Volume 13, Issue 3, March 1972
Index of content:

A Class of Nonlinear Realizations of the Poincaré Group
View Description Hide DescriptionA class of nonlinear space‐time transformations is exhibited, which forms a nonlinear realization of the Poincaré group. The transformations leave the expression I = x ^{2}+f(x ^{μ}/x ^{0}) invariant; f is an arbitrary function of the ratios x ^{μ}/x ^{0}. The infinitesimal generators are constructed as differential operators in the Minkowski space. The transformations are defined only in a restricted region (the allowed region) of the Minkowski space. By introducing auxiliary variables, the transformations can be recast in their usual linear form; this, however, is in general possible only in a region (the linear region) which is different from . The region structure is analyzed in general and given explicitly for a special form of the function f. Among the physical ideas suggested by the nonlinear formalism is the notion of ``relativity of coincidence.'' This expresses the fact that events coincident (or having arbitrary small Minkowski separation) in one frame of reference will not be coincident (or will have finite Minkowski separation) in a transformed frame.

Hamiltonian Formalism for General Lagrangian Systems in an Exceptional Case
View Description Hide DescriptionIt is shown that in the exceptional case of a Lagrangian system, where the highest derivative of a generalized coordinate cannot be expressed in terms of the lower derivatives and the generalized momenta, the procedure to adopt in setting up the Hamiltonian formalism for the system (and consequently the quantization of the system) is to pass to an equivalent Lagrangian which does not have this defect and to use this rather than the original Lagrangian.

Colliding Plane Gravitational Waves
View Description Hide DescriptionThe equations governing the collision of two plane gravitational waves are derived. The general exact solution representing this situation when both waves are linearly polarized are found, and some special solutions of possible physical interest are discussed in detail.

Random Walks in Polytype Structures
View Description Hide DescriptionIt is shown that the total number of walks, starting and ending at the same point and having the same number of steps, is the same for all polytype structures in the fcc, hcp series and in the zincblende, wurtzite series and is independent of the starting point. This result is proved by showing that the eigenvalues of a simple Hamiltonian are the same within the two series considered. A relation is found between random walks in the two series of structures that is useful in extending currently available tables of random walks for the zincblende structure.

Moments and Correlation Functions of Solutions of Some Stochastic Matrix Differential Equations
View Description Hide DescriptionThe (not necessarily linear) vector differential equationis first considered, where M(z) is a finite‐state Markov process which has, in general, a nonstationary transition mechanism. The joint process {u(z), M(z)} is a Markov process, and forward and backward Kolmogorov equations are derived for the transition probability density functions. Attention is then turned to the linear matrix differential equation,where W and γ are n×m matrices and A is an n×n matrix. The forward equations for the corresponding probability density functions are used to obtain two different, but equivalent, formulations for the calculation of the moments of any given order, and of the correlation functions, of the solution. The calculation of the moments and correlation functions is reduced to the solution of systems of linear ordinary differential equations, with prescribed initial conditions. The inhomogeneous matrix equationis also considered. Some applications, in particular to the calculation of the average modal powers in randomly coupled transmission lines, will be given elsewhere.

Projective Representations and the Relation of Internal to Space‐Time Symmetries
View Description Hide DescriptionThe relation of space‐time to internal symmetries in relativistic quantum mechanics is investigated using the Mackey theory of induced projective representations of group extensions. Representation multipliers are found for semidirect products of the Poincaré group and arbitrary internal symmetry groups. Upon investigating a typical example, it is found that when relations such as U _{π} U_{C} =(−1)^{2J } U_{C}U _{π}, obtained from field theory, are assumed, a unique choice of representation multiplier follows; in particular, this multiplier requires for all J. Representations relative to this multiplier are computed.

Position Operators as ``Internal'' Symmetries
View Description Hide DescriptionSpace‐time variables are generated as representation labels of an underlying group, the group itself being combined with the Poincaré group in a manner reminiscent of the way in which internal symmetries are combined with the Poincaré group. After representations of the group are found, a transform is introduced which allows one to pass from spinor to Wigner wavefunctions in a boost independent manner, exhibiting clearly the spin dependence of the wavefunctions.

On the Correlations of the Resonance Parameters for the Overlapping Resonances
View Description Hide DescriptionA statistical study of the correlations of the complex poles of the unitary collision matrix is carried out. It is shown that both for the elastic and the inelastic scattering the correlation coefficient of the two total widths is always very small. A simple relation satisfied by the correlation coefficient of the real parts of the complex poles is given. The distribution of the single width is calculated and compared with the Porter‐Thomas distribution and the one obtained by a numerical calculation. Some other interesting results, like the energy correlation function for the purely elastic scattering cross section and a relation satisfied by the resonance parameters for the fluctuation calculation, are also given.

Solution of the Difference Equations of Generalized Lucas Polynomials
View Description Hide DescriptionBarakat and Baumann have introduced polynomialsU^{(N)} (a _{1}, a _{2}, …, a_{N} ) termed the generalized Lucas polynomials satisfying a difference equation with a set of initial conditions. We show that these polynomials can be obtained directly from the symmetric functions h_{n} , which are of basic importance in combinatorial analysis. Moreover, we extend the definition of V(a _{1}, a _{2}) to V^{(N)} (a _{1}, a _{2}, …, a_{N} ) and establish that these polynomials too can be obtained from the symmetric functions S_{n} . Further, closed expressions for the U and V are obtained.

Normalization of Certain Higher‐Order Phase‐Integral Approximations for Wavefunctions of Bound States in a Potential Well. II
View Description Hide DescriptionUsing a formula derived from equations given by Furry for the normalization integral of the wavefunction corresponding to a bound state, we derive the normalization factor for the higher‐order phase‐integral approximations introduced by N. Fröman. The present treatment is based on the method developed by N. Fröman and P.O. Fröman, in which one uses exact formulas in the calculations and makes the approximations in the final stage. We particularize the resulting general formula to the case of a single‐well potential previously discussed by the present author and to the case of a double‐well potential, which has been treated in a series of papers from this institute.

Constants of Motion and Lie Group Actions
View Description Hide DescriptionA classical Hamiltonian dynamical system with 2N‐dimensional phase space is studied in the case when a Lie algebra of constants of the motion exists which contains 2N −‐ 1 functionally independent elements and when each constant of motion generates a complete integral curve. It is proved that a connected (global) Lie groupG acts on the phase space and acts transitively on each connected component of each surface of constant energy. When G is compact, each component of the space of time orbits corresponding to a fixed energy is shown to be a (2N − 2)‐dimensional compact symplectic manifold diffeomorphic to an orbit of G in the dual of the adjoint representation. It is shown that a (global) Lie group does act in the case of the harmonic oscillator, but does not act in the case of the negative energy Kepler problem.

Scattering from a Periodic Corrugated Structure. II. Thin Comb with Hard Boundaries
View Description Hide DescriptionThe scattered field is calculated for plane wave incidence on a periodic rectangularly corrugated surface (thin comb grating) with hard (Neumann) boundary conditions. Except for the hard boundary (and the consequent representation of the field in the comb wells), the formalism is similar to that of a previous paper [J. Math. Phys. 12, 1913 (1971)]. Reflection coefficients are plotted, grating anomalies illustrated, and a correspondence between reflection coefficients and amplitude phases (as a function of corrugation depth) is illustrated.

On Projective Unitary‐Antiunitary Representations of Finite Groups
View Description Hide DescriptionA generalization of Schur's treatment of projective representations is discussed for a very general class of representations occurring in physical theories: the projective representations by unitary and antiunitary operators. It is shown that for every finite group the projective unitary‐antiunitary (PUA) representations can be obtained from the ordinary unitary‐antiunitary representations of another finite group. The construction of such a representation group is treated. As an example we apply the theory for the determination of irreducible representations of subgroups of the Poincaré group. The classes of PUA representations for all finite crystallographic groups in spaces of dimension up to four are explicitly given.

Commuting Polynomials in Quantum Canonical Operators and Realizations of Lie Algebras
View Description Hide DescriptionIt is pointed out that the problem of realizing Lie algebras through polynomials in quantum canonical operators is not equivalent to its classical counterpart because the polynomialLie algebras taken with respect to the classical and quantum Lie brackets are not isomorphic. Yet there are still many results which are common to both. To show this, the properties of commuting polynomials in quantum canonical operators are analyzed. This makes possible an extension from the classical to the quantum domain of a number of theorems on realizations of semisimple Lie algebras. At the same time it is stressed that differences can arise in the classical and quantum solutions, and some of these are described.

Stochastic Master Equations: A Perturbative Approach
View Description Hide DescriptionLinear stochastic master equations for wave propagation in a continuous random medium are derived along the lines of the resolvent theory used in nonequilibrium statistical mechanics.Equations for the mean and fluctuating fields are subsequently obtained by operating directly on the stochastic master equations with statistical projection operators. The findings are compared with the results determined using the method of renormalization and the method of smoothing.

On Wave Propagation In Inhomogeneous Media
View Description Hide DescriptionFamiliar relations between phase and group velocity for wave propagation in homogeneous media are generalized to the inhomogeneous case. The constant velocity ``c'' is merely replaced by an appropriate weighted average. The key tool lies in the stationary property of the frequency as a functional of the wavefunction.

On the Interaction of the Electromagnetic Field with Heat Conducting Deformable Insulators
View Description Hide DescriptionThe differential equations and boundary conditions describing the behavior of a finitely deformable, polarizable, and magnetizable, heat conducting continuum in interaction with the electromagnetic field are derived by means of a systematic application of the laws of continuum physics to a well‐defined macroscopic model. The model consists of an electronic charge and spin continuum coupled to a lattice continuum, which in itself consists of two interpenetrating ionic continua, which can displace with respect to each other to produce ionic polarization. Since spin angular momentum and electronic and ionic linear momentum are taken into account, magnetic spin resonance and both ionic and electronic polarizationresonances are included in the treatment. Magnetic interaction terms are obtained by regarding magnetization as a consequence of point circulating current densities. When material resonances are suppressed, a simpler model is applicable and a not only smaller but somewhat different system of equations turns out to be convenient.

Quantum Theory of Heat Transport in an Isotopically Substituted, One‐Dimensional, Harmonic Crystal
View Description Hide DescriptionWe present a quantum mechanical treatment of thermal transport in a one‐dimensional isotopically substituted harmonic lattice. This work is an extension of a classical mechanical treatment. We find that the difference between the quantum and classical expressions for the thermal conductivity of a random chain vanishes in the limit N → ∞, where N is the number of isotopes. Thus, as in the classical treatment, the thermal conductivity diverges as N ^{1/2}. For a periodic diatomic lattice, we derive explicit formulas for the heat current as a function of temperature. At very low temperatures, this quantum mechanical current exhibits Kapitsa behavior.

Velocity‐Dominated Singularities in Irrotational Hydrodynamic Cosmological Models
View Description Hide DescriptionWe consider irrotational perfect fluid solutions of the Einstein equations with an equation of statep = γρ. We define ``velocity‐dominated'' singularities of these solutions, a notion previously introduced for dust models. We demonstrate explicitly that uniquely and invariantly defined inner metric tensor, extrinsic curvature tensor, and scalar bang‐time function can be assigned to these singularities, as in the dust case. We study the effects of a time‐varying equation of state and viscosities on these singularities, and show by order‐of‐magnitude estimates that they do not change the structure of the singularity provided γ > 0. Some known exact perfect fluid solutions, both homogeneous and inhomogeneous, are listed as examples.

A Method for Generating New Solutions of Einstein's Equation. II
View Description Hide DescriptionA scheme is introduced which yields, beginning with any source‐free solution of Einstein's equation with two commuting Killing fields for which a certain pair of constants vanish (e.g., the exterior field of a rotating star), a family of new exact solutions. To obtain a new solution, one must specify an arbitrary curve (up to parametrization) in a certain three‐dimensional vector space. Thus, a single solution of Einstein's equationgenerates a family of new solutions involving two arbitrary functions of one variable. These transformations on exact solutions form a non‐Abelian group. The extensive algebraic structure thereby induced on such solutions is studied.