solutions of an SU(2) gauge theory in terms of the positions of 2q constituent particles or “instanton quarks”. Explicit computations of lowest order quantum fluctuations about the q = 1 and q = 2 solutions show that the short-distance interaction between instanton quark pairs is logarithmic. Extending these interactions to arbitrary q, we describe the multi-instanton gas as a plasma of instanton quarks.     

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1?q<2, while the interval 0<q?1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2−q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (−∞,0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.     

Nonextensive formalism and continuous Hamiltonian systems

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function Sq while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy Sq yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q>1. Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q<1, which raises the question of its formal validity range for continuous Hamiltonian systems.     

Nonextensive statistical mechanics of q-bosons based on the q-deformed entropy

Based on the q ↔ q⁻¹ symmetric deformed entropy, we develop a general framework for nonextensive statistical mechanics of ensembles of q-deformed systems. Applying this doubly deformed formalism to q-bosons, a correction to the Planck law is evaluated in the weak deformation regime and its properties are discussed. It is found that at high temperature the dominant part of the correction comes from the deformation of the oscillator dynamics, whereas at low temperature the deformation of the entropy gives a leading contribution. This suggests the nonextensive approach to q-deformed ensembles might be important at low temperature.     

A note on q-weak law of large numbers

q-limit theorems for random variables are arising from non-extensive statistical mechanics. In this note we will prove q-weak law of large numbers using the notions of q-Fourier transform, q-independence, q-weak convergence.     

q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard su _q (2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the u _q (2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0⁺ states in the deformed and nondeformed cases.     

Non-Gaussian effects on quantum entropies

A deduction of generalized quantum entropies within the non-Gaussian frameworks, Tsallis and Kaniadakis, is derived using a generalized combinatorial method and the so-called q and κ calculus. In agreement with previous results, we also show that for the Tsallis formulation the q-quantum entropy is well-defined for values of the nonextensive parameter q lying in the interval [0,2].