Volume 23, Issue 9, September 1982
Index of content:

Molien generating functions, invariants and covariants of magnetic point groups
View Description Hide DescriptionThe general properties of the Molien generating function and invariant and covariant tensors for the corepresentations of nununitary point groups are presented. The generating functions and (D_{1},D_{ m }) (invariant) and (D_{ r },D_{ m }) (covariant) tensors are obtained for the 32 grey point groups and the 58 black‐white point groups.

Generalized SU(2) spherical harmonics
View Description Hide DescriptionThe generating functions for polynomialtensors based on each SU(2) tensor of rank from 7 to 13 (angular momentum 7/2 to 13/2) are given in a ’’positive’’ form suitable for interpretation in terms of an integrity basis. An iterative procedure for extending the results to higher rank tensors is indicated.

Bäcklund generated solutions of Liouville’s equation and their graphical representations in three spatial dimensions
View Description Hide DescriptionThe Bäcklund–Bianchi method is employed to generate, in three spatial dimensions, the following multiple solutions of Liouville’s equation ∇^{2}α = exp α: The three‐wave interaction function α_{3} and the five‐wave interaction function α_{5}. It is verified numerically that α_{3} satisfies Liouville’s equation to an accuracy of one part in 10^{14}, while α_{5} satisfies it to one part in 10^{6}. The construction of α_{5} is conditional upon solving ten nonlinear constraint equations. We analyze the complicated structures of α_{3} and α_{5} with the help of a three‐dimensional plotting routine. It is found that α_{3} is, surprisingly enough, only characterized by a single ring singularity, while α_{5} exhibits three ring singularities. It is speculated that the function tanh α_{3} represents a ring soliton whose shape appears to be preserved in the nonlinear superposition of similar ring solitons. The derivation of Liouville’s solutions α_{3} and α_{5} is intimately connected with the auxiliary functions β_{2} and β_{4} which solve Laplace’s equation. The latter are also derived and plotted in the paper.

On the projections of representation spaces of the symmetry group on the Minkowski space
View Description Hide DescriptionIn this paper we generalize the projection of the representation space of the symmetry group SU(2,2)×U(2) on the Minkowski space to arbitrary internal symmetries U(m). The procedure involves certain restrictions on the coordinates of the representation space. Representations of the symmetry group in the restricted space and in the corresponding restricted Hilbert space are constructed.

On a certain class of two‐sided continuous local semimartingales: Toward a sample‐wise characterization of the Nelson process
View Description Hide DescriptionWorking with the extended framework of stochastic integrals recently discovered by Itô, a complex of stochastic processes inherent in quantum mechanics, the Nelson process, is characterized in terms of sample paths. It is shown that the Nelson process belongs to a certain class of two‐sided continuous local semimartingales. Several basics of stochastic calculus in this class are presented. Stochastic calculus of variations is applied in this class to construct the Nelson process and to further illustrate some details of its sample paths. Examples are the bound states, the two‐slit interference, and the gravity in quantum mechanics.

On Jacobi’s decomposition of the motion of a heavy symmetrical top into the motions of two free triaxial tops
View Description Hide DescriptionJacobi discovered that the motion of a heavy symmetrical top can be decomposed into the motions of two torque‐free triaxial tops. In this paper we investigate the connection between the three sets of the dynamical constants in the three top motions. The formulas connecting these constants are found to be projective transformations (fractional linear transformations).

Vector fields generating analysis for classical dissipative systems
View Description Hide DescriptionA class of vector fiels is identified which (locally) generate first integrals of a dissipative system. The structure of these vector fields and of the corresponding invariants is studied. The relationship with a previously proposed generalization of Noether’s theorem for nonconservative systems, is pointed out.

Root parities and phase behavior in the slow‐fluctuation technique
View Description Hide DescriptionThe slow‐fluctuation technique for integrating autonomous, conservative, nonlinear, near‐resonant oscillatory systems of many degrees of freedom requires ultimately no more than the study of a certain polynomial. It is shown that a paritylike property can be attributed to the roots of this polynomial, which proves helpful in even the most complex situations. It aids to classify the solutions of the equations of motion in terms of ’’representatives’’ which involve only one‐half of the integration constants, the other half being rather unimportant physically, and it allows one to start up the representative motions from representative initial conditions. It also leads to a characterization of phase behavior which in particular describes not only the constant‐amplitude motions but also their dynamical neighborhoods, and in many cases it explains gross features of the motion such as the occurrence of Lissajous‐like patterns and orbit reversals.

Structure results for the Segal quantization of Fermi systems
View Description Hide DescriptionIn Segal’s approach to linear Fermi quantum systems, a one particle picture with linear symmetries (e.g., with a linear dynamics) can be quantized very straightforwardly when a complexification is given for the (real linear) one particle picture. We examine how the symmetries that are embodied in the one particle picture can determine the structure of the family of the possible complexifications. Among other results, we prove that if the symmetries can be represented in a suitably irreducible way then the complexification is essentially unique. Also, when the one particle space is a generalization of the one defined by the Dirac equation, we prove that there are many complexifications, and inequivalent too, as they generate inequivalent representations of the canonical anticommutation relations; however, we find two criteria that single out the ’’physical’’ complexification. We use the general results we prove to discuss a few familiar models.

Measurement systems and Jordan algebras
View Description Hide DescriptionAn axiomatization of the measuring process leads to a Jordan algebrastructure on the observables. The novel features in this development include a proof of the existence of the sum of observables, a proof of the quadratic nature of the square of an observable, the lack of a finite dimensionality assumption, and the exploitation of the change in the measuring process due to a change in the counting observable.

Upper and lower bounds to zeroth order Coulombic hyperangular interaction integrals
View Description Hide DescriptionIn recent years it has been shown that the use of hyperspherical coordinate representation of the Schrödinger equation for electrically charged particles necessitates the evaluation of certain kinds of hyperangular interaction integrals. The analytic evaluation of rather simple cases has also been accomplished. On the other hand certain numerical devices have been utilized and the complications that have arisen have been discussed for their computation. We have attempted in this work to find upper and lower bounds for all types of zeroth order hyperangular interaction integrals having Coulombic potentials. A nesting procedure has been developed for obtaining close bounds which can possibly be used to evaluate the desired value of integrals under consideration. In this context a theorem has been established for the evaluation of a similar type of integral and possible ways towards the generalization of the theorem have also been discussed. For three particle systems some applications have also been presented.

Mass dependence of Schrödinger wavefunctions for an exponential potential
View Description Hide DescriptionIt is shown that the condition G(r)⩾0 for r⩾0, where G(r) = (∂/∂m)∫^{ r } _{0}(u(r))^{2} d r, does not hold in the case of exponential type potentials.

On the effect of space‐time isometries on the neutrino field
View Description Hide DescriptionWe consider a neutrino field in interaction with a space‐time admitting an isometry group and we attempt to derive the symmetries imposed on the neutrino flux‐vector and on the neutrino field for solutions of the Einstein‐Weyl field equations. It is proved that if one of the following two constraints is imposed, (i) the neutrino field is of class E _{1}, (ii) the neutrino flux‐vector is collinear with one of the principal null directions of the Weyl tensor, so that Ψ_{0} = 0, then L_{ n }ξ^{ A } = −(i/2)sξ^{ A } and L_{ n } l ^{μ} = 0, where ξ^{ A } is the neutrino field, l ^{μ} is the neutrino flux‐vector, n ^{μ} is a Killing vector field, and s is a real constant. However, in the cases of a pure‐radiation field with diverging rays and a pure‐radiation field with nondiverging rays and Ψ_{3} = 0 the above formulas become L_{ n }ξ^{ A } = (1/2)(p−i s)ξ^{ A } and L_{ n } l ^{μ} = p l ^{μ}, where now p and s are in general real functions of the coordinates.

The Melnick–Tabensky solutions have high symmetry
View Description Hide DescriptionMelnick and Tabensky recently gave a class of static perfect fluid solutions of Einstein’s equations in which the metric (in co‐moving coordinates) takes the conformastat form. In this paper we point out that all these solutions have spherical or plane or pseudospherical symmetry.

Manifestly covariant equations of motion for a particle in an external field
View Description Hide DescriptionA relativistic variational principle for a particle in an external field is developed both in flat spacetime and in curved spacetime. In flat spacetime Kalman’s equations follow from the variational principle, and their relationship to four‐dimensional Euler–Lagrange equations is clarified. It is shown that Kalman’s equations are uniquely defined and that they may be recast into a generalized Hamiltonian formalism. The equations of motion arising from the curved‐spacetime variational principle are shown to be uniquely defined.

Conditional symmetries in parametrized field theories
View Description Hide DescriptionIn parametrized field theories, spacelike hypersurfaces and fields which they carry are evolved by a Hamiltonian which is a linear combination of the super‐Hamiltonian and supermomentum constraints. We say that a dynamical variable K generates a conditional symmetry of the Hamiltonian when it is linear both in the hypersurface and the field momenta and its Poisson bracket with the Hamiltonian vanishes by virtue of the constraints. Generators are classified by their dependence on the momenta: P‐restricted generators depend only on the hypersurface momenta, π‐restricted generators depend only on the field momenta, while mixed generators depend on both kinds of momenta. Conditional symmetries in a parametrized Hamiltonian theory are then linked either with ordinary symmetries (isometries, conformal motions, or homothetic motions) of the spacetime background, or with internal symmetries of the fields. In particular, we prove that a generic field with nonderivative gravitational coupling and a quadratic energy density has a P‐restricted conditional symmetry if and only if the spacetime background has a Killing vector, while a field with a trace‐free energy–momentum tensor has a P‐restricted conditional symmetry if and only if the background has a conformal Killing vector. An algorithm allowing us to enumerate all possible mixed conditional symmetries in a given parametrized field theory is explained on an example of the Klein–Gordon field. These results complement our previous proof that canonical geometrodynamics does not possess any conditional symmetry.

Time‐dependent embeddings for Schwarzschild‐like solutions to the gravitational field equations
View Description Hide DescriptionAn explicit formula for embedding the Schwarzschild solution in a three‐dimensional flat space with indefinite metric for arbitrary Kruskal timelike coordinate v is presented. The time development of the Schwarzschild solution can then be represented by a succession of spacelike surfaces, each corresponding to a different value of v. It is seen that the standard representation of the Schwarzschild metric, the Flamm paraboloid, is in fact the v = 0 special case of a similar time‐dependent embedding in a three‐dimensional Euclidean space with positive definite metric. However, this embedding is inadequate in that it is not defined for most values of v. Thus, the embedding in a space with indefinite metric is to be preferred. The results for the Schwarzschild case are found to be readily extended to all metrics of a certain class, and a general embedding formula for arbitrary v results. Embeddings for the Schwarzschild, de Sitter, and Reissner–Nordström metrics are then special cases of this general form. It is seen that all such solutions behave similarly as v gets large. This suggests an alternate interpretation of the oscillatory character of the Reissner–Nordström ’’wormhole.’’

Spatial topology and Yang–Mills vacua
View Description Hide DescriptionWe study the canonical vacuum structure of Yang–Mills theories defined on an arbitrary (nonsimply connected) three‐space. We find that the presence of flat Yang–Mills connections with a nontrivial (discrete) holonomy group has profound consequences at the quantum level. In particular, such connections may lead to either an increase or a decrease in the number of quantum vacuum sectors. Our method consists of finding a representation for the space of classical zero‐energy field configurations in terms of a function space D. A simple assumption concerning the physical equivalence of these classical configurations then permits a formal classification of the quantum vacua by the zeroth homotopy set π_{0}(D). Significant progress is made in the analysis of π_{0}(D) for arbitrary three‐spaces and gauge groups, and several specific questions concerning the vacuum states and their diagonalization are answered.

Boson operator representation of Brownian motion
View Description Hide DescriptionIn the framework of the classical theory of Brownian motion the time evolution of the distribution function in the full phase space of a particle immersed in a fluid is governed by a Fokker–Planck equation. The reduced distribution function in coordinate space fulfills the Smoluchowski equation in first approximation. This work improves previous derivations by including higher order corrections and by using an expansion which permits the discussion of the size of the error made by truncating the infinite series. The derivation is based on the adaption of a powerful mathematical tool used in quantum field theory: The Fokker–Planck equation is written in terms of boson operators. Conditional equilibrium averages of operators are defined which play the role of vacuum expectation values. The time‐ordered cumulant expansion is used to calculate the formal diffusion operator in terms of conditional equilibrium averages of powers of the ’’Liouville operator in the interaction picture.’’ It is shown that all these averages can be obtained from a G a u s s i a ngenerating functional which is explicitly calculated using the time‐ordered version of Glauber’s theorem. The resulting diffusionequation, a fourth order partial differential equation in the position space, is obtained by calculating the cumulant expansion up to sixth order. Conditions on the potential are established which guarantee that these equations are dissipative and it is shown that all solutions approach the Boltzmann distribution as t→∞. Curvilinear, non‐Euclidean coordinates are introduced in order to interpret these diffusionequations. Nonlinear diffusionequations and their application regarding the self‐avoiding random walk are discussed.

Lattice dynamics, random walks, and nonintegral effective dimensionality
View Description Hide DescriptionDefinitions of the nonintegral effective dimensionality of recursively defined lattices (fractal lattices) may be based on scaling properties of the lattices, or on the qualitative behavior of cooperative phenomena supported by the lattices. We examine analogs of these definitions for regular (i.e., periodic) lattices supporting long‐range interactions. In particular, we show how to calculate a harmonic oscillator effective dimension, a scaling dimension, and a random walk effective dimension for simple cubic lattices with a class of long‐range interactions. We examine the relationship between these three dimensions for regular lattices, and conjecture a constraint on the analogs of these dimensions for fractal lattices.