Some remarks on the two-body-problem in geometrodynamics

The interest in the two body problem in geometrodynamics and its present status are discussed. Special consideration is given to time-symmetric initial data. An equation relating the “bare mass energy,” the potential energy and the “vibrational energy” at the initial moment is proposed. The significance of coordinate conditions in dynamical equations is discussed next. Conditions implying maximal slicings of spacetime and harmonic coordinates in those slices are considered in detail, as they seem to be well suited for numerical calculations and for interpretation of the results of such calculations.     

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 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 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 the Gribov ambiguity

The set of all connections of a principal bundle over the 4-sphere with compact nonabelian Lie group under the action of the group of gauge transformations is studied. It is shown that no continuous choice of exactly one connection on each orbit can be made. Thus the Gribov ambiguity for the Coloumb gauge will occur in all other gauges. No gauge fixing is possible.     

Some remarks on the paper of Callias

This paper discusses the topological setting of the preceding paper by Callias. In particular, an alternate way of deriving his results is outlined.     

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 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 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 the time-dependent Thomas-Fermi theory

We investigate the Thomas Fermi limit (?=0) of the density operator in phase space for a system of noninteracting fermions evolving from the ground state in a time dependent potential. The semiclassical calculations for model situations in one spatial dimension are compared with the solution of the time dependent Schrödinger equation. The role of time dependent invariants and the relation to the hydrodynamical formulation of quantum mechanics is pointed out.     

Some remarks on the maximum adsorption energy

The Green function for a semi-infinite unsymmetric sp-band monatomic linear chain is derived. The electronic densities of states for the first few layers as well as the energy and the decaying constant of the Shockley surface state have been calculated. We also observe that the Shockley surface state can exist in both the direct and the indirect energy gaps.     

Some remarks on the finiteness of soliton mass

Some subtleties regarding regularizations in computing the soliton energy of degenerate systems are discussed.     

Some remarks on the algebra of Eddington'sE numbers

This paper reviews the algebra of Eddington'sE numbers and identifies those points where Eddington anticipated results of current interest. He discovered the Majorana spinors, and was responsible for the standard ⁵ notation as well as the notion of chirality. Furthermore, Eddington defined Clifford algebras in eight and nine dimensions which are now appearing in grand unified gauge and supersymmetric theories. A point which Eddington cleared up, yet is still misunderstood, is that the Dirac algebra corresponds to afive-dimensional base space.     

Some remarks on the second law of thermodynamics

We show that the existence of a temperature scale implies the existence of the absolute temperature and the entropy. The consequences for the structure of thermodynamics are discussed.     

Some remarks on the geometry of superspace supergravity

In this note, we first give a quick presentation of the supergeometry underlying supergravity theories, using an intrinsic differential geometric language. For this, we adopt the point of view of Cartan geometries, and rely as well on the work of John Lott, who has found a unified geometrical interpretation of the torsion constraints for many supergravity theories, based on the use of H-structures. In this framework, the constraints amount to requiring first-order integrability of H-structures, for a specific supergroup H.The supergroup H used by Lott is not the usual diagonal representation of the Lorentz group on superspace, but an extension of the latter. This extension appears to be natural and it can be related to the super-Poincaré group. We also observe that the constraints arising from the requirement of first-order integrability have basically the same form, in any spacetime dimension.Looking at supergravity from an affine viewpoint (i.e. as a gauge theory for the super-Poincaré group), we show that requiring first-order integrability amounts to requiring the equivalence, up to gauge transformations, between infinitesimal gauge supertranslations acting on the supervielbein and infinitesimal superdiffeomorphisms acting on the supervielbein.The latter action is performed through a covariant Lie derivative, whose expression involves naturally the supertorsion tensor. We use this expression to show that the term added to the spin connection, in the supercovariant derivative of d=11 supergravity, has a natural superspace origin. In particular, the 4-form field strength is related to a specific component of the supertorsion tensor.We conclude by some general remarks concerning Killing spinors in geometry and supergravity, discussing their possible interpretations, as Killing vector fields on a specific supermanifold on one hand, and as parallel spinors for an appropriate connection on the other hand. We show that this last interpretation is very natural from the point of view of Klein and Cartan geometries.