Volume 33, Issue 1, January 1992
Index of content:

Lattice Virasoro algebras, unitarity, and the eight‐vertex model
View Description Hide DescriptionAn infinite set of lattice Virasoro algebras is constructed for the eight‐vertex model, with the logarithm of the corner transfer matrix as the central element. By imposing unitarity, the c=1 algebra is uniquely singled out.

Representations and traces of the Hecke algebras H _{ n }(q) of type A _{ n−1}
View Description Hide DescriptionThe notion of the connectivity class of minimal words in the algebra H _{ n }(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented. Guided by these results, the connection between the Ocneanu trace on H _{ n }(q) and Schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of H _{ n }(q). They appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions. Their evaluation involves the use of the Littlewood–Richardson rule, which is proved to apply to H _{ n }(q) just as it does to S _{ n }. Both representation matrices and characters are tabulated.

On the algebra of coupled SO(3) tensors
View Description Hide DescriptionThe commutation relation for coupled SO(3) tensor operators is constructed. This gives rise to a Lie algebra for which 9j symbols appear in the structure constants. In an example it is shown how an exceptional Lie subalgebra can be realized. The closure requirement for the exceptional Lie algebra does not seem to be related to nontrivial 9j symbols but rather to vanishings of relations involving 6j symbols.

On the quantum differential calculus and the quantum holomorphicity
View Description Hide DescriptionUnder some natural assumptions [less restrictive than in the paper by Wess and Zumino (preprint CERN‐TH‐5697/90, LAPP‐TH‐284/90)] differential calculi on the quantum plane are found and investigated. Complex structure, complex derivatives, and holomorphic functions on the quantum plane are defined. Generalized Cauchy–Riemann equations are given and solved.

Generating functions of multivariable generalized Bessel functions and Jacobi‐elliptic functions
View Description Hide DescriptionIt is pointed out that the Jacobi‐elliptic functions are the natural basis to get generating functions of the multivariable generalized Bessel functions. Analytical and numerical results are given of interest for applications.

Reflection coefficient beyond all orders for singular problems. III. Essential singularities on the real axis
View Description Hide DescriptionIn this paper uses the generalized Bremmer series to calculate the nontrivial behavior of the reflection coefficient for singular problems, which have isolated critical points and essential singularities on the real axis. The generalized Pokrovskii–Khalatnikov is not applicable in this case.

A representation of solutions of the Helmholtz equation with application to crystal Green’s functions
View Description Hide DescriptionFunctions that solve the Helmholtz equation in a bounded region are represented in the form of an integral on the unit sphere. The linear space needed for this, as well as the uniqueness of the representation, are investigated. A specific application of the technique to general spectral calculations for electrons in periodic crystals is included.

On the group aspects of Riemann–Hilbert transformations
View Description Hide DescriptionThe more general algebraic structures of two‐dimensional integrable systems are derived in a model independent manner by using the infinitesimal Riemann–Hilbert transformation. It is found that the resulting algebra decomposes into many families and, for each family, there is a Lie multialgebra structure. Examples of the chiral sigma models, four‐dimensional self‐dual Yang–Mills field, sine–Gordon, and Liouville equations and the BZ gravity are given with two special Lie products equipped therein.

On gauge‐invariant and phase‐invariant spinor analysis. II
View Description Hide DescriptionGranted customary definitions, the operations of juggling indices and covariant differentiation do not commute with one another in a Weyl space. The same noncommutativity obtains in the spinor calculus of Infeld and van der Waerden. Gauge‐invariant and phase‐invariant calculations therefore tend to be rather cumbersome. Here, a modification of the definition of covariant derivative leads immediately to a manifestly gauge‐invariant and phase‐invariant version of Weyl–Cartan space and of the two‐spinor calculus associated with it in which the metric tensor and the metric spinor are both covariant constant.

On the phase diagram for a class of continuous systems
View Description Hide DescriptionSome Gibbsian aspects of the scheme known as the sine‐Gordon transformation are discussed. Dobrushin–Lanford–Ruelle (DLR) equilibrium functional equations are introduced and analyzed. Certain uniqueness theorems are proved. Thermodynamicalproperties of the corresponding Gibbs measures are discussed. In particular, the Gibbs variational principle and the information gain principle are proved.

A note on the Lorentz transformations linking initial and final four‐vectors
View Description Hide DescriptionThe set of all (homogeneous, proper, orthochronous) Lorentz transformations that link two given forward‐timelike four‐vectors with equal norms, as well as the condition under which a unique Lorentz transformation is singled out, is completely determined, and presented in a form suitable for immediate reduction to Galilean transformations by letting c→∞,c being the speed of light in empty space. Analogies to the intuitively well‐understood Galilean transformation group are obvious. Thus, for instance, the ordinary velocity addition operator ‘‘+’’ involved in the determination of a Galilean transformation link becomes ⊕, the relativistic velocity addition operator involved in the determination of a Lorentz transformation link. The analogies shared by Galilean and Lorentz transformation links were overlooked by explorers since, as opposed to the associative–commutative binary operation + in the Euclidean three‐space R^{3}, the binary operation ⊕ in R^{3} _{ c } = {v∈R^{3}:‖v‖<c} is neither associative nor commutative.

The general solution of the three‐dimensional acoustic equation and of Maxwell’s equations in the infinite domain in terms of the asymptotic solution in the wave zone
View Description Hide DescriptionThe general solution of the three‐dimensional scalar wave equation (or acoustic equation) and of Maxwell’sequations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.

Low‐velocity scattering of vortices in a modified Abelian Higgs model
View Description Hide DescriptionThe moduli space metric for hyperbolic vortices is constructed, and their slow motion scattering is calculated in terms of geodesics in this space.

Schrödinger–Fuerth quantum diffusion theory: Generalized complex diffusion equation
View Description Hide DescriptionAccording to Schrödinger–Fuerth quantum diffusion theory the equations of quantum theory can be derived by applying the Fuerth transformation to the equations of diffusion theory. By introducing a complex flux whose real and imaginary parts are the x and y components of the flux, a complex form of the two‐dimensional diffusionequation is obtained and the equation is then generalized to three dimensions. Application of the Fuerth transformation to the generalized complex diffusionequation yields a quantum equation, which is shown to be equivalent to the Dirac equation.

The Barut–Girardello coherent states
View Description Hide DescriptionIt is shown that the completeness problem of the SL(2, R) coherent states proposed by Barut and Girardello leads to a moment problem, not a Mellin transform. This moment problem, which also appears in the theory of para‐Bose oscillators, has been solved following the Sharma–Mehta–Mukunda–Sudarshan solution of the problem. The matrix element of finite transformation in the coherent state basis is shown to satisfy a ‘‘quasiorthogonality’’ condition analogous to the orthogonality condition of the matrix element in the canonical basis. Finally, the Barut–Girardello ‘‘Hilbert space of entire analytic functions of growth (1,1)’’ turns out to be only a subspace of Bargmann’s well‐known Hilbert space of analytic functions. This subspace, which has been called ‘‘the reduced Bargmann space’’ in a previous paper, is an invariant subspace of SL(2,R). With this identification the generators of the group in this realization turn out to be the well‐known boson operators of Holman and Biedenharn.

Quantization of a four‐dimensional generally covariant conformal model
View Description Hide DescriptionA four‐dimensional generally covariant action, which depends on the complex structures of the Hermitian space‐times, is studied. It is also invariant under a gauge group and independent Weyl transformations of the vierbein. These symmetries render it formally renormalizable. It is quantized using the (Dirac) canonical and the BRS quantization procedures. An approximative expansion of the vierbein around background Hermitian manifolds permits explicit calculations. Hermitian manifolds, satisfying trivial reality conditions (z ^{0} = z̄ ^{0}, z ^{0̃} = z̄ ^{0̃}, z ^{1̃} = z̄ ^{1}), cannot be such background manifolds, because the first approximation of the BRS charge is not nilpotent for any number of gauge fields. On the other hand, the expansion may be performed around the Kerr–Newman manifold only if its mass, charge, and angular momentum satisfy a precise relation.

Phase‐space representations of general statistical physical theories
View Description Hide DescriptionIt is shown that Hilbert‐space quantum mechanics and many other statistical theories can be represented on some phase space, in the sense that states can be identified with probability measures and observables can be described by functions. In the general context of statistical dualities, informationally complete observables are introduced and a theorem on their existence is proven. The correspondence between these observables and the injective affine mappings from the states into the probability measures on phase space, i.e., the phase‐space representations, is pointed out. In particular, a description of all observables by functions is presented, such that all expectation values can, in arbitrarily good physical approximation, be calculated as integrals. Moreover, some new aspects of the particular case of those phase‐space representations of quantum mechanics that are related to certain joint position‐momentum observables are discussed.

Limiting Gibbs states and dynamics for thermal photons
View Description Hide DescriptionFor thermal photons, the limiting Gibbs state is investigated in the framework of algebraic quantum theory. For the one‐particle Hamiltonian, the square root of the local Laplacians with Dirichlet boundary conditions is used. The local regions are only demanded to have the segment property and the thermodynamic limit is performed along a nearly arbitrary family of such local regions. Also, the limiting dynamics as a noncontinuous group of Bogoliubov transformations on an extension of the quasilocal Weyl algebra and as a W* dynamics on the GNS–von Neumann algebra is derived. After having performed the limits, the photon algebra is restricted to the physical (transversal) photons. The crucial mathematical method in verifying these results are comparisons of the semigroups of the square root of the Laplacians in different regions of the ν‐dimensional Euclidean space by means of the ordering of positivity preserving operators.

On supersymmetries in nonrelativistic quantum mechanics
View Description Hide DescriptionOne‐dimensional nonrelativistic systems are studied when time‐independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical invariance Lie superalgebras are pointed out. The study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding Schrödinger equation. Three tables collect the even, odd, and total supersymmetries as well as the invariance (super)algebras.

Geometric superalgebra and the Dirac equation
View Description Hide DescriptionA unified mathematical approach to spinors and multivectors or superalgebra is constructed in a form useful to study the mathematical description of matter and its interaction fields. The formalism then encompasses both points of view: m u l t i v e c t o r s for the description of (space‐time) geometry and the description of the integer spin, interaction fields, and s p i n o r representations suitable for the description of half odd integer, matter fields. An application is made to study the change of the Dirac equation under the spinors to multivectors (to scalars) mapping. The physical and geometric content of the multivector solutions of the Dirac–Hestenes equation is clearly shown.