Volume 18, Issue 5, May 1977
Index of content:

A criterion for reducibility of a relativistic wave equation
View Description Hide DescriptionIn general when one writes a relativistic wave equation of the form (−iΓ⋅∂+m) ψ (x) =0, that transforms covariantly under some representation Λ→T (Λ) of SL(2,C), it is nontrivial to determine whether or not the equation is irreducible or to avoid ending up with a reducible equation; especially if T (Λ) contains repeat ing irreducible r epresentations. In this paper a simple (st) criterion is given by which one can determine whether or not an equation is irreducible. It is shwon that if Λμ have any invariant subspace at all, then that subspace must be a representation space of some combination of SL(2,C) representations in T (Λ). Knowing this it is shown that a wave equaiton is reducible if an only if there exists some idempotent projector such that (1−P̃) Γ_{0}P̃=0 other than P̃=O or I. A method for constructing all possible addmissable P̃’s is given. A simple example of the technique is given. A simple example of the technique is also given.

Asymmetric gas thermodynamics
View Description Hide DescriptionWe consider a gas in which the density and temperature fields are spatially and timewise nonuniform. We then show that on the basis of the entropy principle the stress tensor is nonsymmetric. This nonsymmetry of the stress thensor is shown to be the deriving force behind the creation the microrotational fields and dynamic spatial polarizations in the gas. It is also shown that the nonsymmetric part of the stress tensor arises out of the interaction of gradient fields of temperature and density.

Classical systems of infinitely many noninteracting particles
View Description Hide DescriptionIt is proved that the C*‐algebra of observables of an infinite classical system is isomorphic to the group algebra on the test function space D. The physical dynamical system consisting of infinitely many noninteracting particles is studied. A particular class of states, called the quasifree states, is exhibited and their properties are studied. Some results on the spectral properties of monoparticle evolutions are obtained. Finally we give explicitly a solution of the classical KMS condition for these evolutions.

On a family of interior solutions for relativistic fluid spheres with possible applications to highly collapsed stellar objects
View Description Hide DescriptionA one‐parameter family of interor solutions of Einstein’s field equations for a static spherical fluid is given. It is shwon that for various values of the parameter and choices of the constants in ingegration, several previously known solutions for static fluids are contained therein. This family of solutions can be joined continuously to the Schwarzchild exterior solution, and as such may be applicable to be investigation of stellar interior where high central densities and and pressures are of interest.

Group theory of the collective model of the nucleus
View Description Hide DescriptionIn the present paper we extend the group theoreticalanalysis of a previous publication to obtain explicitly, as a polynomical in sinγ, cosγ, the function φ^{λμl } _{ k }(γ) required in the discussion of the quadrupole vibrations of the nucleus. The states appearing in the collective model 〈νλμL V〉=F _{1} ^{λ}(β) Σ_{ K }φ^{λμL } _{ K }(γ) D ^{ L }*_{ M K }(φ_{ i }), l= (ν−λ)/2, are then defined, as F ^{λ} _{ l }(β), D ^{ L }*_{ M K }(φ_{ i }) are well known. All matrix elements required in the collective model of the nucleus are related then with the expression (λμL;λ′μ′L′;λ^{″}μ^{″} L ^{″}= ∂^{π} _{0}Σ_{ K K′K ″ } (^{ L L′L ″ } _{ K K′K ″ }) φ^{λμL } _{ K }(γ) φ^{λ′μ′L′} _{ K′} (γ) φ^{λ″μ″ L ″ } _{ K ″ } (γ)sin 3γdγ, which is a reduced 3j‐symbol in the O(5) O(3) chain of groups.

A systematic investigation of the Petrov G _{4} types
View Description Hide DescriptionIn order to investigate exact solutions in general relativistic cosmology, one usually assumes the spacetime possesses symmetry. Here, we study exact solutions for the Petrov four‐parameter Lie groupsG _{4}, acting on nonnull hypersurfaces where the energy‐momentum tensor is that of a pressureless perfect fluid (a so‐called ’’dust’’). We find that the preponderance of solutions are for a spacelike dust and, in several cases, are able to give their explicit forms. Among these are spacelike versions of previously known timelike matter cosmological models.

Investigations of space‐times with four‐parameter groups of motions acting on null hypersurfaces
View Description Hide DescriptionAn investigation of all metric having a four‐parameter group of symmetrics with null three‐dimensional orbits is made. An attempt to solve the Einstein field equations using various simple energy‐momentum tensors, with on exception, gives incompatible sets of equations. The exception is a solution for a null fluid possessing the G _{4} I _{1} group of symmetries.

Lorentz transformations as space‐time reflections
View Description Hide DescriptionA rank‐two tensor is built out of the 4‐velocities of two inertial observers, which corresponds precisely to the most general Lorentz matrix connecting the two Cartesian frames of the observers. The Lorentztensor is then factorized as the product of two ’’complementary’’ space‐time reflections. It is shown that the first tensorial factor performs the very essential tasks (i.e., FitzGerald contraction and time dilation) of the corresponding Lorentz transformation, while the second factor is just an internal reflection performed in one and the same intertial frame. Thus, in its essential features, a Lorentz transformation between two different inertial frames obtains upon performing just one space‐time reflection. It is also shown that the (same) Lorentztensor of the two inertial observers can be factorized into ’’complementary’’ reflections either by two hyperplanes with spacelike normals, or else by tow hyperplanes with timelike normals, which geometric meaning is rather simple. An application of the presented formalism to Dirac’s 4‐spinor transformation law is also briefly discussed.

Construction of the Yukawa_{2} field theory with a large external field
View Description Hide DescriptionWe consider the Yukawa_{2}model with (relativistic) interaction density λψ̄Γψφ+μφ, where Γ=1 or γ_{5}. For sufficiently large μ, we apply the Glimm–Jaffe–Spencer cluster expansion to construct the infinite volume theory satisfying the Wightman and Osterwalder–Schrader axioms including a positive mass gap.

The bundle boundary in some special cases
View Description Hide DescriptionWe examine a class of two‐dimensional Lorentz manifolds which are ’’singular’’ in a certain sense. It is shown that, for such a manifold (M, g), the bundle boundary is a single point whose only neighborhood is all of [the bundle completion of M; see B. G. Schmidt, Gen. Rel. Gravitation 1, 269–80 (1971)]. The four‐dimensional Schwarzschild and Friedmann–Robertson–Walker solutions are then investigated. We show that the bundle completions of these spaces are not Hausdorff.

Phase‐integral calculation of quantal matrix elements without the use of wavefunctions
View Description Hide DescriptionSimple phase‐integral formulas for the calculation, without the use wavefunctions, of quantal matrix elements of multiplicative and differential operators are given for the case of bound states in a single‐well potential. The matrix elements are obtained to within the accuracy corresponding to any conveniently chosen order of the kind of phase‐integral approximations used.

Energy forms, Hamiltonians, and distorted Brownian paths
View Description Hide DescriptionWe study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇f d’gm, where dμ is a positive, not necessarily finite measure on R ^{ n }. We cover the cases of very singular interactions (e.g., N particles in R ^{3} interacting by two‐body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markovdiffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (R ^{ r },dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect to the constructed Hut processes, rather than with respect to Brownian motion. This is discussed in relation to recent work of Ezawa, Klauder, and Shepp, as well as of Hida and Streit.

Instability of the continuous spectrum: The N‐band Stark ladder
View Description Hide DescriptionIt is shwn that the energy spectrum of the Bloch electron in an external field is continuous. Furthermore, it is shown that all approximations which take into account interband coupling whithin groups of finite number of bands (the N‐band approximation) lead to a pure‐point spectrum of intertwining Wannier‐stark ladders. This instability of the continuous spectrum under the N‐band approximation is related to a theorem due to Weyl and von Neumann. Approximation methods for dealing with interband coupling within a group of finite number of bands are given.

Synchronized solitons
View Description Hide DescriptionWe find that, under certain conditions, two solitons of an integrable system can form a synchronized bound state when the system is perturbed.

T matrix and effective range function for Coulomb plus rational separable potentials especially for I=1
View Description Hide DescriptionThe off‐shell l=1 T matrix in the momentum representation for the pure Coulomb potential and for the Coulomb plus a rational plus a rational separable potential of the Yamaguchi type is obtained in closed form. The amplitude, the effective range function, and the effective range parameters are derived from the T matrix and are given in closed form. For a large number of rational separable potentials we prove thar the effective range function is real analytic at zero energy. We give, however, an axample of potential for which this effective range function has a pole at the orign. From t hese effective range functions a certain functionW is extracted which does not depend either or l or on the particular potential. This functionW is studied in detail. We indicate howe the result s of this paper can be generalized to arbitrary values of l and to all Coulomb plus rational separable potentials.

The number of bound states of the Coulomb plus Yamaguchi potential
View Description Hide DescriptionIt is shown that certain assertions on the number of bound states of a Coulomb plus Yamaguchi potential which Zachary [J. Math. Phys. 12, 1379 (1971); 14, 2018 (1973)] claims to have proved are incorrect. We prove that there are always infinitely many bound states if the Coulomb part of the potential is attractive and that, in case the Coulomb part of the potential is repulsive, there is one bound state only if the Yamaguchi potential is sufficiently attractive.

Regge trajectories for a velocity dependent potential
View Description Hide DescriptionWe study the distribution of the singularities in the complex angular momentum plane of the S matrix for a velocity‐dependent potential, and note some deviations with respect to the general behavior established for static potentials. We analyze the physical implications of our results concerning the existence of bound states and resonances.

Phase‐space approach to relativistic quantum mechanics. I. Coherent‐state representation for massive scalar particles
View Description Hide DescriptionWe construct a family of equivalent representations U _{λ} (λ≳0) of the restricted Poincaré group P^{↑} _{+} for a massive scalar particle on spaces K_{λ} of functions defined over ’’phase space’’ P _{λ}. Each P _{λ} is a submanifold of the forward tube, and K_{λ} consists of restrictions on holomorphic solutions of the Klein–Gordon equation to P _{λ}. Each K_{λ} has a resolution of the identity in terms of ’’coherent states’’ e _{ z }, zεP _{λ}, which are wavepackets characterized by an invariant extremal property.

Curvature invariants and space–time singularities
View Description Hide DescriptionThis paper collects together in a general setting observer dependent curvature invariants for space–time and applies them to an analysis of curvature singularities. Observer dependent quantities, such as energy and momentum densities and tidal stresses, are dependent not only on the space–time point but also on the observer’s 4‐velocity. The properties of these invariants are discussed, and it is shown that they completely describe the behavior of curvature along timelike curves. In particular, curvature singularities can be characterized by unboundedness of these invariants.

Nonstandard vector connections given by nonstandard spinor connections
View Description Hide DescriptionA unique two‐component spinor connection, which we call the standard connection, is determined by the requirement that it be compatible with the spinor inner product and that it give rise to the standard 4‐vector connection. Here we take the most general spinor connection, presuming that the conjugate spinor connection is uniquely determined by it, and examine which 4‐vector connections are thereby determined. We classify such nonstandard spinor connections and the resulting 4‐vector connections and show that the most general torsion tensor can be so generated. However, it is not possible to generate in this way the most general tensor describing incompatibility of the 4‐vector connection with the 4‐vector inner product. These results illuminate the relationship which must exist between nonstandard theories for spinors and for vectors.