Volume 24, Issue 2, February 1983
Index of content:

A generalization of the construction of Ilamed and Salingaros
View Description Hide DescriptionIlamed and Salingaros construct the real and complex algebras with three anticommuting elements which can arise in physics. It is shown here that the ‘‘algebra of color’’ can be similarly constructed with six anticommuting elements. As a consequence of this construction, these algebras are all simple, quadratic algebras.

Representations of the groups Sp(n,R) and Sp(n) in a U(n) basis
View Description Hide DescriptionThe explicit expressions for the infinitesimal operators and the (finite) matrix elements with respect to a U(n) basis are obtained for the representations of the most degenerate series of Sp(n,R) and for the irreducible unitary representations of Sp(n) with the highest weights (M,0,...,0).

Reduction of inner‐product representations of unitary groups
View Description Hide DescriptionA direct method for the reduction of inner products of irreducible representations (irreps) of unitary groups has been proposed using the duality between the permutation and unitary groups. A canonical tensor basis set has been used to obtain a closed expression for the Clebsch–Gordan coefficients of U(n). This expression involves the subduction coefficients arising in the outer‐product reduction of S _{ N 1 }⊗S _{ N 2 }→S _{ N 1+N 2 } of the permutation groups, the symmetrization coefficients of U(n), and matrix elements of the standard representation of S _{ N }. The expression holds good for an inner‐product reduction of irreps of U(n), and is independent of n. The method has been illustrated with examples.

An integral transform related to quantization. II. Some mathematical properties
View Description Hide DescriptionWe study in more detail the mathematical properties of the integral transform relating the matrix elements between coherent states of a quantum operator to the corresponding classical function. Explicit families of Hilbert spaces are constructed between which the integral transform is an isomorphism.

On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender–Wu formula for x ^{2N }‐anharmonic oscillators
View Description Hide DescriptionRigorous results on the Laplace expansions of conditional Wiener integrals with functional integrands having a finite number of global maxima are established. Applications are given to the Bender–Wu formula for the x ^{2N } ‐anharmonic oscillator.

Generalized Fokker–Planck equation for non‐Markovian processes
View Description Hide DescriptionThis paper deals with a system of linear non‐Markovian–Langevin equations with memory functions that are not constant in time and a nonzero initial instant of time. A set of statistical means, based on the application of a generalized Furutsu–Novikov formula, was used to derive a generalized Fokker–Planck equation corresponding to this system and holding for both long and short instants of time. Considered as an example is the Brownian motion of a particle in a viscoelastic fluid with a particular relaxation time.

Practical use of the Hamilton–Jacobi equation
View Description Hide DescriptionWe show by means of several examples how the Hamilton–Jacobi equation can be used to solve nonlinear ordinary differential equations whose direct integration is otherwise difficult.

Covariance and geometrical invariance in *quantization
View Description Hide DescriptionSeveral notions of invariance and covariance for * products with respect to Lie algebras and Lie groups are investigated. Some examples, including the Poincaré group, are given. The passage from the Lie‐algebra invariance to the Lie‐group covariance is performed. The compact and nilpotent cases are treated.

Interaction‐set scattering equations
View Description Hide DescriptionThe generalization of the pair‐labeled Rosenberg equations for many‐particle scattering are found in the case where there are arbitrary multiparticle interactions. These are called interaction‐set equations because they involve auxiliary transition operators which are labeled by the same set of partitions which characterizes the various connectivities of the interparticle interactions. The technique which we employ also provides the analogous extension of a recently proposed set of connected‐kernel multiple scatteringequations for the Watson‐type transition operators. Further, the structure of the interaction‐set equations leads to the identification of an entire class of interaction‐set connected‐kernel scattering integral equations, each of which is based upon a distinct choice of unperturbed Green’s function and its associated connectivity structure. The generalized Rosenberg equations and the connected‐kernel Watson‐type multiple scatteringequations, which are limiting members of this class, correspond to the choice of the N‐free‐particle and two‐cluster‐channel unperturbed Green’s functions, respectively.

Improvement of the Froissart–Martin bound for complex scattering angles
View Description Hide DescriptionExtension of the Froissart–Martin bound for complex scattering angles is improved using the solution of the Dirichlet boundary value problem for doubly connected domains. The Froissart–Martin bound for physical scattering angles is used as input value on one of the two boundaries. The obtained bound is valid in an ellipse smaller than the Lehmann–Martin one. Possibilities for further improvements and applications are discussed.

Conditions of existence for a class of self‐dual solutions of the Einstein equations
View Description Hide DescriptionWe study the integrability of the Einstein equations for a class of empty homogeneous space‐times (Bianchi class A), once the self‐duality constraints are imposed on the space‐time itself. This system is integrable in the case of Bianchi type I and is a subset of Bianchi types VI _{0} and VII_{0}. Bianchi type II space‐times do not admit self‐dual solutions and for the case of Bianchi VIII and IX we were unable to find a general integrability condition.

On Riemannian spaces with conformal symmetries or a tool for the study of generalized Kaluza–Klein theories
View Description Hide DescriptionThe Riemann geometry of a space with conformal symmetries is written in terms of intrinsic objects defined from the action of the symmetries. Its application in the study of generalized Kaluza–Klein theories is discussed.

Convergence of multitime correlation functions in the weak and singular coupling limits
View Description Hide DescriptionFor a system coupled to a thermal bath we prove the convergence of the multitime correlation functions of system observables in the weak and singular coupling limits. The limiting correlation functions are given by the quantum regression law. Therefore, our result implies that in the limit the dynamics of the system are governed by a quantum stochastic process in the sense of Lindblad.

Rational von Neumann lattices
View Description Hide DescriptionA rational von Neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers. Zeros of harmonic oscillator functions in the k q representation on such lattices are found. It is shown that the number of zeros of the k q function determines the number of states by which a set on a von Neumann lattice is overcomplete. Interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the k q representation.

A geometrical theory of energy trajectories in quantum mechanics
View Description Hide DescriptionSuppose f(r) is an attractive central potential of the form f(r)=∑^{ k } _{ i=1} g ^{(i)} ( f ^{(i)}(r)), where {f ^{(i)}} is a set of b a s i s p o t e n t i a l s (powers, log, Hulthén, sech^{2}) and {g ^{(i)}} is a set of smooth increasing transformations which, for a given f, are either all convex or all concave. Formulas are derived for bounds on the energy trajectories E _{ n l } =F _{ n l }(v) of the Hamiltonian H=−Δ+v f(r), where v is a coupling constant. The transform Λ( f)=F is carried out in two steps: f→f̄→F, where f̄(s) is called the k i n e t i c p o t e n t i a l of f and is defined by f̄(s)=inf(ψ,f,ψ) subject to ψ∈D⊆L ^{2}(R ^{3}), where D is the domain of H, ∥ψ∥=1, and (ψ,−Δψ)=s. A table is presented of the basis kinetic potentials { f̄^{(i)}(s)}; the general trajectory bounds F _{*}(v) are then shown to be given by a Legendre transformation of the form ( s, f̄_{*}(s)) →( v, F _{*}(v)), where f̄_{*}(s) =∑^{ k } _{ i=1} g ^{(i)}× ( f̄^{(i)}(s)) and F _{*}(v) =min_{ s>0}{s+v f̄_{*}(s)}. With the aid of this potential construction set (a kind of Schrödinger Lego), ground‐state trajectory bounds are derived for a variety of translation‐invariant N‐boson and N‐fermion problems together with some excited‐state trajectory bounds in the special case N=2. This article combines into a single simplified and more general theory the earlier ‘‘potential envelope method’’ and the ‘‘method for linear combinations of elementary potentials.’’

Exact results for the diffusion in a class of asymmetric bistable potentials
View Description Hide DescriptionWe solve the Fokker–Planck equations with drifts deriving from a class of asymmetric nonharmonic potentials which include bistable cases. An analytical expression for the probability current over the potential barrier is obtained. Finally, we compare our exact results with those obtained by Kramers’ approximation.

Quantization of spinor fields. III. Fermions on coherent (Bose) domains
View Description Hide DescriptionA formulation of the c‐number classics‐quanta correspondence rule for spinor systems requires all elements of the quantum fieldalgebra to be expanded into power series with respect to the generators of the canonical commutation relation (CCR) algebra. On the other hand, the asymptotic completeness demand would result in the (Haag) expansions with respect to the canonical anticommutation relation (CAR) generators. We establish the conditions under which the above correspondence rule can be reconciled with the existence of Haag expansions in terms of asymptotic free Fermi fields. Then, the CAR become represented on the state space of the Bose (CCR) system.

Verification of the global Markov property in some class of strongly coupled exponential interactions
View Description Hide DescriptionWe verify the global Markov property in some class of strongly coupled exponential interactions in two‐dimensional space‐time. To obtain this result we apply the Albeverio and Ho/egh‐Krohn strategy. The basic ingredients we use in order to employ this strategy are the Fortuin–Kastelyn–Ginibre correlation inequalities.

Multiple fiber bundles and gauge theories of higher order
View Description Hide DescriptionWe investigate a theory of gauge fields over multiple bundles, i.e., principal bundles constructed over base spaces which are principal bundles themselves. The Higgs–Kibble field is introduced geometrically, together with the quartic potential. Results of Forgács and Manton are interpreted in this scheme. We also discuss a spherically symmetric ansatz which yields the quartic potential introduced by ‘t Hooft.

Graded gauge theories over supersymmetric space
View Description Hide DescriptionWe generalize the usual gauge theories, as well as the supergauge theories, in the following way. We construct a graded group associated with a compact semisimple Lie groupG. This graded group contains G and the linear space of anticommuting G‐spinors on which G acts through a highly reducible representation. The graded group generalizes the notion of the super‐Poincaré group. Next we construct a fiber bundle the basis of which is the superspace, the structural group being the graded group. Then we introduce the connection, curvature, and calculate the corresponding Yang–Mills Lagrangian. The nontrivial content of such a theory is put forward if we impose the Grassmann parity condition on our connection and curvature; we supposed here that both Grassmann parities (i.e., the one in the superspace and that in the graded group) add up to define the Grassmann parity of the corresponding field components. Together with the Hermiticity condition this supergauge leaves almost no room for arbitrariness in the expansion of the superconnection; it contains only the usual gauge field, the adjoint Higgs multiplet, and the spinor multiplet belonging to the spinorial representation of G. The conformal symmetry of the Lagrangian is broken, and the mass terms appear for the Higgs scalar and the spinor multiplet. The Yukawa and current–current interactions are also obtained, together with the Fermi four‐point interaction term. The theory yields the ratio of the Higgs scalar mass versus the bare spinor mass equal to 27/40; the strengths of other couplings depend on the group via the decomposition of the spinor multiplet into the irreducible representations.