Some formal aspects of subdynamics

The theory of subdynamics is formulated assuming the existence of a spectral representation of the collision operator. This approach avoids perturbation schemes; however the presentation is formal. It may be used to develop further the theory as well as a starting point for a rigorous mathematical discussion. The construction of the operators introduced in the theory of subdynamics is presented in detail. Some questions related to the transformation theory leading to the so-called “physical representation” are briefly discussed.     

Some remarks concerning stability for nonstationary quantum systems

The problem of characterizing stability and instability for general nonstationary quantum systems is investigated. Some characterizations are reported and some elementary properties of a topological characterization are established. Then, it is proven, by considering a simple example, that there are nonperiodic driven systems whose orbits are neither precompact nor leave on average any compact set. Autocorrelation measures are computed and the possible roles of the generalizes quasienergy operator and energy growth are briefly discussed.     

Vacuum subdynamics in large quantum systems

We present a constructive procedure to deal with large quantum systems in the thermodynamic limit. Starting with a discrete spectrum, we perform a complete decomposition of the evolution into one-dimensional subdynamics. We then go to the limit of a continuous spectrum after collecting them into global subdynamics of given degrees of correlation. Previously obtained results for the vacuum subdynamics are justified. The procedure is applied to the problem of potential scattering.It is a privilege for the authors to dedicate this paper to Prof. I. Prigogine. They are both genuine offspring of the Brussels school. As such, most of the ideas they profess owe their inspiration to I. P., their mentor, colleague, and friend through fascinating teaching, stimulating discussions, and illuminating speculations, but they take full responsability for their obvious mistakes, pernicious misconceptions, and malignant deviations from the orthodoxy they have modestly helped to create.     

Some remarks on nonequilibrium dynamics of infinite particle systems

Classical mechanics of infinitely many particles in dimensions one and two is considered, particles interacting by a superstable pair potential of finite range. The group of motion generated by Newton's equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity. A core of the Liouville operator is also described. Results of Dobrushin and the author and of Marchioro-Pellegrinotti-Pulvirenti are improved.     

Some remarks on the dynamical systems approach to fourth order gravity

Building on earlier work, we discuss a general framework for exploring the cosmological dynamics of Higher Order Theories of Gravity. We show that once the theory of gravity has been specified, the cosmological equations can be written as a first-order autonomous system and we give several examples which illustrate the utility of our method. We also discuss a number of results which have appeared recently in the literature.     

Some remarks on the location of zeroes of the partition function for lattice systems

We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.     

Some results in the theory of interacting electron systems

The expressions for the longitudinal dielectric function, spin and orbital susceptibilities in the static, long wavelength limit are evaluated by solving the corresponding linearized vertex functions exactly in this limit. The plasma dispersion relation to leading order in the long wave limit is similarly obtained. These are compared with the corresponding results obtained previoulsy by us by a variational solution to the same vertex equations. It is established that the variational method gives the exact results in the static, zero wave vector limit, involving the proper renormalizations. The plasma dispersion relation is found to be the same as in the exact calculation whereas the coefficient of q² in the static density correlation function has an important additional contribution to the variational result. Applications of these results are briefly discussed.     

Some remarks on the ground state of infinite systems in statistical mechanics

We investigate the ground states of infinite quantum lattice systems. It is shown in particular that a positive energy operator is associated with these states.     

Some remarks on Yang-Baxter algebras

We point out the existence of an alternative algebraic structure in Yang-Baxter algebra with trigonometric R-matrix, which appears to be the generalization of the Yangian in Yang-Baxter algebras with rational R-matrix and which is described most naturally by q-commutators. Some properties are presented, in particular in the case of the well-known symmetric six-vertex model. Received: 13 February 1998 / Revised: 16 March 1998 / Accepted: 17 April 1998     

Some remarks on Lie superalgebras

We propose an approach to the theory of Lie superalgebras based on what we call a Lie algebra square root. Every Lie algebra square root has a Lie algebra as its square, but many different Lie algebra square roots may have the same square.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Some remarks on reflection positivity

The requirement of reflection positivity is investigated and its general applicability to different physical theories is pointed out. Its role is illustrated on an example from electrostatics and on several simple examples of field theories. Then, after presenting an abstract construction of the concept, the role of reflection positivity in classical lattice systems is discussed.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.For valuable discussions I should like to thank J.Löffenholz.     

Some remarks on dynamical semigroups

We deal with time evolution of a finite quantum system given by a dynamical semigroup Λ_t. For the semigroup we define and give some properties of the convex Λ_t-invariant subset of states “pathological” in some aspect evolving in strictly reversible manner independently of the stochastic surroundings of the system.     

Some remarks on orthonormal tetrad transport

Some general features of orthonormal tetrad transport along time-like curves are considered while emphasising the kinematical meaning of the ‘vierbein’ transport process. The issue is further illustrated by comparing Frenet-Serret transport with Fermi-Walker transport.     

Some remarks on producing Hopf algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to aq-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure ofsl _q (2) can be completely determined by requiring theq-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to defineSL _q (2) as a symmetry algebra of the bosonic and fermionic quantum planes.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.This work was supported in part by International Sciences Foundation (grant RFF-300) and by Russian Basic Research Foundation (grant 95-02-05679).I acknowledge helpful discussions with A. Isaev, P. Kulish, V. Lyakhovsky, O. Ogievetsky, P. Pyatov, and V. Tolstoy.     

Some remarks on generalised multipole expansions

Models for the mutual potential energy between two molecules proposed in the scientific literature often contain a sum of inverse-power interactions involving pairs of sites belonging to the two particles; in turn, these quantities are functions of a few scalar invariants involved in the problem at hand, and one is often interested in directly obtaining an explicit expression of the potential in terms of the latter; the extensively studied two-centre multipole expansion for the mutual electrostatic energy between two charge distributions is a well-known example of this procedure and of its restrictions. We consider here another, less widely known and possibly complementary, approach, proposed by ebek some years ago [J. ebek, Czech. J. Phys. B 38, 1185 (1988)]; the resulting formulae show that this procedure can become computationally favourable for sufficiently high molecular symmetry.     

Some remarks on the quantum double

We recall the abstract theory of Hopf algebra bicrossproducts and double cross products due to the author. We use it to develop some less-well known results about the quantum double as a twisting, as an extension and asq-Lorentz group.     

Some remarks on the particle multiplicities

Recent ALICE data for the multiplicity distributions in the central rapidity bins at LHC energies are compared with the results from two default versions of the PYTHIA 8 generator. We find that, contrary to the earlier versions of PYTHIA, the model overestimates the increase of average multiplicity with energy. Tuning two of the model parameters one obtains reasonable agreement with data. The dependence of the normalized moments of the distribution on the rapidity bin width and on energy is also qualitatively correct.     

Some remarks on the quantum geometry

In this paper we study the Schrödinger equation and the integrability problem at the quantum level from the geometric quantization point of view.     

Some remarks on defects and T-duality

The equations of motion for a conformal field theory in the presence of defect lines can be derived from an action that includes contributions from bibranes. For T-dual toroidal compactifications, they imply a direct relation between Poincaré line bundles and the action of T-duality on boundary conditions. We also exhibit a class of diagonal defects that induce a shift of the B-field. We finally study T-dualities for S¹$S^1$-fibrations in the example of the Wess–Zumino–Witten model on SU(2)$\mathrm{SU}(2)$ and lens spaces. Using standard techniques from D-branes, we derive from algebraic data in rational conformal field theories geometric structures familiar from Fourier–Mukai transformations.