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Volume 26, Issue 1, January 1985

Tensor product of quantum logics
View Description Hide DescriptionA quantum logic is the couple (L,M) where L is an orthomodular σ‐lattice and M is a strong set of states on L. The Jauch–Piron property in the σ‐form is also supposed for any state of M. A ‘‘tensor product’’ of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ‐lattices. The existence and uniqueness of the tensor product in special cases of Hilbert spacequantum logics and one quantum and one classical logic are studied.

Triality principle and G _{2} group in spinor language
View Description Hide DescriptionThe presented paper is aimed at providing a systematic study of a relation between octonions and spinors corresponding to S ^{7} 7‐sphere, starting from a natural point of view, enabling us to endow spinor space S _{+}(8,0) with octonion algebrastructures. As a result we arrive at formulations of triality principle in its finite form in terms of vector and fundamental representations of Spin (8) group—both for spinors and vectors. The group of automorphisms of octonion algebra, as well as its Lie algebra, gains clear interpretation in the context. The method proposed is purely algebraic and could be applied as well to C(4,4) Clifford algebra corresponding to _{4} S ^{7} indefinite 7‐sphere geometry.

The Gel’fand realization and the exceptional representations of SL(2,R)
View Description Hide DescriptionIt is shown that the canonical representation space of Gel’fand and co‐workers is particularly appropriate for problems requiring explicit reduction under the noncompact SO(1,1) and E(1) bases for both the principal and exceptional series of representations of SL(2,R). We use this realization to set up complete orthonormal sets of eigendistributions corresponding to the three subgroup reductions, namely, SL(2,R)⊇SO(1,1), SL(2,R)⊇E(1), and SL(2,R)⊇SO(2), and evaluate the unitary transformations connecting these reductions. These overlap matrix elements appear as the applications of these distributions to a set of well‐defined test functions. Using the rigorous theory of analytic continuation we show that the results for the exceptional representations have the same analytic forms as the corresponding results for the principal series. Some of these results are essential prerequisites for the solution of the Clebsch–Gordan problem (series and coefficients) of SL(2,R) in the SO(1,1) basis.

A note on the multiplicity‐free unitary irreps of SL(3,R)
View Description Hide DescriptionUnitary irreps of ∼(SL(3,R)) which contain a representation D ^{ j } of SU(2) at most once are known to exist with SU(2) content j=k+2n (n=0,1,2,.. and k=0, 1/2 ,1). Güler, in a recent paper, claims that there also exist multiplicity‐free unitary representations with SU(2) content j=k _{0}, k _{0}+1, k _{0}+2,..., for k _{0}>3. We show that such representations do n o t exist for k _{0}>1.

Multiplicity‐free, unitary and nonunitary irreducible representations of SL(3,R)
View Description Hide DescriptionMultiplicity‐free, irreducible representations of the group ∼(SL(3,R)) are obtained from SU(2) subgroup representations by a constructive method. It is observed that there exist two series of unitary representations with k contents {k _{0},k _{0}+1,k _{0}+2,...}k _{0}≥3, {k _{0},k _{0}+2,k _{0}+4,...} k _{0}=0, 1, 1/2 and finite‐dimensional representations with k content {k _{0}+1,k _{0}+3,...,k _{0}+2n+1} k _{0}=1, 1/2 {2,4,6,...,2n}, n=1,2,...,k _{0}=0.

Unitary representations of the (4+1) de Sitter group on unitary irreducible representation spaces of the Poincaré group: Equivalence with their realizations as induced representations
View Description Hide DescriptionIn a previous work we have constructed realizations of the principal continuous series of unitary irreducible representations of the simply connected covering group of the (4+1) de Sitter group on unitary irreducible representation spaces of the simply connected covering group of the Poincaré group. In this work we demonstrate the equivalence of the representations constructed in the previous work with their realizations as induced representations.

Linearly and nonlinearly transforming fields on homogeneous spaces of the (4,1)‐de Sitter group
View Description Hide DescriptionScalar functions on the homogeneous spaces H^{ R } of the de Sitter group G=SO(4,1) are studied, where the spaces H^{ R } are of the form G/K with K being a subgroup of the Lorentz groupH=SO(3,1) contained in SO(4,1). The spaces H^{ R } can be regarded as fiber bundles E ^{ R }=E ^{ R }(G/H,H/K), with the base V ^{′} _{4} =G/H being a space of constant negative curvature characterized by a fundamental length parameter R[(4,1)‐de Sitter space], and the fiber S=H/K being a homogeneous space of the Lorentz group. The action of G on the spaces E ^{ R } is a linear action on V _{4} and a nonlinear action on S, where the latter action is defined by a generalized Wigner rotation. A gauge theory based on the (4,1)‐de Sitter group is investigated with matter represented in terms of a generalized wave function Φ(x;ξ,ỹ) [with x∈U _{4} (Riemann–Cartan space‐time), ξ∈V ^{′} _{4}, and ỹ∈S] which is defined as a map from a cross section on the bundle E=E ( U _{4}, F=E ^{ R }, G=SO(4,1)) over space‐time U _{4} with fiber F=E ^{ R } =G/K and structural group G=SO(4,1) into the complex numbers. The introduction of purely nonlinearly transforming fields _{(N)}Φ(x;ỹ) is discussed as well as the nonlinear realization of the SO(4,1) symmetry in terms of transformations of the Lorentz subgroup H (generalized Wigner rotations). The geometric implications of symmetry breaking are pointed out.

Classical and quantum symmetry groups of a free‐fall particle
View Description Hide DescriptionSymmetry of a free‐fall particle is studied in quantum as well as classical mechanics. The quantum symmetry group is shown to be a central extension of the classical one. In the case of two degrees of freedom, the action of the quantum symmetry group is expressed in the form of integral transform as a unitary operator on the space of wave functions.

Factors of the Fock functional
View Description Hide DescriptionCan the fields φ and π of a representation of the CCR’s be written as φ=1/{φ_{1}×1+1⊗φ_{2}} and similarly for π, such that φ_{ i } and π_{ i } satisfy the CCR’s? What are the possible φ_{ i }’s and π_{ i }’s? This is equivalent to a factorization of the corresponding generating functionals (scaled by 1/). Generalizing this question somewhat we show a noncommutative analog of Cramér’s theorem of probability theory. If φ and π are Fock fields then so are φ_{ i }, π_{ i }, i=1,2; similarly for quasifree representations of the CCR’s. As an application we show that the fields of a representation of the CCR’s whose generating functional differs from a Fock functional by a phase factor only are just shifted Fock fields.

Symmetric‐tensor eigenspectrum of the Laplacian on n‐spheres
View Description Hide DescriptionThe eigenvalues and degeneracies of the covariant Laplacian acting on symmetric tensors of rank m≤2 defined on n‐spheres with n≥3 are given.

Invariants for dissipative nonlinear systems by using rescaling
View Description Hide DescriptionA rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=d t/A ^{2}(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)x ^{ m+1}/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.

On the analysis of relaxation in electrolytes
View Description Hide DescriptionDebye and Falkenhagen [Phys. Z. 2 9, 121 (1928)] analyzed a linear initial/boundary value problem for a differential equation of diffusion type to model the phenomenon of relaxation in electrolytes; specifically, they sought to characterize the disappearance in time of the radially symmetrical and static charge distribution surrounding an individual motionless ion after the latter is instantaneously removed. A detailed reexamination discloses the existence of multiple solutions for the posed problem, with agreement as regards the initial condition and disparity as regards behavior at the central location. A regular solution during the entire relaxation regime is exhibited and offered in place of the classical one, due to Debye and Falkenhagen, which retains a singular nature at the site originally occupied by the reference ion.

Cauchy system for resolvent of Milne’s integral equation with anisotropic scattering
View Description Hide DescriptionIt has been shown previously that the resolvent of the inhomogeneous truncated Milne’s integral equation with isotropic scattering enables us to reduce dimensionally the above auxiliary equation via Chandrasekhar’s X‐ and Y‐functions. In the present paper, extending the above procedure to the case of anisotropic scattering, we show how to use effectively the Bellman–Krein–Sobolev‐like formula for the dimensional reduction of the Cauchy system of the source function via the generalized Chandrasekhar X‐ and Y‐functions.

Bound and radiation fields of a pointlike SO(3) monopole
View Description Hide DescriptionThe macroscopic field of a pointlike, arbitrarily moving monopole is split up into its bound and radiation parts. The SO(2) fiber bundle, having the total monopole field as its curvature, is embedded into an SO(3) bundle (Georgi–Glashow model) such that the SO(2) monopole field is composed in an SO(3) invariant manner by the microscopic gauge and Higgs fields: The Higgs field constitutes the bound part and the gauge field (Yang–Mills field) is responsible for the radiation part of the macroscopic monopole field. The role played by the field equations for the validity of separate local energy‐momentum conservation laws is discussed extensively.

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform
View Description Hide DescriptionThis paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

Jet extension of a classical particle field: Principles of covariance and minimal coupling
View Description Hide DescriptionJet theory of a classical particle field is developed through a systematic transition from a local to a global formulation. Emphasis is laid on the role of gauge covariance and minimal coupling principles in producing a global geometrical framework which faithfully generalizes the one of gravitationtheories.

The action‐angle variables for the massless relativistic string in 1+1 dimensions
View Description Hide DescriptionIn this paper the Poisson bracket algebra for the open massless relativistic string in the one‐space‐ and one‐time‐dimensional case is considered. In order to characterize the orbit of the system the directrix function, i.e., the orbit of one of the endpoints of the string, is used. It turns out that the Poisson bracket algebra is of a very simple form in terms of the parameters of the directrix function. We use these results to construct action‐angle variables for the general motion of the string. The variables are different for different Lorentz frames, with a continuous dependence. The action‐angle variables of the center‐of‐mass frame and of the light‐cone frames are of particular interest with respect to the simplicity of the Poincaré generators and the physical interpretation. For the light‐cone frame variables the equivalence to a set of indistinguishable oscillators is shown, for which an excitation corresponds to an instantaneous momentum transfer to an endpoint of the string.

Mass eigenfunction expansions for the relativistic Kepler problem and arbitrary static magnetic field in relativistic quantum theory
View Description Hide DescriptionWe investigate the existence of orthogonality and completeness relations for the eigenvalue problem associated with the differential operator Λ=−Π_{μ}Π_{μ} −i eσ ⋅ (E+i B), Π_{μ}=−i ∂_{μ}−e A _{μ}. The operator Λ acts on 2×1 Pauli‐type spinor fields defined over all Minkowski space, and may be interpreted as the square of the mass of a charged Dirac particle moving in an external c‐number electromagnetic field. We show that Λ is self‐adjoint with respect to the not positive‐definite inner product (φ_{ b }; φ_{ a })=∫ d ^{4} x φ̄_{ b }φ_{ a }, where φ̄_{ b } is defined as φ̄_{ b }=φ^{†} _{ b }(−iΠ↙_{4}−σ⋅Π↙). A proof is provided for the Coulomb case that the mass eigenfunctions form a complete set in spite of the indefinite metric in Hilbert space. The mass eigenfunction expansion of the propagator is worked out explicitly for the Kepler case. This mass eigenfunction expansion is expected to be quite useful for bound state calculations in quantum electrodynamics, since it involves the covariant denominators (m′)^{2}−(m)^{2}.

A general approach to the systematic derivation of SO(3) shift operator relations. I. Theory
View Description Hide DescriptionA new technique is established for the construction of relations between products consisting of two, three, or more SO(3) shift operators, within the framework of the Lie algebras of SO(3) tensor operators constructed out of an underlying single‐boson structure.

A general approach to the systematic derivation of SO(3) shift operator relations. II. Applications
View Description Hide DescriptionThe technique reported on in the preceding paper is applied to construct shift operator relations in the SU(3) and SO(5) Lie algebras. Comments are made concerning the integrity basis for SO(3) scalar operators appearing in their respective enveloping algebras.