Volume 31, Issue 2, February 1990
Index of content:

Normally ordered Fermi operator realization of the SU_{n} group
View Description Hide DescriptionThe mapping of the SU_{n} group transformation in Grassmann number space into unitary normally ordered and antinormally ordered Fermi operator realizations in Hilbert space are investigated. The unitary Fermi operators are evaluated in fermion coherent state representation using the ‘‘integration within ordered product’’ technique for a fermionic system. Some new Fermi operator identities are thus obtained.

On continuity of mean values of unbounded observables
View Description Hide DescriptionAn unbounded observable whose mean value at some state changes discontinuously under the action of a *‐weakly continuous one‐parameter symmetry group is exhibited. Also, a sufficient condition for differentiability of a mean value of a not necessarily bounded observable that evolves under the action of a *‐weakly continuous one‐parameter automorphism group is given.

Kronecker products, minuscule representations, and polynomial identities
View Description Hide DescriptionThe special role of the highest long and highest short roots in the derivation of extremum properties of the weights of irreducible representations of semisimple Lie algebras is pointed out. These properties are used to give an intrinsic and unifying reformulation, as well as a new proof of Klimyk’s theorem on Kronecker products [Ukrain. Mat. Z. 1 8, 19 (1966)]. This new form of Klimyk’s theorem reveals the special position of the minuscule representations in Kronecker products; as an immediate consequence, the explicit formula for the Kronecker product of an arbitrary representation with a minuscule representation is obtained. Explicit expressions for the weights of the minuscule representations are given. New derivations of theorems of Dynkin [Trudy Mosk. Obsch. 1, 39 (1952)] and Feingold [Proc. Am. Math. Soc. 7 0, 109 (1978)] on Kronecker products, as well as the necessary condition for a Kronecker product to decompose in two irreducible components are obtained. The proofs are based on a theorem of Parthasarathy, Ranga Rao, and Varadarajan [Ann. Math. 8 5, 383 (1967)].

On left invariant Brownian motions and heat kernels of nilpotent Lie groups
View Description Hide DescriptionLeft‐invariant Brownian motions on nilpotent Lie groups are studied. Their characterization is given through Ito or Stratonovich stochastic differential equations, their generators are exhibited and the associated heat semigroups are studied. A reduction formula is given for these semigroups and their kernels, as integrals of products of normalized random Gaussian densities.

Inhomogeneous boson realization of indecomposable representations of Lie algebras
View Description Hide DescriptionBy making use of the differential realization of Lie algebras in the space of inhomogeneous polynomials of a certain number of variables, the corresponding inhomogeneous boson realization of Lie algebras is given. A new kind of indecomposable representations of Lie algebras are studied on the universal enveloping algebra of Heisenberg–Weyl algebra, its subspaces and its quotient spaces. The finite‐dimensional representations are naturally obtained on the subspaces of Fock space. As an example, the indecomposable and irreducible representations of the Lie algebra su(2) are discussed in detail.

Classical non‐Abelian Berry’s phase
View Description Hide DescriptionThe classical non‐Abelian Berry’s phase is defined for a parameter‐dependent dynamical system that is collective with respect to a Hamiltonian G‐action when the parameters are fixed. It is shown that the corresponding angular two‐form of Berry is given in terms of the momentum mapping of the G‐action and is induced naturally by the symplectic form on the phase space. Moreover, this angular two‐form can be seen as the correction term in the effective symplectic form on the parameter space, with respect to which the Hamiltonian is to be quantized.

The group theoretical analysis of gravitational instanton equations
View Description Hide DescriptionGroup theoretical properties of certain nonlinear partial differential equations playing a distinguished role in the gravitational instanton theory and in complex relativity are studied. It is demonstrated that, in general, the groups of contact transformations admitted by these equations appear to be the first prolongations of appropriate point transformation groups. An exceptional case leading to the Gibbons–Hawking metric is examined in detail.

Form perturbations of the Laplacian on L ^{2} (R) by a class of measures
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Noncommutative differential geometry of matrix algebras
View Description Hide DescriptionThe noncommutative differential geometry of the algebraM _{ n } (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebraC (sl(n,C),M _{ n } (C))=M _{ n } (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on M _{ n } (C). A canonical symplectic structure is exhibited for M _{ n } (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for M _{ n } (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.

Noncommutative differential geometry and new models of gauge theory
View Description Hide DescriptionThe noncommutative differential geometry of the algebraC ^{∞}(V)⊗M _{ n }(C) of smooth M _{ n }(C)‐valued functions on a manifoldV is investigated. For n≥2, the analog of Maxwell’stheory is constructed and interpreted as a field theory on V. It describes a U(n)–Yang–Mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential. The Euclidean action, which is positive, vanishes on exactly two distinct gauge orbits, which are interpreted as two vacua of the theory. In one of the corresponding vacuum sectors, the SU(n) part of the Yang–Mills field is massive. For the case n=2, analogies with the standard model of electroweak theory are pointed out. Finally, a brief description is provided of what happens if one starts from the analog of a general Yang–Mills theory instead of Maxwell’stheory, which is a particular case.

On bi‐Hamiltonian structures
View Description Hide DescriptionThe definition of a bi‐Hamiltonian structure is reviewed, and it is shown that for systems of differential equations of the form ẋ=v(x) on even‐dimensional manifolds, there always exists locally a bi‐Hamiltonian structure. If this structure is ‘‘global,’’ then the system of equations is integrable. Furthermore, the geometry and canonical forms for such structures are discussed.

On the convergence of the Magnus expansion in the Schrödinger representation
View Description Hide DescriptionThe convergence properties of the Magnus expansion in the Schrödinger representation are investigated. A quite general result is rigorously derived from first‐order perturbation theory. A finite matrix representation is presented for obtaining the exponential time‐evolution operator more easily. Two time‐dependent models, an oscillator and a spin system, are considered as illustrative examples.

On the convergence of the Feynman path integral for a certain class of potentials
View Description Hide DescriptionA direct proof is given of the convergence of the discretized Feynman path integral to the fundamental solution of the (time‐dependent) Schrödinger equation, for potentials which are bounded, integrable and continuous on the real line. No further smoothness assumptions are required.

Untenability of a proposed constrained dynamics for the damped harmonic oscillator
View Description Hide DescriptionIt is argued that the method introduced recently by Wang [J. Phys. A: Math. Gen. 2 0, 4745 (1987)] to quantize the damped harmonic oscillator is untenable because it does not reproduce the standard results for the quantum oscillator in the limit of arbitrarily small damping. The method is also shown to be inconsistent if the friction coefficient is allowed to take any positive value.

Novel correspondence of the fixed‐seniority and the fixed‐isospin averages in their reduction formulas
View Description Hide DescriptionIt is shown that known reduction formulas for partial averages of two different kinds, the fixed‐seniority and the fixed‐isospin averages, are transcribed into each other by replacement of arguments. The average here is that of a general k‐body operator in a finite space of fermions (bosons). Description of the formulas in a common form in terms of eigenvalues of Casimir operators is also exploited.

Geometry and uncertainty
View Description Hide DescriptionIn the context of the Wigner–Weyl phase space formulation of quantum mechanics, a version of the uncertainty relations invariant under affine canonical transformations is derived. For a fixed Wigner distribution function possessing a finite covariance, ‘‘directions’’ of minimal uncertainty are found. The geometry of the Wigner ellipsoid and its Legendre transform, the dual Wigner ellipsoid, both of which are associated with the same covariance, is discussed. The results obtained are generalizations of the well‐known fact that, for one degree of freedom, the area of the Wigner ellipse must be of order ℏ or larger. Instead of area, which is an invariant only when n=1, these results involve Poincaré invariants of certain curves and surfaces.

On nonperfect fluid cosmologies evolving towards perfect fluid cosmological models
View Description Hide DescriptionThe problem of entropy production in near‐equilibrium situations and their evolution towards a state of thermodynamic equilibrium is considered in the cosmological context. A physically realistic model (i.e., satisfying the energy conditions) describing such a situation is constructed. From a few hypotheses and considerations, it is seen that the metric tensors of the space‐time at both equilibrium and nonequilibrium configurations are conformally related. The material content described by the energy‐momentum tensor is interpreted as a viscous fluid smoothly evolving into a perfect one.

Inflation in a spatially closed anisotropic universe
View Description Hide DescriptionThe effects of shear on the occurrence of inflation are studied on the basis of a simple model for a spatially closed universe which enters an inflationary era. It is assumed that the universe enters a vacuum‐dominated phase in an abrupt transition that occurs everywhere at the same time. The space‐time geometries, before and after the phase transition, are matched to each other via the Lichnerowicz junction conditions. The Einstein field equations are solved exactly for a viscous universe of the Kantowski–Sachs type. It is found that the inclusion of (positive) shear retards the occurrence of the vacuum phase transition. The magnitude of this effect depends on the mass of the universe at the time of the phase transition. For a universe with a mass of about 10 kg (which is a value usually associated with the mass of the region from which our universe originated), it is found that the inclusion of shear does not really have a large effect on the time at which the vacuum phase transition occurs. The generality of the results is also discussed.

On the Legendre transformation for a class of nonregular higher‐order Lagrangian field theories
View Description Hide DescriptionIn the framework of higher‐order calculus of variations, the generalized Legendre transformation for a wide class of Lagrangians is considered, which depend in a nonregular way on the derivatives of maximal order. A rigorous theory is discussed for Lagrangians depending on a constant rank set of affine combinations of these derivatives. This allows the reduction of the Poincaré–Cartan formalism and the Hamiltonian formalism to the appropriate constraint in the appropriate phase space of the problem. The case considered here covers many important physical examples, such as the Yang–Mills theories (at order one) and relativistic metric theories of gravitation (at order two).

On the nth‐order structure of solitonic wormholes
View Description Hide DescriptionThe structure of the equations describing a solitonic wormhole in the Einstein–Yang–Mills–Higgs system for an arbitrary compact and connected gauge group G and representation of an arbitrary Higgs field Q is studied. The general structure of these equations and its use in deriving the first‐order equations, which result by applying the operator O _{ N }=lim_{ N→0} ∂/∂N to the original equations, where N is the lapse function, is discussed. It is also possible to write down the nth‐order equations, which are obtained by applying n times the operator O _{ N }, for a certain class of solutions. These nth‐order equations are then specialized to the model with G=SU(2) and Q in the adjoint representation. These nth‐order equations on the background of a non‐Abelian zero‐order solution, which can be gauge transformed to satisfy the ’t Hooft ansatz at the internal infinity of the hole, are solved. The technique used to solve the system is that of harmonic expansion, yielding an algebraic system of equations which can be interpreted as an eigenvalue problem. The eigenvalues are functions of the parameters of the model and zero‐order quantities. Thus, depending on the values of these parameters, nontrivial nth‐order solutions may exist or not. Those cases in which there exist nontrivial solutions are stated. Only for those cases can it be expected that global non‐Abelian solitonic wormholes are found.