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Volume 7, Issue 11, November 1966

Transformation of Proca‐Type System of Partial Differential Equations
View Description Hide DescriptionThe Proca‐type system of partial differential equations [−curl curl A + k A = τ grad φ; ▿^{2}φ − Kφ = τ div A, with k, K, and τ constants] is transformed into a new system such that the vector potential A and the scalar potential φ satisfy separate differential equations. It is also shown that, for the solution of the system, there are only two different modes, div A = 0 with φ = 0, and curl A = 0 with φ ≠ 0, respectively.

Relativistic Effects in Atomic Fine Structure
View Description Hide DescriptionOperators are obtained which can be evaluated with respect to nonrelativistic wavefunctions to produce the same result as obtained by evaluating the Breit equation with respect to relativistic wavefunctions. This greatly simplifies calculations involving the Breit equation by allowing the calculations to be made within the more familiar framework of nonrelativistic theory. The operators are classified according to their angular dependence; a comparison with the angular dependence of each fine‐structure operator leads to the relativistic equivalents of the fine‐structure interactions. The operators are expanded in a power series in (v/c)^{2}, and the lowest nonvanishing terms are shown to be the fine‐structure interactions.

The Validity of Perturbation Series with Zero Radius of Convergence
View Description Hide DescriptionAlthough the perturbation expansion for the S‐matrix of the Peres‐model field theory has zero radius of convergence, it uniquely defines the S‐matrix and is easily summable by the method of Padé approximants.

Unitary and Antiunitary Ray Representations of the Product of n Commuting Parity Operators
View Description Hide DescriptionThe unitary and antiunitary ray representations of the group of n commuting operators Γ_{ i }, i = 1, 2, …, n, satisfying Γ_{ i } ^{2} = 1, Γ_{ i }Γ_{ j } = Γ_{ j }Γ_{ i }, have been determined explicitly. The ray representations are shown to be homomorphic to the representations of a class of generalized Clifford algebras defined in the text. The enumeration of different algebras gives a complete classification of inequivalent ray representations. Whereas the irreducible vector representations of the group are all one‐dimensional, the irreducible ray representations lead to various multiplet structure of states ranging in dimensions from 1 to 2^{½n } (n even), or 2^{½(n−1)} (n odd), in powers of two. The arbitrary antiunitary case can be reduced to the case with only one antiunitary Γ.

On Irreducible Representation of a Class of Algebras and Related Projective Representations
View Description Hide DescriptionThe number and the dimensions of irreducible representations of a general class of algebras occurring in the projective representations of finite groups have been determined. The Lie subalgebras have been found, and the isomorphism between the quantum‐mechanical ray representations of finite groups and the fundamental representations of Lie algebras is shown.

Invariance and Conservation Laws in Classical Mechanics. II
View Description Hide DescriptionIn this paper, the invariances of the equation of motion of a classical particle, to coordinate and time translations, to scale transformations and inversions, and to Galilean transformations, are considered individually. Resultant conditions on the equation of motion are given, and, for invariance to the one‐parameter continuous transformations, it is shown that the equation of motion can be reduced from second to first order. Associated with each such reduction is a conservation law. The implications of the invariance of the system Lagrangian to these transformations are indicated, and the conservation laws, if any, associated with them. Some requirements on the Lagrangian for invariant equations of motion are also presented, and it is shown that the invariance of an equation of motion derived from a Lagrangian does not imply the invariance of that Lagrangian to the transformation. It is also shown that time‐translation invariance of the equation of motion does not always require conservation of the Hamiltonian.

Relativistic Spinor Formulation of Stokes Parameters with Application to the Inverse Compton Effect
View Description Hide DescriptionA formulation of the Stokes Parameters for light is found in terms of second‐rank antisymmetric spinors, and arbitrary Lorentz transformations are made by using the unimodular representation of the Lorentz group. A three‐component spinor form is established along with its accompanying representation of the Lorentz group, and the connection with the photon spin is explicitly shown. Differential Compton cross sections for any Lorentz frame are calculated. These are seen to be invariant only for particular polarization reference directions.

The Solution of a Stark‐Effect Model as a Dynamical Group Calculation
View Description Hide DescriptionA model for the Stark mixing of orbital states in the annihilation of particles at rest is solved by using the dynamical group techniques. The Schrödinger equation for the problem is given meaning in a representation of the three‐dimensional rotation group and reduced to a two‐dimensional linear homogeneous differential equation.

On the Connection between External and Internal Symmetries of Strongly Interacting Particles
View Description Hide DescriptionThe relation between the mass and spin‐parity J ^{p}, and internal quantum numbers of elementary particles, hints at a nontrivial connection between the external symmetry, namely the Poincaré group or its Lie algebra, and a so‐called internal symmetry. Studying this connection mathematically, we find that any extension of (resp. by) the Poincaré Lie algebraP by (resp. of) a semisimple Lie algebraX is equivalent to the trivial one, P ⊕ X. Moreover, if we are looking for a Lie algebra containing P and X in an economical and nontrivial way, namely what we call a (nontrivial) unification of P and X, we find restrictions on the possibilities of choice of X, which exclude compact internal symmetries. An explicit treatment of SL(3, C) as internal symmetry, related to the external symmetry by the unification process of Lie algebras, gives a mass formula which is in very good accordance with the experimental data, and can be theoretically interpreted by means of a so‐called ``classification principle''.

A Manifestly Covariant Description of Arbitrary Dynamical Variables in Relativistic Quantum Mechanics
View Description Hide DescriptionThe concepts of instantaneous observables and dynamical variables are analyzed and generalized to arbitrary spacelike hyperplanes. A formalism is developed which gives the basic equations of relativistic quantum mechanics for dynamical variables on arbitrary hyperplanes a manifestly covariant form. A covariant linear transformation on the Poincaré generators introduces the hyperplane generators which yield commutation relations displaying a clear separation of the kinematical and dynamical properties of dynamical variables. An axiomatic study of the center‐of‐mass position operator yields the uniqueness of the operator and completes the physical interpretation of the hyperplane generators. The Poincaré invariance and hyperplane independence of the scattering operator is related to asymptotic conservation laws in the hyperplane formalism, and finally, a nonlocal, hyperplane‐dependent, field theory of free spinless particles is considered.

The Asymptotic Theory of Cerenkov Radiation
View Description Hide DescriptionRecently developed methods of asymptotic analysis are applied to the problem of Cerenkov radiation. The mathematical description of this physical phenomenon is given by the integro‐differential system of equations for the electromagnetic field in a dispersive medium. The parameter λ is introduced into these equations, where λ is a characteristic frequency of the medium. It is for large λ that the asymptotic expansion of the electromagnetic field is sought. Isotropic, uniaxial crystalline and gyrotropic media are treated in detailed. The source function which appears in the field equations is taken to be quite general, e.g., it may be used to represent the current associated with any moving ``multipole'' source. By applying the method of stationary phase to an integral representation of the solution, the leading term of the asymptotic expansion of the electromagnetic field is obtained. More precisely, a parametric representation of the expansion is found in which certain space‐time curves called ``rays'' play a key role. An expression for the total energy of the radiation is then determined.

Complex Temperatures and Phase Transitions
View Description Hide DescriptionThe thermodynamic limit is considered for complex temperatures, and a picture of a phase transition, similar to the Yang‐Lee picture, is proposed. For certain cases a representation of the partition function as an infinite product is obtained. Some simple models are considered.

Characterizing Coherent States of the Radiation Field
View Description Hide DescriptionSimple proofs are given for two properties of a Bose field discovered recently in the quantum‐theoretic description of optical coherence. The first is the theorem of Glauber and Titulaer that first‐order coherence means that only one mode is excited. The second is the theorem of Aharanov, Falkoff, Lerner, and Pendleton that eigenstates of the annihilation operators are characterized by their ability to factor when the system is divided into two channels. The restriction of the latter to the case of a single excited mode is removed.

The Discontinuities of the Triangle Graph as a Function of an Internal Mass. II
View Description Hide DescriptionWe consider the discontinuities of the triangle‐graph amplitude as a function of an internal mass variable. These discontinuities are important, since they form the kernel of the Aitchison‐Anisovich integral equation, which is derived from the Khuri‐Treiman three‐body final‐state‐interaction dispersion relation. We evaluate the discontinuities by explicitly performing the Feynman α integrations. We also discuss their analytic continuations. Finally we consider the applicability of the Cutkosky rules to such an internal mass variable discontinuity. It is argued that these rules must be modified in two ways. One of these is straightforward, having to do with the appearance of spacelike masses. The other is more involved and is a consequence of the results of homology theory. We apply the modified Cutkosky rules to the triangle‐graph discontinuities and obtain the same results as found by the direct method, so confirming the modifications which we have made.

Continuous Degenerate Representations of Noncompact Rotation Groups. II
View Description Hide DescriptionThree principal continuous series of most degenerate unitary irreducible representations of an arbitrary noncompact rotation group SO(p, q) have been derived and their properties discussed in detail. The corresponding harmonic functions have been constructed.

Bi‐Orthogonality Relations for Solving Half‐Space Transport Problems
View Description Hide DescriptionThe method of singular eigenfunction expansions is applied to time‐independent, one‐speed, half‐space transport problems with anisotropic scattering. Adjoint eigenfunctions are constructed such that a set of half‐range bi‐orthogonality relations is valid. These relations lead to the expansion coefficients in a direct manner. The adjoint eigenfunctions are also used to express the half‐space albedo operator which relates the emerging angular density to the ingoing one.

Theory of Paramagnetic Impurities in Semiconductors
View Description Hide DescriptionIn this paper, a model of a paramagnetic impurity in a semiconductor (or of an F′ center in an alkali halide) is proposed. It is an exactly soluble form of the quantum‐mechanical 3‐body problem. Specifically, we deal with 2 interacting particles in any number of dimensions in an attractive external potential, and present the qualitative features of the resulting eigenvalues and eigenfunctions. We find algebraically the conditions for a magnetic moment to appear (e.g., for an F′ center to become unstable with respect to an F center) and discover that even a large 2‐body electronic repulsion U does not cause a moment to appear when the one‐electron bound state orbits about the impurity are sufficiently great. Conversely, in the case of small, tightly bound orbits, beyond a certain value of U, the impurity does in fact become magnetic in the ground state. Using the exact ground‐state solution, we show that a perturbation‐theoretic expansion in powers of U has a finite radius of convergence.

Acoustic Scattering from an Interface between Media of Greatly Different Density
View Description Hide DescriptionThe problem of acoustic scattering from a curved interface between two homogeneous media is formulated as two integral equations relating the normal velocity and the velocity potential at the interface. The equations are so chosen as to minimize coupling. When γ, the ratio of the densities, goes to zero, the equations decouple, one becoming the equation of a soft‐boundary problem, the other of a hard‐boundary problem. For small γ, an approximate solution is constructed by perturbation methods from solutions to the related soft‐ and hard‐boundary problems.

Integral Forms for Quantum‐Mechanical Momentum Operators
View Description Hide DescriptionThe usual differential form P _{0} for the quantum‐mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P _{0} = − i(g ^{−½}) ∂/∂q (g ^{½}), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P _{0} is not always self‐adjoint on the domain D of physically acceptable bound‐state wavefunctions, as a proper quantum‐mechanical operator should be. An integral form is proposed for P, defined by,where.The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P _{0} only at the end‐points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta‐function terms to P _{0}, but it suffers from none of the defects, since delta‐functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain D is presented as an appendix.

Existence of Particlelike Solutions to Nonlinear Field Theories
View Description Hide DescriptionA pseudovirial theorem is derived for time‐independent particlelike solutions of finite energy (singularity‐free and spatially localized time‐independent solutions) to field theories associated with an action principle. It is shown that a useful necessary condition for the existence of such particlelike solutions is generally obtainable as a corollary to the pseudovirial theorem. This necessary condition is in fact sufficient to preclude existence of any well‐localized particlelike solution for all but special field theories with the more common forms of algebraic interaction. On the other hand, strong satisfaction of the necessary condition can lead to modelfield theories with rigorous closed‐form particlelike solutions, as shown by example for a class of Lorentz‐covariant theories which feature a real scalar field in interaction with a two‐component complex Weyl spinor field. Some of the latter particlelike solutions to the scalar‐spinor theory are energetically stable with respect to spatial dilatations, hence likely to be stable in the dynamical sense. A counter example to the more general sufficiency of the strong satisfaction condition is presented, showing that strong satisfaction of the pseudovirial theorem's corollary does not always guarantee the existence of singularity‐free particlelike solutions.