Volume 24, Issue 7, July 1983
Index of content:

Casimir invariants, characteristic identities, and Young diagrams for color algebras and superalgebras
View Description Hide DescriptionThe generalized commutation relations satisfied by generators of the general linear, special linear, and orthosymplectic color (super) algebras are presented in matrix form. Tensor operators, including Casimir invariants, are constructed in the enveloping algebra. For the general, special linear and orthosymplectic cases, eigenvalues of the quadratic and higher Casimir invariants are given in terms of the highest‐weight vector. Correspondingly, characteristic polynomial identities, satisfied by the matrix of generators, are obtained in factorized form. Classes of finite‐dimensional representations are identified using Young diagram techniques, and dimension, branching, and product rules for these are given. Finally, the connection between color (super) algebras and generalized particle statistics is elucidated.

Character generators for unitary and symplectic groups
View Description Hide DescriptionCombinatorial algorithms for computing the character generators of U(n), SU(n), and Sp(2n, C) are described. These algorithms produce relatively compact, nested expressions for the character generators. Moreover, the terms appearing in these expressions all have positive coefficients. This feature is not shared by the expression for the character generator which uses the Weyl character formula.

Missing label operators in the reduction O(p)↓O(p−2)×O(2)
View Description Hide DescriptionI consider the ‘‘missing label’’ problem for basis vectors of an O(p) representation corresponding to a group reduction chain with links O(p)↓O(p−2)×O(2). A chain with these links is required if the basis vectors are to be of definite weight. I obtain two different sets of missing label operators, which together with the Casimir operators of group and subgroups from a complete set of labeling operators whose eigenvectors provide a canonical basis in the O(p) representation space. The problem is solved for both the even‐ and odd‐dimensional orthogonal groups.

Construction of N‐dimensional indecomposable representations for scale and special conformal transformations
View Description Hide DescriptionThe theories of N‐dimensional indecomposable representations for the group of semidirect products of D (dilatation group) and K (special conformal group) are investigated through the studies on space‐time inversions.

Casimir invariants, characteristic identities, and tensor operators for ‘‘strange’’ superalgebras
View Description Hide DescriptionWe define a class of (super) subalgebras of gl(m/n) realized as the set of fixed points of a (graded) endomorphism of gl(m/n). This class includes the superalgebras gp(m) and gq(m) [related to the so‐called ‘‘strange’’ simple superalgebras p(m) and q(m)], as well as osp(m/n). General covariant, contravariant, and mixed tensor operators are defined for this class in terms of appropriate module homomorphisms. Traces of certain tensors give the usual sequence of Casimir invariants. For gp(m), these are shown to vanish identically, while for gq(m), eigenvalues of the quadratic and cubic Casimir invariants are derived in terms of highest weights and a polynomial characteristic identity is exhibited.

Inequalities and local uncertainty principles
View Description Hide DescriptionInequalities for Fourier transforms are developed which describe local uncertainty principles in the sense that if the uncertainty of momentum is small, then so is the probability of being localized at any point. They give estimates for essentially all states and lead to lower bounds for Hamiltonians.

Superposition rules for nonlinear coupled first‐order differential equations
View Description Hide DescriptionPairs of coupled nonlinear differential equations of the polynomial type have been studied. No higher powers than quadratic were considered. Lie’s theorem provides a superposition rule exists if certain operators generate a finite‐dimensional Lie algebra. The Lie algebras possible were divided into twenty categories. For each case the general form of the coupled differential equations is obtained. The coupled differential equations are then separated where possible. Solutions are obtained expressed in terms of a finite number of particular solutions.

Nonlinear evolution equations and nonabelian prolongations
View Description Hide DescriptionA systematic analysis of the class of nonlinear evolution equations u _{ t } +u _{ x x x } +φ(u, u _{ x })=0 is carried out within the Estabrook–Wahlquist prolongation scheme.

Characterization of canonical Bose–Fermi systems by ‘‘anti‐Hermitian’’ symplectic forms
View Description Hide DescriptionA complexification of graded manifold theory is given, following Kostant’s procedure, but with a reality concept defined by ‘‘classical’’ correspondent to Hermitian conjugation in quantum mechanics. Presented herein are definitions of graded manifolds with ‘‘Hermite’’ coordinates and of ‘‘Hermiticity’’ on graded differential forms and graded vector fields, all in the coordinate independent way, and characterization of ‘‘classical’’ Bose–Fermi systems by graded symplectic forms ω which are, here, ‘‘anti‐Hermitian’’ nonsingular closed 2‐forms of z _{2} grading 0. Also given are Frobenius’ theorem on the graded manifold with ‘‘Hermiticity,’’ and Darboux’s theorem, ω=∑_{ k } d p _{ k }Λd q _{ k }+i∑_{ j } +i∑_{ j }(ε_{ j }/2)d s _{ j }Λd s _{ j }, where all coordinates are ‘‘Hermite,’’ p _{ k } ^{†} =p _{ k }, q _{ k } ^{†}=q _{ k }, s _{ j } ^{†} =s _{ j }. Naive quantization procedures fit in with these systems.

Cartan structures on Galilean manifolds: The chronoprojective geometry
View Description Hide DescriptionA new geometry is constructed over Galilean manifolds expressing the compatibility requirement between the conformal equivalence notion of two Galilean structures and the projective equivalence notion of two affine connections. It is shown that it is the very geometry of the Newtonian cosmology (chronoprojective flatness is equivalent to isotropy of Newtonian cosmological models); moreover, it also explains various ‘‘accidental’’ symmetries in classical mechanics.

Time‐dependent vector constants of motion, symmetries, and orbit equations for the dynamical system r̈=î_{ r }{[Ü(t)/U(t)]r −[μ_{0}/U(t)]r ^{−} ^{2}}
View Description Hide DescriptionThe most general t i m e‐d e p e n d e n t, central force, classical particle dynamical systems (in n‐dimensional Euclidean space,n=2 or 3) of the form (a) r̈=î_{ r } F(r, t), (r ^{2}≡r ⋅ r, r=î_{ k } x ^{ k }, k=1,...,n), which admit vector constants of motion of the form (b) I=U(r, t)(L×v)+Z(r, t)(L×r) +W(r, t)r (L≡r×v, v≡ṙ) are obtained. It is found that the only class of such dynamical systems is (c) r̈=î_{ r }(ÜU ^{−} ^{1} r−μ_{0} U ^{−} ^{1} r ^{−} ^{2}), for which the concomitant vector constant of motion (b) takes the form (d) I=U(L×v)−U̇(L×r)+μ_{0} r ^{−} ^{1} r, where in (c) and (d) U=U(t) is arbitrary (≠0). The dynamical system (c) includes both the time‐dependent harmonic oscillator and a time‐dependent Kepler system. Based upon infinitesimal v e l o c i t y‐i n d e p e n d e n t mappings the complete symmetry group for the dynamical system (c) is obtained. This complete group of [2+n(n−1)/2] parameters contains a complete Noether symmetry subgroup of [1+n(n−1)/2] parameters. In addition to the n(n−1)/2 angular momenta, there is an energy‐like constant of motion also associated with the Noether symmetries. By means of the vector constant of motion (d), the orbit equations of the dynamical system (c) are obtained. A one‐dimensional procedure for obtaining constants of motion developed by Lewis and Leach is applied to the effective one‐dimensional system concomitant to (c). Relations between constants of motion so obtained and those mentioned above are determined.

Deformation of Hamiltonian dynamics and constants of motion in dissipative systems
View Description Hide DescriptionA necessary condition for the existence of arcs of vector fields with constants of motion is found. The result is applied to arcs obtained by deformation of Hamiltonian dynamics and illustrated in the Van der Pol and Lorenz models.

Irreversible quantum dynamics and the Hilbert space structure of quantum kinematics
View Description Hide DescriptionGeneral dynamics compatible with the Hilbert space structure of quantum kinematics are considered. The general form of dynamics which preserve the set of closed linear submanifolds (i.e., properties) is deduced. Since the orthogonality relation is not necessarily preserved, the result generalizes Wigner’s theorem and provides a model of some irreversible phenomena like spin relaxation, damped oscillator, etc. Connections with quantum logic and with statistical mechanics are presented.

The Fierz identities—A passage between spinors and tensors
View Description Hide DescriptionAll possible Fierz identities among 16 elements in the Diracalgebra have been obtained. These relations impose various constraint conditions on Hermitian and non‐Hermitian bilinear currents. Independent relations are sought out from the highly redundant system of constraint conditions. Relations between derivatives of a spinor and tensor currents are also obtained.

Superoperator perturbation theory for propagators
View Description Hide DescriptionA well‐defined superoperator perturbation theory for propagators is developed, based on equivalence classes of operators, which avoids the ambiguity of approaches based on a degenerate inner product. The Van Vleck formalism provides a natural tool for such a theory when self‐consistent propagator approximations are chosen as zeroth‐order approximations.

Scattering theory for the dilation group. I. Simple quantum mechanical scattering
View Description Hide DescriptionA theory of scattering based on the dilation group is developed for quantum mechanical one‐particle systems. A scattering operator is defined that agrees with the usual scattering operator, whenever the usual wave operators exist and are asymptotically complete.

Non‐Grassmann quantization of the massive Thirring model
View Description Hide DescriptionA direct quantization of the c‐number (semi) classical massive Thirring model in the inverse scattering formalism leads to the Bose massive Thirring model, which is equivalent to the conventional Fermi one, both having identical S‐matrices and bound‐state spectra.

An exactly solvable quantum field theory in three dimensions
View Description Hide DescriptionWe construct a quantum field theoretic model in three space dimensions and show that its spectrum can be exactly calculated. We also show that all the eigenvectors of the Hamiltonian can be obtained by a recursive procedure.

Dirac gravitational magnetic monopoles do not exist
View Description Hide DescriptionWe show that the gravitational analog of the Diracmagnetic monopole does not exist for a gauge theory of gravitation with either a T _{4} or an SL(2, C) × T _{4} gauge group.

Remarks on the Tomimatsu–Sato metrics
View Description Hide DescriptionAfter introducing a new way of writing the Tomimatsu–Sato solutions of Einstein’s field equations, we consider the geometry in the neighborhood of the ‘‘poles.’’ We also show that nonequatorial timelike and null geodesics can reach none of the ring singularities.