Volume 23, Issue 10, October 1982
Index of content:

Generalized quaternions and spacetime symmetries
View Description Hide DescriptionThe construction of a class of associative composition algebrasq _{ n } on R ^{4} generalizing the well‐known quaternions Q provides an explicit representation of the universal enveloping algebra of the real three‐dimensional Lie algebras having tracefree adjoint representations (class A Bianchi type Lie algebras). The identity components of the four‐dimensional Lie groups GL(q _{ n },1) ⊆q _{ n } (general linear group in one generalized quaternion dimension) which are generated by the Lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree Lie algebra adjoint representations. In almost all cases the complete group of isometries of such a spacetime is isomorphic to a subgroup of the group of left and right translations and automorphisms of the appropriate generalized quaternion algebra. Similar results hold for the single class B Lie algebra of Bianchi type V, characterized by its ‘‘pure trace’’ adjoint representation.

Some comments on finite subgroups of SU(3)
View Description Hide DescriptionA recently published set of finite subgroups of SU(3) is shown to contain some groups which are not subgroups of SU(3). The others are subgroups of one of the dihedral‐like family of SU(3) subgroups Δ, of order 3n ^{2}. Some comments are made also on the structure of other finite subgroups previously listed.

Some spectral properties of the Kostant complex
View Description Hide DescriptionIn this paper we establish some spectral properties of an elliptic complex introduced by Kostant in the context of geometric quantization.

Analytic continuation from data points with unequal errors
View Description Hide DescriptionThe problem solved in this paper is that of constructing zero‐free holomorphic functions which will (a) assume specified values at a set of discrete data points in a data region Γ_{1} inside the holomorphy domain, and which at the same time will (b) provide the optimum solution to various stabilizing (boundedness or smoothness) conditions on the boundaries Γ_{ R }. The immediate motivation to this problem arose from the need to renormalize data with unequal errors by using a holomorphic weight function to bring all the errors to the same value: this was a preliminary step before making an analytic continuation off the data region Γ_{1}. Since a stabilizing condition has to be imposed on the boundaries Γ_{ R }, the weight function must be chosen so as to introduce the minimum additional instability on Γ_{ R }. Although this was the specific motivation, other interesting applications suggest themselves and some of these are discussed. The stability conditions on Γ_{ R } which are treated may all be expressed in terms of the real parts, or of the normal derivative of the real parts. Γ_{1} is taken to be on the real axis and the functions considered satisfy a reflection principle which means that the data values are real. It follows that the results obtained may be expressed in terms of the real parts alone—in other words the problem solved here is in fact that of obtaining harmonic functions which take specific values inside their harmonicity domain and which satisfy the appropriate extremum condition on the boundary.

Initial‐boundary‐value problem for diffusion of magnetic fields into conductors with external electromagnetic transients
View Description Hide DescriptionThe initial‐boundary‐value problem for the diffusion of an initially homogeneous magnetic field into a slab of conductivity σ<∞ and width Δx=2a is solved, under consideration of the electromagnetic wave pulses generated at the surfaces of the conductor by its interaction with the external magnetic field, which propagate into the surrounding vacuum. The analytical solutions show that (i) the external electromagnetic transients are necessary in order to correctly satisfy the boundary conditions for the tangential electric and magnetic field components, and (ii) the spatial and temporal development of the electromagnetic field and electric current in the conductor is quantitatively determined by a new dimensionless parameter group R=μ_{0}σa c[c=(μ_{0}ε_{0})^{−} ^{1} ^{/} ^{2}]. This ‘‘magnetic Reynolds number of the vacuum’’ determines the coupling between the transient fields in the conductor σ>0 and the ambient space (σ=0).

Some comments about the tensor virial theorem and orthogonal linear transformations
View Description Hide DescriptionThe tensor virial theorem is analyzed in relation to orthogonal linear transformations. The physical implications are discussed.

Gradient theorem for completely integrable Hamiltonian systems
View Description Hide DescriptionFor evolution equations which can be written in Hamiltonian form two ways, there exists a relation between two functions Q ^{(} ^{1} ^{)} and Q ^{(} ^{2} ^{)}, both of which are gradients of conserved functionals. The relation can be extended to define (recursively) functions Q ^{(n)}. It is shown that the Q ^{(n)} corresponding to the general evolution equation associated with the Zakharov–Shabat eigenvalue problem are all gradients of conserved functionals. This in turn implies all these functionals are in involution.

Wigner approach to quantization. Noncanonical quantization of two particles interacting via a harmonic potential
View Description Hide DescriptionFollowing the ideas of Wigner, we quantize noncanonically a system of two nonrelativistic point particles, interacting via a harmonic potential. The center of mass phase‐space variables are quantized in a canonical way, whereas the internal momentum and coordinates are assumed to satisfy relations, which are essentially different from the canonical commutation relations. As a result, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. In particular, the distance between the constituents is preserved in time and can take at most four different values. The orbital momentum is either zero or one (in units ℏ/2). The operators of the coordinates do not commute with each other and, therefore, the position of any one of the constituents cannot be localized; the particles are smeared with a certain probability in a finite space volume, which moves together with the center of mass. In the limit ℏ→0 the constituents ‘‘fall’’ into their center of mass and the composite system behaves as a free point particle.

Quantum and classical mechanics on homogeneous Riemannian manifolds
View Description Hide DescriptionStarting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous Riemannian manifolds, the classical theory is derived as a consequence, complete with its phase space realized as the space of pure classical states, a generalized version of the Wigner–Moyal correspondence rule, the Jordan and Lie algebrastructures of functions on the cotangent bundle, given by point‐wise multiplication and Poisson bracket, and the momentum map. A comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature.

Quantum evolution in the presence of additive conservation laws and the quantum theory of measurement
View Description Hide DescriptionIn this paper we derive in a completely rigorous way a family of inequalities holding for proper combinations of the squared norms of the states generated by the quantum evolution of a compound quantum system in the presence of additive conservation laws. The application of these inequalities to the quantum theory of measurement yields lower bounds for the malfunctioning of a measuring apparatus, which are valid under more general mathematical conditions and for a larger variety of physical situations than those considered up to now in the literature.

Constructing measures for spin‐variable path integrals
View Description Hide DescriptionBy exploiting the overcompleteness of the spin‐coherent states we derive expressions for spin‐kinematics path integrals (specifically for spin 1/2 and spin 1) in terms of genuine (Wiener) measures on continuous paths lying on the unit sphere and for certain dynamical systems which when projected onto the subspace spanned by the proper spin‐coherent‐state matrix elements yield the appropriate quantum‐mechanical propagator.

Constructing measures for path integrals
View Description Hide DescriptionThe overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum‐mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase‐space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.

Properties of solutions for N‐body Yakubovskii–Faddeev equations
View Description Hide DescriptionWe give a revised presentation of the Yakubovskii–Faddeev formalism based on a systematic study of the N‐body system chain structure. Completeness properties of the corresponding equations in differential form are considered. The expressions of physical and spurious solutions are given in terms of the N‐body asymptotic partition Hamiltonians eigenvectors.

Perturbative technique as an alternative to the WKB method applied to the double‐well potential
View Description Hide DescriptionWe give an explicit and complete perturbationtheoretical analysis of the solutions and the eigenvalues of the Schrödinger equation for the double‐well potential. In particular we demonstrate the matching of various branches of the solutions over the entire range of the independent variable, and we calculate the splitting of eigenvalues due to the finite height of the central hump of the potential.

Demiański‐type metric in Brans–Dicke theory
View Description Hide DescriptionA Demiański‐like metric is obtained by means of a complex coordinate transformation in the Brans–Dicke theory.

On the invertibility of Mo/ller morphisms
View Description Hide DescriptionLocal perturbations of the dynamics of infinite quantum systems are considered. It is known that, if the Mo/ller morphisms associated to the dynamics and its perturbation are invertible, the perturbed evolution is isomorphic to the unperturbed one, and thereby shares its ergodic properties. It was claimed by V. Ya. Golodets [Theor. Math. Phys. 2 3, 525 (1975)] that the above condition holds whenever the observable algebra is asymptotically abelian for the unperturbed evolution, and the perturbed evolution has a KMS state. The present paper contains a counterexample to this statement, and a construction of a spatial representation of the Mo/ller morphisms.

On the relativistic Bose–Einstein integrals
View Description Hide DescriptionTwo integrals which appear in the study of the relativistic Bose gas are analyzed. The complete low‐temperature and high‐temperature expansions are computed.

Rigorous iterated solutions to a nonlinear integral evolution problem in particle transport theory
View Description Hide DescriptionAfter a preliminary functional study of the operator associated with the relevant Boltzmann equation, which is shown to be a contraction operator, a nonlinear integral evolution problem occurring in the diffusion of the particles of a mixture is solved by resorting to a rigorous iterative scheme, in the case without removal. According to this scheme, an explicit recursive representation for the general iterated solution of order n is developed. Structure and behavior of the solution so obtained are investigated and commented on.

On the inconsistency of a photon creation mechanism in an expanding universe
View Description Hide DescriptionWe show here that if the quantum equivalence principle (QEP), as it was formulated in previous papers, is applied to the massless vector field, an inconsistent unphysical photon creation is found. This timelike and longitudinal photon creation is obtained when the 4‐potential A ^{μ} is quantized in a covariant way.

On mapping approaches in axiomatic quantum field theory
View Description Hide DescriptionWe here state a collection of results in axiomatic quantum field theory obtained under the general philosophy of ‘‘mapping approaches.’’ It is hoped that these results will stimulate further investigations along this direction, especially in connection with the problem of the existence and the construction of nontrivial four‐dimensional quantum field theories.