Volume 17, Issue 8, August 1976
Index of content:

Quantization from the algebraic viewpoint
View Description Hide DescriptionWe use the Weyl quantization W in a general context valid for any finite‐dimensional Lie algebraG, to derive an explicit formula for (P _{1},P _{2}) ≡W ^{−1}([W (P _{1}),W (P _{2})]), P _{1},P _{2}polynomials. In the particular case of the Heisenberg Lie algebra, this formula reduces to the familiar Moyal bracket.

Cross sections for screened potentials
View Description Hide DescriptionWe consider scattering by screened Coulomb and screened short‐range potentials. We prove, in Born approximation, that the cross section for a sharply cutoff Coulomb potential does not converge to the Rutherford cross section as the screening radius ρ→∞. On the other hand, for certain ’’smooth’’ screening functions we show that the screened Coulomb cross section does approach the Rutherford cross section as ρ→∞. In the case of short‐range potentials, we prove (exactly, not just in Born approximation) that the screened cross section approaches the unscreened cross section as ρ→∞, whether the screening function is smooth or not.

Branching rules and Clebsch–Gordan series for E _{8}
View Description Hide DescriptionBranching rules for the second lowest dimensional representation of the exceptional simple Lie algebra of type E _{8} are given with repsect to all its 14 maximal semisimple subalgebras. This representation of E _{8} is of dimension 3875, the maximal subalgebras are of types A _{8}, D _{8}, A _{1}‐A _{7}, A _{1}‐E _{7}, A _{2}‐E _{6}, A _{3}‐D _{5}, A _{4}‐A _{4}, A _{1}‐A _{2}‐A _{5}, G _{2}‐F _{4}, A _{1}‐A _{2}, C _{2}, and three nonconjugate subalgebras all of type A _{1}. The Clebsch–Gordan series, necessary for decomposition of the direct product of three representations of dimension 248, are given.

The state labeling problems for SO(N) in U(N) and U(M) in Sp(2M)
View Description Hide DescriptionIt is shown that, in a boson representation, the operators whose eigenvalues serve to label representations of SO(N) in U(N) also serve to label representations of U(M) in Sp(2M). The problem of labeling U(2) in Sp(4) is considered in detail, and it is shown how to find labeling operators with rational eigenvalues, depending, however, on the representation. The solution of this problem is shown to provide a solution of the equivalent problem of the labeling of SO(3) in U(3).

Particle permutation symmetry of multishell states. I. Two shells
View Description Hide DescriptionA method is developed for constructing N‐particle states of definite symmetry from n _{1}‐particle and n _{2}‐particle states of definite symmetry where N=n _{1}+n _{2}. A canonical resolution of the attendant multiplicity question is given. The results, which are a first step toward the construction of appropriate coefficients of fractional parentage, do not rely upon any particular form for the N‐particle Hamiltonian. Rather, the results are based entirely upon properties of the symmetric groups S_{ n } _{ 1 }, S_{ n } _{ 2 }, and S_{ N }. The group theoretic problem which is the construction of irreducible representations of S_{ N } from those of S_{ n } _{ 1 } ×S_{ n } _{ 2 } is solved using induced representation theory together with projection operator techniques.

Symmetry and invariance properties of the Boltzmann equation on different groups
View Description Hide DescriptionThe introduction of the group theory in the treatment of the Boltzmann equation shows the reducibility of the collision integral operator on the invariant subspaces of Klein V or SO_{2} group. Especially we prove the equality of matrices representing the collision integral operator between inequivalent subspaces first in its linear form and then in its general form. These results are finally expanded to the full Boltzmann equation when we consider its properties as a whole in the phase space (E_{r}×E_{v}). This brings back Boltzmann equation following the Chapmann–Enskog process to the differential equation system depending solely on the variable ‖r‖. The examination of the Boltzmann equation symmetries allows us to obtain the selection rules which lead to an important simplification in theoretical as well as numerical calculations of the distribution function.

One‐component plasma in 2+ε dimensions
View Description Hide DescriptionThe one‐component plasma (ocp) model with neutralizing background is extended to real dimensionality ν=2+ε with −2⩽ε⩽2. The equilibrium properties (pair correlation and thermodynamic functions) investigated within the Debye approximation, up to the second‐order in the plasma parameter e ^{2}/k _{ B } Tλ^{ε} _{ D }, with the aid of the Wilson quadratures, interpolate between two‐ and three‐dimensional results for 0<ε<1, and extend the ν=3 behavior to all ν⩽2. The dimensionality ν=2 is shown to play a special role. Quantum diffraction corrections are included in the high temperature limit through a temperature‐dependent effective Coulomb interaction. As a by‐product, the particle diffusion coefficient (Bohm) of the strongly magnetized two‐component plasma taken in the fluid limit may be given a finite volume‐independent expression in the thermodynamic limit when ν=2, provided due attention is paid to the Tauberian properties of the Coulomb potential for −2⩽ε⩽0.

Stability and instability conditions for nonlinear evolutional equations in Hilbert spaces
View Description Hide DescriptionSufficient conditions for stability, global asymptotic stability, and explosive instability are established for a class of nonlinear evolutional equations defined in Hilbert spaces by using certain relations between an abstract function and its Gateaux differential. These results are applied to specific forms of nonlinear evolutional equations arising from physics, in particular, a finite‐dimensional system of complex ordinary differential equations, functional differential equations, and systems of complex partial differential equations describing nonlinear diffusion or wave phenomena.

Relativistic multiple scattering of electromagnetic waves
View Description Hide DescriptionA general, relativistically exact formulation of scattering of electromagnetic waves by two or more objects in relative, uniform motion is presented. Coupled and separated integral equations are obtained for far‐field scattering amplitudes of bodies within a system in terms of known far‐field scattering amplitudes of the isolated obstacles. The integral equations are accessible for iteration analogous with the case of nonmoving objects. Explicit expressions and numerical results are given for the two‐dimensional problem of two parallel, perfectly conducting cylinders.

Nonspreading solutions to a class of differential equations
View Description Hide DescriptionExplicit nonspreading (i.e., characteristically propagating) solutions to a certain class of two‐dimensional, hyperbolic, differential equations are found. This class of equations is a generalization of equations that arose in a study of radiation in cosmological backgrounds.

Radiation in cosmological backgrounds
View Description Hide DescriptionThe purpose of this investigation is to find the conditions for characteristic propagation of multipole radiation in Friedmann backgrounds. The radiation fields studied are Klein–Gordon scalar fields, conformally invariant scalar fields,electromagnetic fields, and gravitational fields. The behavior of electromagnetic and conformally invariant scalar radiation is similar to that of the corresponding radiation in flat space‐time, since both fields satisfy conformally invariant equations and the Friedmann backgrounds are conformally flat. Thus characteristically propagating solutions are possible for both fields in any Friedmann background. For the Klein–Gordon and gravitational fields, it is found that characteristic propagation is possible only for special Friedmann backgrounds. Two physically important Friedmann backgrounds, those for which P=0 and P=ρ/3 (where P is pressure and ρ is density), are among these special backgrounds for both types of radiation. In the course of this study, all Friedmann backgrounds for which P=αρ, where α is an arbitrary constant, are found; the methods used and the resulting solutions are much simpler than those previously given by Tauber.

Symmetry breaking interactions for the time dependent Schrödinger equation
View Description Hide DescriptionA systematic study of the symmetry porperties of the Schrödinger equationu _{ x x } + i u _{ t } = F (x,t,u,u*) is performed. The free particle equation (for F=0) is known to be invariant under the six‐dimensional Schrödinger group S_{1}. In this paper we find all continuous subgroups of S_{1} and for each subgroup we construct the most general interaction term F (x,t,u,u*), reducing the symmetry group of the equation from S_{1} to the considered subgroup. Since we allow for an arbitrary dependence of F on the wavefunctionu (and its complex conjugate u*) the considered Schrödinger equation is in general a nonlinear one [the ordinary Schrödinger equation with a time dependent potential is recovered if F (x,t,u,u*) =u G (x,t)]. For each symmetry breaking interaction F the remaining symmetry group is used to obtain special solutions of the equations or at least to separate variables in the equation and to obtain some properties of the solutions.

SU(2) × SU(2) scalars in the enveloping algebra of SU(4)
View Description Hide DescriptionWe build an integrity basis for the SU(2) × SU(2) scalars belonging to the enveloping algebra of SU(4). We prove that it contains seven independent invariants in addition to the Casimir operators of SU(4) and SU(2) × SU(2). We form a complete set of commuting operators by adding to the latter two linear combinations of the former the operators Ω and Φ first introduced by Moshinsky and Nagel. We then solve the state labeling problem that occurs in the reduction SU(4) ⊆ SU(2) × SU(2) by diagonalizing simultaneously Ω and Φ. Their eigenvalues are calculated numerically in all irreducible representations of SU(4) that are encountered in light nuclei up to and including the s–d shell. Finally we build the propagation operators for the widths of the fixed supermultiplet, spin and isospin spectral distributions by taking appropriate linear combinations of SU(2) × SU(2) invariants of degree less than or equal to four, and we tabulate the averages of these operators in the above‐mentioned irreducible representations of SU(4).

Canonical realizations of the Poincaré group. II. Space–time description of two particles interacting at a distance, Newtonian‐like equations of motion and approximately relativistic Lagrangian formulation
View Description Hide DescriptionThe physical meaning of the relativistic action‐at‐a‐distance dynamics for two particles in a canonical framework is investigated on the basis of a general formalism introduced in previous works. Starting from the well‐known prescription given by Bakamjian and Thomas in terms of ’’center‐of‐mass’’ (Q,P) and ’’internal’’ (ρ,π) canonical coordinates, we show how to construct p h y s i c a l, i.e., covariant, position vectors x_{τ} (Q, P, ρ, π) (τ=1,2) which approach the free particle coordinates in the limit ρ→∞ for short range forces; this procedure is actually performed by means of a 1/c ^{2} power expansion for any interaction potential U (ρ,π). In force of the zero‐interaction theorem the physical coordinates, which do satisfy the world‐line condition to any order in 1/c ^{2}, cannot play the role of canonical variables, i.e., the ’’localizability,’’ {x _{τi }, x _{τj }}=0 (τ=1,2), and the ’’causality’’ conditions {x _{τi }, x _{τ′j }}=0 (τ,τ′=1,2; τ≠τ′) cannot be simultaneously satisfied. It is possible, however, to satisfy the former set of equations to any order in 1/c ^{2} by exploiting the arbitrariness lying in the definition of x_{1} and x_{2}. By means of a suitable choice of a ’’gauge’’ for the internal variables, the remaining freedom is then shown to consist of the appearance of a single scalar function Λ (ρ,π). This function, entering the defining relations of x_{1}, x_{2} in terms of the canonical variables Q, P, ρ, π, plays the role of an additional interaction potential which is effective for the space–time description of the particles in the interaction region, but does not affect the scattering properties of the system. On the other hand, assuming a static nonrelativistic limit of the canonical potential, U ^{(0)}=U ^{(0)}(ρ), the ’’causality’’ conditions are necessarily violated at the order of the radiation effects (1/c ^{4}). In terms of x_{1}, x_{2}, the equations of motion assume a Newtonian‐like structure m _{τ}ẍ_{τ}=F_{τ}[x_{1}−x_{2},v_{1},v_{2}] (τ=1,2), of the Currie type or a variety of manifestly covariant forms m _{τ} d ^{2} x ^{μ} _{τ}/d s ^{2} _{τ}=S ^{μ} _{ν} f ^{ν} [x _{1}(s _{1}),x _{2}(s _{2}),u _{1}(s _{1}),u _{2}(s _{2})], where S ^{μ} _{ν} is the Lorentz transformation which connects the laboratory frame with the Lorentz frame in which x _{1}(s _{1}) and x _{2}(s _{2}) are simultaneous. A final point is the derivation of the Newtonian‐like equations of motion from a true Lagrangianvariational principle δFL [x_{1},x_{2},v_{1},v_{2}]d t=0. It is shown in general that if U ^{(0)}(ρ) ≡0, this can be done only up to the post‐Newtonian approximation, essentially because of the violation of the ’’causality’’ conditions at the order 1/c ^{4}. Then a general form of approximately relativistic Lagrangian for two particles is derived which actually contains a l l the examples quoted in the literature, among which the well‐known Darwin–Breit and the Einstein–Infeld–Hoffmann Lagrangians. This investigation appears to disprove the widespread opinion according to which the zero‐interaction theorem prevents the existence of invariant world lines and/or renders the relativity principle vacuous within a Hamiltonian framework.

Asymptotic radiation from spinning charged particles
View Description Hide DescriptionA manifestly covariant expression for the asymptotic energy–momentum and angular momentum emitted by a charged spinning particle in arbitrary motion is found. A center of energy theorem is discussed and it is also shown that, for spinning particles, the radiation rate is not an invariant.

Exact closed evolution equation for the electron density operator averaged over impurity configurations
View Description Hide DescriptionThe exact Markoffian evolution equation for the electron density operator n (t) averaged over impurity configurations (describing a grand ensemble of non‐interacting electrons in the potential fields of impurities fixed in space) is not closed. The interaction term, that is, the integral involving the electron‐impurity interaction and the electron‐impurity density operator is analyzed with the aid of diagrams after its evolution operator is expanded in perturbation and its initial density operators are expanded in terms of correlation operators. A closed non‐Markoffian equation for n (t) is obtained without introducing any approximations. This equation contains infinite sets of collision and initial correlation terms which are most conveniently represented by connected diagrams, and which can be expressed in terms of n (t) and initial correlation operators arbitrarily given. The equation is found to be identical with the closed evolution equation obtained earlier in the fixed‐particle‐number ensemble theory with the bulk limit.

Sum rules for the optical constants
View Description Hide DescriptionA number of sum rules for the optical constants, in particular, the refractive index of a nonconducting medium, are obtained. Some of the sum rule constraints are highly damped for large frequencies, exponentially in one particular case. Formal integral relationships for the index of refraction at complex frequencies are presented. Sum rules based on known experimental points for which n (ω)−1 has zeros are indicated. An outline of a modified derivation of some recently presented sum rules for the optical constants is given.

Ruelle’s cluster property for nonstrictly localizable fields derived from symmetry of their Wightman functions
View Description Hide DescriptionHermitian fields are considered which fulfill all the Wightman axioms, except local commutativity, formulated for the Gel’fand space S ^{1}(R^{4}) instead of the Schwartz space S (R^{4}). For fields of this class, containing the Jaffe class and even the larger class of essentially local fields on S ^{1}(R^{4}), it is shown that symmetry of the Wightman functions and mass gap imply Ruelle’s cluster property. Hence the Haag–Ruelle–Hepp scattering formalism applies to these fields, which need not be local.

Representations of the universal covering group of SU(1,1) and their bilinear and trilinear invariant forms
View Description Hide DescriptionThe unitary irreducible and many nonunitary representations of the universal covering group of SU(1,1) are given in a realization on certain spaces of functions. We discuss intertwining operators for these representations and their connection with the discrete series. The tensor product decomposition is performed by means of an integral transformation. A completeness relation for these integral kernels is derived.

Lagrange multipliers and gravitational theory
View Description Hide DescriptionThe Lagrange multiplier version of the Palatini variational principle is extended to nonlinear Lagrangians, where it is shown in the case of the quadratic Lagrangians, as expected, that this version of the Palatini approach is equivalent to the Hilbert variational method. The (nonvanishing) Lagrange multipliers for the quadratic Lagrangians are then explicitly obtained in covariant form. It is then pointed out how the Lagrange multiplier approach in the language of the (3+1) ‐formalism developed by Arnowitt, Deser, and Misner permits the recasting of the equations of motion for quadratic and general higher‐order Lagrangians into the ADMcanonical formalism. In general without the Lagrange multiplier approach, the higher order ADM problem could not be solved. This is done explicitly for the simplest quadratic Lagrangian (g ^{1/2} R ^{2}) as an example.