Volume 20, Issue 5, May 1979
Index of content:

Covariant n ^{2}‐plet mass formulas
View Description Hide DescriptionUsing a generalized internal symmetry group analogous to the Lorentz group, we have constructed a covariant n ^{2}‐plet mass operator. This operator is built as a scalar matrix in the (n;n*) representation, and its SU(n) breaking parameters are identified as intrinsic boost ones. Its basic properties are: covariance, Hermiticity, positivity, charge conjugation,quark contents, and a self‐consistent n ^{2}−1, 1 mixing. The GMO and the Okubo formulas are obtained by considering two different limits of the same generalized mass formula.

A critique of the major approaches to damping in quantum theory
View Description Hide DescriptionWe examine the two major approaches that have been suggested for the quantum mechanical treatment of the damped motion of a particle as a one‐body problem. These are the linear, but time dependent, Kanai Hamiltonian, and the more recent nonlinear potentials which have been introduced to simulate the damping force. The most important criticism that has been leveled at the Kanai Hamiltonian is that its solutions seem to violate the uncertainty relations. We show that this Hamiltonian actually represents a particle of variable mass, whose classical behavior is identical to that of a damped particle of constant mass. But quantum mechanically, its changing mass does lead to unphysical behavior when misinterpreted as a constant mass particle. So this Hamiltonian cannot directly describe a constant mass damped quantum particle. The nonlinear model has been interpreted in terms of the hydrodynamical analogy of quantum theory, and a well behaved decaying wavepacket solution has been produced. However we generalize this result to produce solutions that ’’decay’’ to arbitrarily high energy. Thus it is not clear that this model specifically treats dissipation. Rather it seems to seek out any stationary state. At any rate, its physical interpretation is obscure at present. However we show, by analyzing the physical problem of damping at low energies, that one can modify the Kanai Hamiltonian to eliminate its unphysical features, so that this modified Kanai Hamiltonian can in fact be interpreted as representing a constant mass damped particle with physically reasonable solutions.

A new approach to the problem of dissipation in quantum mechanics
View Description Hide DescriptionThe usual treatment of damping forces in quantum mechanics starts from the introduction of the explicitly time dependent Kanai Hamiltonian, which actually represents a variable mass particle, and the misinterpretation of this Hamiltonian as representing a particle of constant mass leads to certain physical difficulties. However the Hamiltonian can be modified so that it can be reinterpreted as describing a constant mass particle. Here we explicitly introduce the mass as a new dynamical variable, which allows us to write a linear, time independent Hamiltonian for the system, which can be solved by conventional methods. The damped harmonic oscillator and damped free particle are treated in detail, both for the Kanai Hamiltonian and for our case, and the solutions are compared. Our solution can be reduced to the Kanai one in appropriate circumstances, but in general it has a much greater versatility, as a result of which it can be more easily reinterpreted as describing a constant mass particle subject to a damping force, which reinterpretation is of course necessary if the method is to have practical applicability. We also show how such a reinterpretation can be carried out in detail by introducing a ’’dissipation variable’’, in terms of which one may avoid the concept of a variable mass altogether.

Upper bounds for the many‐channel Marchenko transformation operator and its derivatives
View Description Hide DescriptionWe specify the class of perturbative complex matrix potentials for which the corresponding many‐channel Marchenko type transformation operators are bounded and integrable. Our reference matrix potential contains Coulomb interactions, different threshold energies, and centrifugal potentials with different angular momenta. Estimates for the transformation operator and its derivatives are obtained; they enable us to improve our recent results and are necessary for the establishment of a unique solution to the ’’generalized Marchenko fundamental equation.’’ From the existence of an integrable transformation operator, the analyticity of the Jost solution as a function of k _{1} is deduced in the upper‐half of the physical k _{1} plane.

On conserved quantities in general relativity
View Description Hide DescriptionRecently, definitions of total 4‐momentum and angular momentum of isolated gravitating systems have been introduced in terms of the asymptotic behavior of the Weyl curvature (of the underlying space–time) at spatial infinity. Given a space–time equipped with isometries, on the other hand, one can also construct conserved quantities using the presence of the Killing fields. Thus, for example, for stationary space–times, the Komar integral can be used to define the total mass, and, the asymptotic value of the twist of the Killing field, to introduce the dipole angular momentum moment. Similarly, for axisymmetric space–times, one can obtain the (’’z‐component’’ of the) total angular momentum in terms of the Komar integral. It is shown that, in spite of their apparently distinct origin, in the presence of isometries, quantities defined at spatial infinity reduce to the ones constructed from Killing fields. This agreement reflects one of the many subtle aspects of Einstein’s (vacuum) equation.

Thermodynamical properties of a class of solvable statistical models for hadronic matter
View Description Hide DescriptionThe exact evaluation for the thermodynamicalproperties of relativistic quantum gas models for excited hadronic matter using a statistical picture is carried out from the analysis of the grand partition function for a general class of excitation spectra. Some special forms of these energy spectra for hadronic matter are discussed for the physical content and the relationship to the known classical limiting cases.

Applications of an algebraic quantization of the electromagnetic field
View Description Hide DescriptionWe quantize the full component electromagnetic potential so that a version of the Haag–Kastler axioms applies. The resulting quasilocal theory is viewed as a gauge theory in the spirit of Doplicher, Haag, and Roberts. Interaction with a classical current is described and found to yield no surprises. Lacking a rigorous scattering theory for the electromagnetic field, we construct a rigorous analog of the asymptotic Hilbert spaces for quantum electrodynamics proposed by Kulish and Faddeev and Zwanziger. On this basis we are able to find the representations of the gauge group that are associated with each charge sector in the asymptotic space. These representations correspond, in a sector of charge q, to the subsidiary condition ∂^{μ} A _{μ}(x) =−q D (x).

Formal analytic continuation of Gel’fand’s finite dimensional representations of gl(n,C)
View Description Hide DescriptionThe article contains three results: I. It is shown that among the 2^{−n } n! (n+1) ! discrete series of representations of the Lie algebra gl(n,C) of complex n×n matrices described in the literature, the majority are not representations at all. Thus for n=3 and 4 one has respectively 12 and 45 series of representations instead of 18 and 180. II. In addition to the p+1 discrete unitary series of representations of u(p,q) [the Lie algebra of the group U(p,q), p?q, and p+q=n] there exist other discrete series of gl(n,C) which become unitary when restricted to its real subalgebra u(p,q). For n=3 there are four such series all corresponding to the chain u(2,1) ⊆u(1,1) ⊆u(1); for n=4 there exist six such series for u(3,1) and four series for u(2,2). Furthermore, some of the gl(n,C) series whose restriction to the real case do not provide unitary representations in general, do contain (infinitely many) particular representations which are unitary. Such unitary representations are contained inside of two of the four series for n=3 and inside of seven of the 27 series for n=4. III. Some properties of indecomposable representations of the Lie algebras for the groups of inhomogeneous transformations are shown using the discrete series of gl(n,C).

The scaling limit of the φ^{2} field in the anharmonic oscillator
View Description Hide DescriptionWe prove that the rescaled and renormalized q ^{2} process for H _{ g }=[p ^{2}+g q ^{4}+(1−g) q ^{2}]/2 tends to the Gaussian process for the Harmonic oscillator as g tends to infinity.

Radiating Kerr–Newman metric
View Description Hide DescriptionA complete generalization of the Kerr–Newman solution to the nonstationary case is given. The possibility of associating the energy–momentum tensor with the electromagnetic field is discussed.

The semiclassical expansion of the anharmonic‐oscillator propagator
View Description Hide DescriptionThis paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum‐mechanical propagator corresponding to the quartic anharmonic‐oscillator potential, V=mω^{2} q ^{2}/2 +λq ^{4}/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt‐Morette, which does not entail the usual time‐slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small‐disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.

A general setting for reduction of dynamical systems
View Description Hide DescriptionThe reduction of dynamical systems is discussed in terms of projecting vector fields with respect to foliations of the manifold on which the dynamics take place. Examples of established reduction procedures are presented and are shown to be special cases of the general procedure described in this paper. Instances of other types of analysis of dynamical systems related to our projection procedure are briefly discussed.

On an inhomogeneous Schrödinger equation and its solutions in scattering theory
View Description Hide DescriptionWe prove that ψ_{ s,l }, the partial‐wave projection of the irregular Coulomb wavefunction ψ_{ s }, is a solution of an inhomogeneous Schrödinger equation. New expressions for ψ_{ s,l } and ψ_{ s } are obtained in terms of the Coulomb Green functionsG _{ C,l } and G _{ C }, respectively. We discuss irregular solutions, the analogs of ψ_{ s }, for Coulomb‐like and short‐range potentials. We find that in general these functions do not approach asymptotically the scattering amplitude times an outgoing spherical wave, in contrast to the pure Coulomb function ψ_{ s }.

Electromagnetic quadripotential for the pure‐radiation field generated by a classical charged point‐particle
View Description Hide DescriptionWe obtain a simple expression for the electromagnetic quadripotential corresponding to the pure‐radiation field generated by a classical charged point‐particle. The solution does not satisfy the Lorentz condition, and has interesting properties. It propagates inside the light cone of the particle and has a discontinuity across the sheet of the light cone itself. This discontinuity is responsible for the correct propagation of the electromagnetic effects with the velocity of light. A similar result is also obtained for the velocity‐dependent field.

On the complete integrability of the stationary, axially symmetric Einstein equations
View Description Hide DescriptionA linear eigenvalue problem in the spirit of Lax is constructed for the nonlinear differential equations describing stationary, axially symmetric Einstein spaces. In suitable variables these equations yield a generalization of the well‐known sine‐Gordon equation. The similarity of the system to the nonlinear σ model is pointed out.

An attempt to separate the long and short range forces by Gaussian method
View Description Hide DescriptionIn the study of phase transition problems, short range forces (SRF) play a dominant role. A constructive and rigorous study of the effects of short range forces has yet to be given. It is suggested in the present paper that by separating long range forces (LRF) from the short range forces, it would be possible to estimate contributions to the virial expansion of the collective oscillations due to short range forces. The method of Stratonovich o r the functional integration technique is employed in the treatment of the interaction term of the partition function.

S‐wave off‐shell T and K matrices for the Yukawa potential by Ecker–Weizel approximations
View Description Hide DescriptionExpressions for s‐state off‐shell wavefunctions associated with outgoing and standing wave boundary conditions are derived for the Yukawa potential by using Ecker–Weizel approximations. The results are used to relate the T and K matrix elements to tabulated transcendental functions.

The classical limit of quantum nonspin systems
View Description Hide DescriptionThe classical limit of operators X belonging to any compact Lie algebra g is computed. If X∈g, the classical limit in the representation Γ^{Λ}, whose highest weight is Λ, is lim Γ^{Λ}(X/N) =Σs _{ i } g (f_{ i },X,Ω), where the limit is taken as N→∞, the sum runs from i=1 to r=rank g, Λ=Σμ_{ i }f_{ i },f_{ i } are the highest weights of the r fundamental representations of g,s _{ i }=lim μ_{ i }/N, and g (f_{ i },X,Ω) is the expectation value of X with respect to the coherent states ‖f_{ i }, Ω〉 in the representation Γ^{f} _{ i }. Examples and applications are given.

Path integrals for waves in random media
View Description Hide DescriptionThe problem of wave propagation in a random medium is formulated in terms of Feynman’s path integral. It turns out to be a powerful calculational tool. The emphasis is on propagation conditions where the rms (multiple) scattering angle is small but the log‐intensity fluctuations are of order unity—the so‐called saturated regime. It is shown that the intensity distribution is then approximately Rayleigh with calculable corrections. In an isotropic medium, the local or Markov approximation which is commonly used to compute first and second (at arbitrary space–time separation) moments of the wave field is explicitly shown to be valid whenever the rms multiple scattering angle is small. It is then shown that in the saturated regime the third and higher moments can be obtained from the first two by the rules of Gaussian statistics. There are small calculable corrections to the Gaussian law leading to ’’coherence tails.’’ Correlations between waves of different frequencies and the physics of pulse propagation are studied in detail. Finally it is shown that the phenomenon of saturation is physically due to the appearance of many Fermat paths satisfying a perturbed ray equation. For clarity of presentation much of the paper deals with an idealized medium which is statistically homogeneous and isotropic and is characterized by fluctuations of a single typical scale size. However, the extension to inhomogneous, anisotropic, and multiple scale media is given. The main results are summarized at the beginning of the paper.

Higgs mechanism and the inverse Einstein–Infeld–Hoffman problem
View Description Hide DescriptionThe Guralnik–Hagen model for a self‐coupled spin −1/2 field is minimally coupled to a gauge gravitational field. The corresponding free gravitational Lagrangian is not explicitly introduced. It is shown that, using spontaneous breakdown of Lorentz invariance and the generalized Gordon decomposition, the Higgs mechanism generates a gravitational Lagrangian which leads to the usual linearized Einstein gravitational fieldequations.