Volume 25, Issue 6, June 1984
Index of content:

Exact solutions of unitarily invariant matrix models in zero dimensions
View Description Hide DescriptionExplicit constructions of Green functions for U(N) invariant‐matrix Φ^{4}theories are given in zero space‐time dimensions. The results are expressed in terms of a new class of orthogonal polynomials, which are orthogonal on the interval [−∞,∞] with respect to the weight function exp{−m ^{2}φ^{2}−λφ^{4}}. An explicit construction of the coefficients of these generalized Hermite polynomials is presented. We briefly discuss the ‘‘thermodynamic’’ limit N→∞ of the corresponding one‐dimensional statistical system of N classical particles with logarithmic pair interactions and subjected to an external anharmonic potential.

Dynamical group of microscopic collective states. III. Coherent state representations in d dimensions
View Description Hide DescriptionThe present series of papers deals with various realizations of the dynamical group Sp_{ c }(2d,R) of microscopic collective states for an Anucleon system in d (=1,2, or 3) dimensions, when these collective states are assumed to be invariant under the orthogonal group O(n) associated with the n=A−1 relative Jacobi vectors. In this paper, we further study the Barut–Girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to Sp_{ c }(2d,R) a class of Sp(2,R) coherent states introduced by Barut and Girardello. For such purpose, our starting point is another coherent state representation, namely the Perelomov one, previously considered by Kramer. We also propose a third, new class of coherent states leading to Holstein–Primakoff representation in a straightforward way. We review various properties of these three classes of coherent states, such as their reproducing kernel and measure explicit forms, their generating function properties, and the representations they lead to for both the collective states and their dynamical group.

Mixed supertableaux of the superunitary groups. I. SU(n‖1)
View Description Hide DescriptionYoung supertableaux of the graded unitary groups SU(n‖1) are classified and interpreted in terms of representations of the group. A distinction between typical and atypical representations arises naturally, and we propose a new type of generalized atypical supertableaux for non‐fully‐reducible representations.

Differentiable vectors and sharp momentum states of helicity representations of the Poincaré group
View Description Hide DescriptionThis paper discusses some mathematical difficulties in handling sharp momentum eigenvectors for a massless helicity representation of the Poincaré group, related to the non‐nuclearity of the space of differentiable vectors, and to the existence of singularities in the Lorentz group generators. A simple characterization of the nuclear space of differentiable vectors of the extension of the representation to a representation of the conformal group is given in terms of functions on the space R^{4}− {0}. Using a fibration of this space over the forward light cone (in momentum space), the singularity in the generators is shown to be related to the fact that the standard presentation of the helicity representations should be reformulated in terms of nontrivial sections over the light cone. The problem is partly identical with the one encountered in the study of an electron in a magnetic monopole field, the generator singularity taking the place of the Dirac string.

Gradient property of bifurcation equation for systems with rotational symmetry
View Description Hide DescriptionWe show that any polynomial equation covariant with respect to any representation of SO(2) is a gradient equation. The same holds for the fundamental representation of SO(3). For the other representations of SO(3) we give a simple necessary and sufficient condition an equation has to satisfy in order to be a gradient one. As a side result, we obtain a formula for the decomposition of the symmetrized power of an irreducible representation of SO(3) in a sum of irreducible representations. We apply our results to symmetric bifurcation theory.

Construction of extremal vectors for Verma submodules of Verma modules
View Description Hide DescriptionIn this paper we discuss certain aspects of the theory of Verma submodules of Verma modules of simple Lie algebra. We describe a simple algorithm for the complete determination of the highest weights which are associated with the Verma submodules. Moreover, we give the proof for a method which permits an explicit determination of the extremal vectors which define the Verma submodules. For the case of the simple Lie algebraA _{ l } the Verma modules are obtained in explicit form.

Lattice degeneracies of fermions
View Description Hide DescriptionWe present a detailed description of the minimal degeneracies of geometric (Kähler) fermions on all the lattices of maximal symmetries in n=1,...,4 dimensions. We also determine the isolated orbits of the maximal symmetry groups, which are related to the minimal numbers of ‘‘naive’’ fermions on the reciprocals of these lattices. It turns out that on the self‐reciprocal lattices the minimal numbers of naive fermions are equal to the minimal numbers of degrees of freedom of geometric fermions. The description we give relies on the close connection of the maximal lattice symmetry groups with (affine) Weyl groups of root systems of (semi‐) simple Lie algebras.

Orbit spaces of low‐dimensional representations of simple compact connected Lie groups and extrema of a group‐invariant scalar potential
View Description Hide DescriptionOrbit spaces of low‐dimensional representations of classical and exceptional Lie groups are constructed and tabulated. We observe that the orbit spaces of some single irreducible representations (adjoints, second‐rank symmetric and antisymmetric tensors of classical Lie groups, and the defining representations of F _{4} and E _{6}) are warped polyhedrons with (locally) more protrudent boundaries corresponding to higher level little groups. The orbit spaces of two irreducible representations have different shapes. We observe that dimension and concavity of different strata are not sharply distinguished. We explain that the observed orbit space structure implies that a physical system tends to retain as much symmetry as possible in a symmetry breaking process. In Appendix A, we interpret our method of minimization in the orbit space in terms of conventional language and show how to find all the extrema (in the representation space) of a general group‐invariant scalar potential monotonic in the orbit space. We also present the criterion to tell whether an extremum is a local minimum or maximum or an inflection point. In Appendix B, we show that the minimization problem can always be reduced to a two‐dimensional one in the case of the most general Higgs potential for a single irreducible representation and to a three‐dimensional one in the case of an even degree Higgs potential for two irreducible representations. We explain that the absolute minimum condition prompts the boundary conditions enough to determine the representation vector.

On the Hankel transform of a generalized Laguerre polynomial and on the convolution involving special Bessel functions
View Description Hide DescriptionIn this paper we calculate two of three simple integrals which are present neither in Bateman nor in Gradshtein and Ryzhik. These integrals are useful to treat some specific physics problems.

Explicit expressions of the nth gradient of 1/r
View Description Hide DescriptionExplicit expressions of the nth gradient tensor and partial derivatives of the function 1/r are given, in the absolute and in the local bases associated, respectively, with the Cartesian and the spherical coordinates of space. They may be used in various multipolar developments of r ^{−} ^{1} interaction energies.

A recursive generation of local higher‐order sine–Gordon equations and their Bäcklund transformation
View Description Hide DescriptionA new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse. It is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine–Gordon and Liouville equations in laboratory coordinates. A generalization of the well‐known Bäcklund transformation and nonlinear superposition formula for the sine–Gordon equation is also obtained.

Quantum solitons of the nonlinear Schrödinger field as Galilean particles
View Description Hide DescriptionThe quantum nonlinear Schrödinger field with attractive coupling is considered through the quantum inverse scattering method. The algebra of scattering data operators is formulated in terms of an infinite family of independent boson fields. It is shown that the unitary representation induced by the Galilean invariance of the model is equivalent to a direct product of simple representations. As a consequence, the quantum solitons turn out to be associated with representations describing quantum elementary Galilean particles. The characterization of quantum solitons as stable asymptotic fragments arising in the scattering of the fundamental bosons of the model is also analyzed.

Fermi pseudopotential in higher dimensions
View Description Hide DescriptionThe Fermi pseudopotential is generalized from three to five dimensions, and the case of an infinite, uniform, equidistant, linear chain of such pseudopotentials is studied in detail. Similar to the three‐dimensional case, zero‐width resonances are also present in five dimensions. While this generalization is natural and can be carried through formally when the strength is negative, there are basic changes in the underlying structure. These results in five dimensions also apply in four dimensions.

General nonlinear realization of chiral SU_{2}×SU_{2} symmetry in curved isospin space
View Description Hide DescriptionThe geometric method for constructing the nonlinear chiralSU_{2}×SU_{2} invariant Yang–Mills Lagrangian in curved isospin space given by Meetz for a particular choice of coordinates is generalized to an arbitrary system of coordinates in curved isospin space. The resulting Lagrangian coincides with the general nonlinear chiralSU_{2}×SU_{2} invariant Yang–Mills Lagrangian constructed previously using the matrix method.

Explicit evaluation of a path integral with memory kernel
View Description Hide DescriptionAn explicit expression is presented for the path integral of an harmonic oscillator interacting with itself in the past. The time dependence of the memory kernel is assumed to be exponential. The relation with the Feynman trial action for the variational calculation of the polaronground state energy is discussed.

The occupation statistics for indistinguishable dumbbells on a rectangular lattice space. I
View Description Hide DescriptionThe method of Hock and McQuistan used recently to solve the occupation statistics for indistinguishable dumbbells (or dimers) on a 2×2×N lattice is extended further to obtain, for the L×M×N lattice, general expressions for the normalization, expectation, and dispersion of the statistics, and their limit as N becomes very large. In particular, an explicit expression of the partition function in the thermodynamic limit Ξ(x) is obtained for any value of the absolute activity x of dimers. The developed mathematical formalism is then applied to planar lattices, 1×M×N, with M=1, 2, 3, and 4. The known results for M=1 and 2 are recovered, and some new ones are obtained. The recurrence relation for the number A(q,N) of arrangements of q dumbbells on a 1×M×N lattice which has 3 and 5 terms when M=1 and 2, respectively, is found to have 15 and 65 terms for M=3 and 4. Analysis and extrapolation of the results enable one to predict the expectation 〈θ〉_{1M N } on a planar 1×M×N lattice to be 63.4%, in the limit as both M and N become infinite. We also find an upper bound on the quantity M N[〈θ^{2}〉−(〈θ〉)^{2}] in the limit as both M and N become infinite. In the thermodynamic limit (M&N→∞) the partition function Ξ(1), for the absolute activity x=1, is found to be equal to 1.95. By limiting the number M of rows of infinite extent (N→∞) to just 4, we find that the error in determining Ξ(1) for the infinite two‐dimensional lattice is just 4.5%. In this paper Ξ(x) is obtained for any value of the absolute activity x for M=1 and 2. A more thorough study of Ξ(x), and its fast convergence with increasing values of M, and applications will be presented in a forthcoming article.

Weak solutions of a quasistatic model of plasmas
View Description Hide DescriptionWe study an analytical structure of a quasistatic model of magnetically confined plasmas. Applying the fixed point theorem, we construct global‐in‐time weak solutions.

First‐order equivalent Lagrangians and conservation laws
View Description Hide DescriptionWe present a theorem for (first‐order) Lagrangian theories which associates several conserved quantities to o n e (s‐equivalence) symmetry transformation.

The scattering of an obliquely incident surface wave by a submerged fixed vertical plate
View Description Hide DescriptionThe problem of scattering of surface waves obliquely incident on a submerged fixed vertical plate is solved approximately for a small angle of incidence by reducing it to the solution of an integral equation. The correction to the reflection and transmission coefficients over their normal incidence values for a small angle of incidence are obtained. For different values of the incident angle these coefficients are evaluated numerically, taking particular values of the wave number and the depth of the plate, and represented graphically.

Variational principle for electromagnetic problems in a linear, static, inhomogeneous anisotropic medium
View Description Hide DescriptionA variational principle δ∫∫L d V d t=0 is formulated for electromagnetic problems in the time domain for a medium which is linear, static, inhomogeneous, and anisotropic. The principle involves all four electromagnetic field vectors (E,H,D,B) and no ‘‘adjoint‐type’’ fields. Both the two ‘‘curl’’ Maxwell equations as well as the two constitutive equations emerge as the four associated Euler–Lagrange equations. The problem of possible variations in the field functions at the temporal boundaries is solved by using the constitutive equations as a constraint at one of the temporal boundaries.