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.     

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.     

A remark on quasi-triangular Lie algebras

Quasi-triangular quantum Lie algebras are algebras of quantum Lie derivatives. We show how to associate a quasi-triangular quantum Lie algebra to any quantum group defined by exchange relations. This provides a systematical way for constructing bi-covariant differential calculus on exchange quantum groups.     

A remark on neutron slowing down

Using the integral representation of Dirac-function the densityW _n(U) of the neutron lethargy probability distribution after then ^th collision with nuclei of a moderator has been found. The assumption has been made that the collisions are elastic and lethargy increments in individual collisions are independent. This procedure leads to expression ofW _n(U) by an improper integral which is in turn converted to a finite sum. Further the asymptotic behaviour ofW _n(U) for great number of collisionsn has been determined.Ronov pod Radhotm, Czechoslovakia.     

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.     

A remark on off-shell phase-shift analysis

We consider the ππ → ππ amplitude that is extracted from πN → ππN experiments, and show that off-mass-shell unitarity can only be used to resolve phase-shift ambiguities if no inelasticity is present. If there is inelasticity in the two-pion subsystem, the off-mass-shell structure does not reduce the on-mass-shell continuum ambiguities in any way. Similar remarks apply to electroproduction, considered as off-shell photoproduction.     

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 further remark on uncertainty relations

As pointed out earlier, the commutation relation 1/i[A, B]=C for quantum mechanical observablesA, B andC by itself does not imply the uncertainty relation ΔA·ΔB≧1/2¦¯C¦. In this note,A, B andC are assumed to be generators of a unitary representation of a suitable Lie group, such that is implied by the group structure. This assumption is then sufficient to yield.     

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 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     

An alternative analytic treatment of the Gutzwiller ansatz

An alternative analytic approach is formulated for treating Gutzwiller's variational ansatz. In contrast to the diagrammatic treatment recently published by Metzner and Vollhardt, our approach is purely algebraic and mainly based on expansions in terms of cluster integrals. The present theory is carried out for arbitrary dimension and band filling, but is applied here only to the Hubbard interaction term. In one dimension we find exact agreement between our results and those obtained by Metzner and Vollhardt.     

A remark on kinks and time machines

We describe an elementary proof that a manifold with the topology of the Politzer time machine does not admit a nonsingular, asymptotically flat Lorentz metric.     

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 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.