A remark on the connection between affine Lie algebras and soliton equations

The Lax equations of Drinfeld-Sokolov are derived in the framework of the Fock representation of Clifford algebras. The derivation is based on the bilinear identities for -functions.     

A remark on coefficients of connection with two gauge fields

Recently a geometrical gravitational theory based on the coefficients of connection with two gauge fields was proposed. The objective of the present paper is to show that the geometry based on the coefficients of connection with two gauge fields can be obtained by the successive operations of conformal and projective transformations on Riemannian metric.     

A remark on the structure of non-stoichiometric copper-manganese ferrite

Czechoslovak Journal of Physics -     

A short remark on the band structure of free-edge platonic crystals

A corrected version of the multipole solution for a thin plate perforated in a doubly periodic fashion is presented. It is assumed that free-edge boundary conditions are imposed at the edge of each cylindrical inclusion. The solution procedure given here exploits a well-known property of Bessel functions to obtain the solution directly, in contrast to the existing incorrect derivation. A series of band diagrams and an updated table of values are given for the resulting system (correcting known publications on the topic), which shows a spectral band at low frequency for the free-edge problem. This is in contrast to clamped-edge boundary conditions for the same biharmonic plate problem, which features a low-frequency band gap. The numerical solution procedure outlined here is also simplified relative to earlier publications, and exploits the spectral properties of complex-valued matrices to determine the band structure of the structured plate.     

Nuclear structure on a Grassmann manifold

Products of particlelike representations of the homogeneous Lorentz group are used to construct the degrees of spin angular momentum of a composite system of protons and neutrons. If a canonical labeling system is adopted for each state, a shell structure emerges. Furthermore the use of the Dirac ring ensures that the spin is characterized by half-angles in accord with the neutron-rotation experiment. It is possible to construct a Clebsch-Gordan decomposition to reduce a state of complex angular momentum into simpler states which can be identified with and particles, multipole operators, etc. Finally, ground-state energy levels are calculated for all the even-even nuclei by using a differentiable manifold that is spin-graded and gauge-invariant by construction. It is shown that this manifold is Grassmann.     

A remark on the boson-fermion correspondence

We introduce the space of skew-symmetric functions depending on an infinite number of variables and give a simple interpretation of the boson-fermion correspondence.     

On the twistor space of a quaternionic contact manifold

In this note, we prove that the CR manifold induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.     

A remark on the Dirac equation

We give a simple deductive derivation of the Dirac equation for a free particle. Our construction provides a clear distinction between what are the physical contents and what are purely mathematical expedients in the formalism of quantum mechanics (QM).     

A remark on the glueball-gluon coupling

The QCD trace anomaly motivates the consideration of a low energy glueball-gluon coupling . We point out that this should constitute the leading kinetic term for the gluons at low energies. Anti-screening of the gluons by the glueball then induces a classical Coulomb potential of color charges which increases at large distances and motivates the inclusion of a corresponding term in the inter-quark potential. Received: 16 March 1999 / Published online: 20 May 1999     

A remark on the cluster theorem

An improved version of the cluster theorem of relativistic quantum field theory with the correct exponential decay is derived by function theoretical methods.Heisenberg fellowLaboratoire propre du Centre National de la Recherche Scientifique     

A remark on the generalized Goldberg-Sachs theorem

Despite the elegant formulations of Kundt and Thompson[1], and Robinson and Schild[2], it is not obvious how general the generalized GoldbergSachs theorem[3] really is. A spacetime satisfying Einstein's equations with a null fluid source, for example, can elude the generalized theorem if, and only if, the null direction of the fluid is a fourfold repeated principal null direction of the Weyl tensor. An example of such a spacetime is presented.     

A remark on the concavity of entropy

We investigate to what extent theorems about quantum mechanical or classical entropy can be generalized to functionals of the type ρ→Tr f(ρ), or ψ→∫f(ψ)dμ, respectively, wheref is an arbitrary concave function.     

A remark on null infinity

It is shown that to each choice of conformal factor for the conformal metric on future null Infinity (₊) there is associated an equivalence class of slicings of ₊. Each slicing in turn gives rise to a family of null hypersurfaces in the physical space-time.Supported by NSF Grant No. GP-35773X1.     

A remark on SU4

Several of the interesting new relations in elementary particle physics discovered from the hypothesis of SU₆ symmetry of strong interactions, are shown to follow from the less restrictive assumption of SU₄ symmetry alone. Since isospin invariance is a much more closely obeyed symmetry than SU₃, these relations may be the best tests for the validity of combining internal symmetries with spin.     

On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

In this work we study in detail the connection between the solutions to the Dirac and Weyl equations and the associated electromagnetic four-potentials.First,it is proven that all solutions to the Weyl equation are degenerate,in the sense that they correspond to an infinite number of electromagnetic four-potentials.As far as the solutions to the Dirac equation are concerned,it is shown that they can be classified into two classes.The elements of the first class correspond to one and only one four-potential,and are called non-degenerate Dirac solutions.On the other hand,the elements of the second class correspond to an infinite number of four-potentials,and are called degenerate Dirac solutions.Further,it is proven that at least two of these fourpotentials are gauge-inequivalent,corresponding to different electromagnetic fields.In order to illustrate this particularly important result we have studied the degenerate solutions to the forcefree Dirac equation and shown that they correspond to massless particles.We have also provided explicit examples regarding solutions to the force-free Weyl equation and the Weyl equation for a constant magnetic field.In all cases we have calculated the infinite number of different electromagnetic fields corresponding to these solutions.Finally,we have discussed potential applications of our results in cosmology,materials science and nanoelectronics.     

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

A remark on transition probability

Uhlmann's transition probability P(ψ, φ) of two normal states of a von Neumann algebra M, which is the supremum of |(Ψ, Φ)|² for all possible choices of representative vectors Ψ and Φ of ψ and φ, is shown to be the infimum of (∫d(μ_{ψ, e})^1/2)² for the induced measures μ_{ω, e}(B)=ω(e(B)) (B: Borel set in ℝ, ω=ψ, φ) for all possible projection-valued measures e belonging to M.     

A remark on quantum gravity

We discuss the structure of Dyson-Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub-Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov-Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson-Schwinger equations.