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Volume 9, Issue 1, January 1968

Symmetrized Tensor ``Trace'' Operations
View Description Hide DescriptionExtracting the ``trace'' from a Cartesian tensor requires working with the whole tensor. Extracting the trace from a symmetrized tensor is shown to allow one to work with separate invariant subspaces (hence yielding more convenient analytical results), and thus, reduces the number of coupled equations to be solved. Symmetrized trace extraction is described for extended symmetrization procedures based respectively on modified Young symmetrizers, Wigner projection operator symmetrizers, and Young symmetrizers. Symmetrized traceless tensors usually need to be further symmetrized to obtain fully symmetrized traceless tensors. Also described is a method where each rank of trace extraction is performed in a separate step and is accompanied by a step of symmetrization. This method yields fully symmetrized traceless tensors and the least coupling of equations.

Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries
View Description Hide DescriptionIn this paper we develop in detail a unified method, referred to as the Related First IntegralTheorem, for obtaining ``derived'' first integrals (i.e., constants of the motion) of mass‐pole test particles with geodesic trajectories in a Riemannian spece. By this method, which is based upon a process of Lie differentiation, additional conservation laws in the form of mth order first integrals can be generated from a given mth order first integral (conservation law), provided the space admits symmetries in the form of continuous groups of projective collineations (which include affine collineations and motions as special cases). We give in tensor form a reformulation of the well‐known Poisson'stheorem on constants of the motion for particles with geodesic trajectories. We then show for this class of trajectories that, as a method for generating mth order first integrals from a given mth order first integral,Poisson'stheorem is a special case of the Related First IntegralTheorem. It is also shown that dependency relations between generically related first integrals obtained by the Related First IntegralTheorem are expressible in terms of the structure constants of the underlying continuous group of symmetries.

Reduction of Relativistic Wavefunctions to the Irreducible Representations of the Inhomogeneous Lorentz Group. Part II. Zero‐Mass Components
View Description Hide DescriptionIn a previous paper, we showed how wavefunctions which transform in a relativistic manner in configuration space can be expanded in terms of amplitudes, which for nonzero mass transform like the wavefunctions for irreducible representations of the proper, orthochronous, inhomogeneous Lorentz group. A simple algorithm was given to obtain the expansion. In the present paper, we extend the results to include zero‐mass amplitudes. It is shown that for wavefunctions which are required to transform under the homogeneous Lorentz group such that the matrices which involve the spinor indices are finite dimensional, the zero‐mass amplitudes transform under nonunitary representations of the inhomogeneous Lorentz group. However, it is possible to split up each such nonunitary representation into a part which corresponds to a unitary representation for finite spin and into a part which corresponds to an unphysical change of wavefunction. As examples of the technique, we consider wavefunctions which transform as an antisymmetric real tensor (i.e., as an electromagnetic field) as a four‐vector with and without the Lorentz condition, and as a Dirac spinor. The results offer interesting contrasts with the reductions of Part I where only nonzero‐mass components were considered. It is shown that the expansion of the present paper, when applied to the solution of Maxwell's equations, leads to an expansion in terms of photonwavefunctions and that the unphysical change of wavefunction is zero. For a real vector potential with the Lorentz condition (i.e., the electromagnetic vector potential), the expansion corresponds to the sum of an expansion in terms of photonwavefunctions and a wavefunction which sets the gauge of the vector potential. The nonphysical part of the transformation of the electromagnetic vector potential is merely a gauge change. Finally, solutions of the massless Dirac equation are expanded in terms of wavefunctions for massless particles of spin ½ for which the nonphysical part of the change is zero. In the present paper, we also show how invariant inner products are to be introduced, how negative‐energy representations can be replaced by positive‐energy representations (''antiparticles''), and show the connection with the usual canonical formalism. Finally, second quantization of the theory is given.

Padé Approximants and Bounds to Series of Stieltjes
View Description Hide DescriptionSeries of Stieltjes with nonzero radius of convergence R have been considered in this paper. It is well known that sequences of Padé approximants to these series may be defined which converge in the complex plane cut from −R to − ∞. It is shown that the Padé approximants satisfy inequalities between 0 and −R which are much more general than those already proved on the positive real axis. A new sequence of approximants is defined, which are closely related to the Padé approximants and which have very similar properties. The two sets of approximants may be used to determine the series of Stieltjes within certain limits for points on the real axis between 0 and −R, given only the first few coefficients of the power series expansion. The result is then extended to all points in the interior of a circle with center at the origin and radius R.

Invariants and Scalars of Compact Inhomogeneous Unitary Algebras
View Description Hide DescriptionThe independent invariants of the fundamental and adjoint inhomogeneous algebras are explicitly constructed. Some classification of inhomogeneous algebras is given and their scalars and invariants are discussed.

Number of Polynomial Invariants of Adjoint and Fundamental Compact Inhomogeneous Unitary Algebras
View Description Hide DescriptionThe number of invariants for two types of inhomogeneous Lie algebras is computed.

Unitary Representations of the Homogeneous Lorentz Group in an O(2, 1) Basis
View Description Hide DescriptionUnitary irreducible representations of the homogeneous Lorentz group,SO(3, 1), belonging to the principal series and containing integral angular momenta, are reduced with respect to the subgroup SO(2, 1). It is found that the representation {j _{0}, ρ} of SO(3, 1), where j _{0} is a nonnegative integer and ρ a real number, contains each representation of SO(2, 1) of the continuous class (nonexceptional and integral) twice, and each of the discrete representations of SO(2, 1) once, for k = 1, 2, ⋯, j _{0}. The latter representations are absent for j _{0} = 0. It is shown that the basis states of the representation (for k ≥ 2) lie in the domain of those generators of SO(3, 1) that are outside the SO(2, 1) subalgebra, while the states of the representations do not lie in this domain. It is further shown that from the point of view of the nature of this domain, the representations of SO(2, 1) are very intimately connected to the continuous class representations of SO(2, 1), and that these two discrete representations act as a bridge between the remaining discrete representations on the one hand, and the continuous class representations on the other.

n‐Particle Kinematics
View Description Hide DescriptionWe suggest a very simple method for handling relativistic n‐body kinematics. In fact the method works well for studying the n‐fold Kronecker products of the representations of many groups, including all compact Lie groups as a simple case. The method is explicitly covariant, and treats all particles symmetrically.

Weyl Transformation and the Classical Limit of Quantum Mechanics
View Description Hide DescriptionThe Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum‐mechanical expectation values as phase‐space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.

Gauge Fields with Noninvariant Interactions
View Description Hide DescriptionGauge fields interacting with source fields in a gauge‐noninvariant way are examined in a purely classical framework. The existence of conserved currents for the solutions of such theories follows from the properties of the free gauge fields rather than from the usual requirement of global gauge covariance. One of the simplest gauge‐noninvariant theories of electromagnetism is shown to be an extension of Dirac'stheory of classical clouds of charge. In this extension the clouds exhibit some quantum features. Gauge‐noninvariant interactions for Einstein's theory of gravitation are presented in which the properties of the sources are carried by the metric field. In the case of the Yang‐Mills field, internal symmetries for a real scalar source field are a consequence of the properties of the gauge field.

Green's Functions for the One‐Speed Transport Equation in Spherical Geometry
View Description Hide DescriptionSeveral problems in one‐speed neutron transport theory for spherically symmetrical systems are discussed. The singular eigenfunction expansion technique is used to construct a solution for a specific finite‐slab Green's function. This slab solution is then used to construct the finite‐medium spherical Green's function by extending the point‐to‐plane transformation concept. For the general case, the expansion coefficients are shown to obey a Fredholm equation, and first‐order solutions are obtained; however, in the infinite‐medium limit the solution is represented in closed form. In addition, the solution for the angular density in an infinite‐medium due to an isotropic point source is developed directly from the set of normal modes of the transportequation. A proof that the result so obtained obeys the proper source condition at the origin is given.

Values of the Potential and Its Derivatives at the Origin in Terms of the S‐Wave Phase Shift and Bound‐State Parameters
View Description Hide DescriptionThe framework of this paper is the quantum‐mechanical theory of nonrelativistic scattering of a particle by a spherically symmetrical, local, and energy‐independent potential. The principle results are explicit expressions for the values at the origin of the potential and of its derivatives in terms of the S‐wave scattering phase shift and of the parameters of the S‐wave bound states (if any). Previous work in this direction is reviewed, and different routes to the establishing of these results are discussed. Some remarks concerning the characterization of S‐wave physically equivalent potentials (i.e., potentials having the same S‐wave phase shift and S‐wave binding energies) are presented. The properties of the S‐wave phase shift and bound state parameters which are necessary and sufficient to ensure that the corresponding potential be an even function of r are ascertained. The analysis is restricted to potentials represented by holomorphic functions of r.

Relativistic Kinetic Equations Including Radiation Effects. I. Vlasov Approximation
View Description Hide DescriptionIn a preceding series of papers, we developed relativistic statistical mechanics of electromagnetically interacting particles. Here we derive a hierarchy of manifestly covariant equations for the reduced densities. This hierarchy is valid at order one in a parameter which is closely related to radiation phenomena. At the lowest order (absence of correlation between particles), we derive a kinetic equation which is a modification of the relativistic Vlasov equation. The added term introduces the effects of the emission of collective radiation. A dimensional analysis shows that this term is important when mc ^{2} ∼ kT, i.e., when the plasma is relativistic. The modified Vlasov equation is shown to be irreversible: an H theorem is proved. Several applications are studied: dispersion relations, hydrodynamical equations. We have also discussed the Einstein‐Ritz controversy on the connection between retarded actions and irreversibility.

Cluster Expansions in Many‐Fermion Theory. I. ``Factor‐Cluster'' Formalisms
View Description Hide DescriptionCluster development may furnish a powerful device for the calculation of the expectation values of the observables of a many‐fermion system with respect to dynamically correlated state vectors. The generalized normalization integral, a generating function for the required expectation values, is defined, and four of the many possible decompositions of this function into cluster integrals are explored. Two of these decompositions are slight extensions of the conventional ones of Iwamoto and Yamada and Aviles, Hartogh, and Tolhoek. The other two are product decompositions, leading to new ``factor‐cluster'' formalisms. A factor‐cluster expansion is applied to the evaluation of the n‐particle spatial distribution function.

Cluster Expansions in Many‐Fermion Theory. II. Rearrangements of Primitive Decomposition Equations
View Description Hide DescriptionThe ``factor‐cluster'' formalisms introduced in the preceding paper are used as a tool for a further development of the cluster theories proposed by Iwamoto and Yamada and by Aviles, Hartogh, and Tolhoek. The primitive decomposition characterizing each of the older formalisms is rearranged into the exponential of a series with a uniform N dependence in the limit of large N. In addition a ``linked cluster'' theorem is proven for the series comprising the exponent in the Iwamoto‐Yamada formalism. Our derivations, unlike those of earlier authors, are valid for all N.

Electromagnetic Scattering from an Inhomogeneous, Collisionless Plasma Cylinder
View Description Hide DescriptionAn analytical investigation is presented of the coherent electromagneticscattering at normal incidence from a quiescent, infinitely long, radially inhomogeneous, collisionless plasma cylinder. The inhomogeneity is characterized by an index of refractionn = [1 − (α/r)]^{½} such as would result from a line plasma source issuing plasma at a constant flow rate. Exact scatteringsolutions are obtained for both transverse electric and transverse magnetic polarizations of an incident plane wave. With k _{0} and a representing the free‐space wavenumber and cylinder radius, respectively, the partial radial wavefunctions and the scattered field are shown to depend markedly on the two nondimensional parameters β and η, where β = k _{0} a and η = k _{0}α. Numerical calculations of the differential scattering cross section for the TM mode show that forward scattering increases sharply with either increasing β or η although the increase is most sensitive to variations in β. The total scattering cross section is also shown to increase monotonically in the same fashion. Finally, it is observed that the scattering efficiency factor asymptotically approaches 2 in the limit as β and η both approach infinity, which is the same as that obtained for the homogeneous cylinder in the limit as β approaches infinity.

Scattering of Massless Scalar Waves by a Schwarzschild ``Singularity''
View Description Hide DescriptionThis paper investigates the scattering and absorption of scalar waves satisfying the equation in the Schwarzschild metric. This problem has been previously considered by Hildreth. We find, for a Schwarzschild mass m, the following cross sections in the zero‐frequency limit for s‐waves: σ(absorption) = 0, dσ/dΩ ≃ [c + ⅓(2m) ln (2mω)]^{2}, where c is a constant of order m. These results disagree with the previous calculation. We exhibit a method of solution for the equation. Its limiting (Newtonian) form, with suitable identification of the coefficients, is the problem of Coulomb scattering in non‐relativistic quantum mechanics. By demanding coordinate conditions which for large l allow the usual Coulomb results in a partial‐wave expansion, we are able to define a partial‐wave cross section. The (summed) differential cross section for small frequencies inherits the logarithmic behavior of the s‐wave part, which is the only contribution explicitly calculated. (The l ≠ 0 contributions and the behavior of the cross sections for ω ≠ 0 are qualitatively indicated.) Cosmological considerations are given which cut off this divergence.

Pair Distribution Function of a One‐Dimensional Hard Rod Gas
View Description Hide DescriptionThe pair distribution functionD _{2}(x _{1}, x _{2}) for an infinite system of one‐dimensional hard rods, each of length d, is derived directly from the expression for a finite system. In addition, for a finite system of N rods, it is shown that D _{2}(x _{1}, x _{2}) is constant in the region of translational invariance if x _{2} − x _{1} ≥ (N − 1)d. A relationship between D _{2}(x _{1}, x _{2}) and D _{1}(x) is also discussed.

Further Generalized Bose Condensation, Anisotropic ODLRO and Thin ^{4}He Films
View Description Hide DescriptionThis paper contains a detailed analysis of the Bose condensation of an ideal Bose gas, finite in one dimension and infinite in the other two. Limiting processes, which were treated questionably in previous work, are here re‐examined, and some seemingly paradoxical conclusions of the past work are explained. Two new concepts emerge—those of ``further generalized Bose condensation'' (FGBC) and ``anisotropic off‐diagonal long‐range order'' (AODLRO). The two concepts are shown to be connected to each other, and both are shown to exist at T = 0 for the model discussed.