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Volume 8, Issue 12, December 1967

Some New General Relativistic Dust Metrics Possessing Isometries
View Description Hide DescriptionIn a four‐dimensional Lorentzian manifold in which Einstein's gravitational field equations hold with the field produced by pressure free dust, a significant set of new solutions is found. Assuming that the manifold possesses a four parametric group of isometries of type 5 in Bianchi's classification, which act on a three‐dimensional negative definite subspace, all metrics are found. The solutions are unusual in that the geodesics which the particles follow are not orthogonal to the three‐dimensional negative‐definite subspace so that the space will not appear homogeneous to these observers. The isotropic expansion is nonzero for almost all these solutions.

Invariant Imbedding as a Generalization of the Resolvent Equation
View Description Hide DescriptionInvariant imbedding equations for the Green's function of a general linear operator are shown to derive from a generalization of the resolvent equation of the theory of operators. Two applications are given: the neutron, or photon,transport in an arbitrary finite body, and the quantum‐mechanical theory of collision. In the last case, a nonlinear differential equation for the amplitude of transition T ^{βα} from a channel α to a channel β with conservation of the total energy is derived from the general equation.

Causality and the S Matrix
View Description Hide DescriptionA formalism is presented in order to impose causality conditions for finite‐range interactions which do not require the existence of wave packets with a sharp front. The absence of bound states and particle production is assumed. Two different causality conditions—referred to as strong and weak conditions—are studied. It is shown that in general the convolution kernel that connects the ingoing with the outgoing wave is not zero in the past. Some analytic properties of the S matrix are deduced.

On the Determination of the Many‐Channel Potential Matrix and S Matrix from a Single Function
View Description Hide DescriptionIt is shown that, for a certain model of many‐channel potential scattering, there exists, under rather general conditions, a single function from which both the potential matrix V and the corresponding S matrix can be constructed. This function is the Fredholm determinant of the Lippmann‐Schwinger equation for the physical wavefunction of a system having a potential matrix which is identical with V in the interval from the origin out to a distance r, but which vanishes identically beyond this distance. A one‐channel and a two‐channel example are discussed.

Statistical Mechanics of Dimers on a Quadratic Lattice
View Description Hide DescriptionThe partition function for the square lattice completely filled with dimers is analyzed for a finite n × m rectangular lattice with edges and for the corresponding lattice with periodic boundary conditions. The total free energy is calculated asymptotically for fixed ξ = n/m up to terms o(1/n ^{2−δ}) for any δ > 0. The bulk terms proportional to nm, the surface terms proportional to (n + m) which vanish with periodic boundary conditions, and the constant terms which reveal a parity and shape dependence are expressed explicitly using dilogarithms and elliptic theta functions.

Solution of the Dimer Problem by the Transfer Matrix Method
View Description Hide DescriptionIt is shown how the monomer‐dimer problem can be formulated in terms of a transfer matrix, and hence in terms of simple spin operators as was originally done for the Ising problem. Thus, we rederive the solution to the pure dimer problem without using Pfaffians. The solution is extremely simple once one sees how to formulate the transfer matrix.

Functional Derivatives and Vector Meson Fields
View Description Hide DescriptionMore general functional derivatives for vector mesons are introduced. As an application, a finite theory is formulated for neutral vector fields based on the ``strong'' Lorentz condition together with the other usual assumptions of asymptotic quantum field theory.

Generalization of the Determinantal Method to Continuous Channels
View Description Hide DescriptionThe procedure by which all elements of the S matrix in multichannel scattering problems are expressed in terms of the Fredholm determinant of the Lippmann‐Schwinger equation, is generalized to the case of ``continuous channels.'' This is a preliminary step toward its generalization to three (or more)‐particle problems, in which the possibility of ``ionizing'' an initial bound state always introduces such a continuum. It is found that, whereas in the case of discrete channels the necessary functions can be obtained from the Fredholm determinant either by analytic continuation in the total energy, or by a ``substitution rule,'' for continuous channels only the latter procedure works, and it is not equivalent to an analytic continuation.

Existence and Bifurcation Theorems for the Ginzburg‐Landau Equations
View Description Hide DescriptionThe second‐order transition of a superconducting material from normal to superconducting state according to the Ginzburg‐Landau theory is rigorously discussed. The bifurcation of a superconducting state is proved for both the Abrikosov mixed state and the case of a film in a parallel magnetic field when the flux or external field is slightly less than critical. The existence of a mixed state for all values of flux below the critical value is also proved.

Properties of the Wavefunction for Singular Potentials
View Description Hide DescriptionFor potentials more singular than the inverse square at the origin and with complex strength, the nature of the solutions to the Schrödinger equation is investigated. The difficulties occurring with attractive real potentials in the three‐dimensional equation are discussed, and it is shown that no solutions exist in a domain including the origin. For complex or repulsive singular potentials with radial form varying as the inverse fourth or sixth power, a relatively simple series solution exists. This is also true for the singular Yukawa with inverse fourth power. These series are shown to be asymptotic by means of general theorems from the theory of ordinary differential equations. Other inverse powers have much more complicated solutions. In the Dirac case, all inverse powers and singular Yukawa forms have asymptotic series solutions, and the series is given explicitly for the singular powers. A form of the S‐wave scattering length for the singular inverse fourth‐power Yukawa V = g ^{2} e^{−iΔ}e^{−μr}/r ^{4} is derived which is valid for small values of μg.

Bethe‐Salpeter Equation in Momentum Space
View Description Hide DescriptionThe Wick transformation in momentum space is modified to include all scattering energies by use of a coordinate surface which possesses limited detours into the complex relative energy plane. This device retains the simple form of the equation for the Bethe‐Salpeter amplitude ψ(p, p _{0}). It is shown that the transformation is valid if ψ(p, p _{0}) has the cut structure indicated by field theory, and this structure is shown to be consistent with the Bethe‐Salpeter equation provided the interaction V(x) satisfies a simple causality condition. It is further shown that the cut structure of a solution to the transformed equation can be deduced from the causal structure of the interaction alone without reference to field theory. Basic properties of the transformed equation are derived and a numerical treatment for purely elasticscattering is presented.

Dynamical Content of Differential Geometry
View Description Hide DescriptionA formulation of the notion of a Cartesian basis for a vector space suitably adapted to the local differential vector space at a space‐time point leads to this description of and derivation for the law of motion of a mass point: There exists a class of equivalent 4‐dimensional Cartesian frames in which the trajectory is a straight line, the velocity being constant along the trajectory. The lived‐in coordinate system, in which the trajectory is in general not a straight line, is in general not a Cartesian coordinate system. The equation for the motion of the point is purely a statement of differential geometry—an integrability condition—and the forces which are responsible for the deviation from constant velocity are immediately derivable from the metric tensor of the non‐Cartesian system. The equation of motion and the relations among dynamical variables are more relativistic than Newtonian, but the metric tensor is positive definite; no indefinite metric is needed. A Newtonian equation of motion is an approximation based on small velocity and it covers the general case of motion in a noninertial system in an external field of force. Intrinsic mass is identified as a constant of the motion and, although the relevancy of mass is somewhat limited in this description of the motion of a single point in an immutable force field, it is seen that the effective mass of the point contains a contribution by the potential energy.

Weak Correspondence Principle
View Description Hide DescriptionThe weak correspondence principle (WCP) for a scalar field states that the diagonal matrix elementsof a quantum generator necessarily have the form of the appropriate classical generatorG in which f(x) and g(x) are interpreted as the classical momentum and field, respectively. For a field operator φ(x) and its canonically conjugate momentum π(x) the states in question are given by,where 0〉 denotes the vacuum. The validity of the WCP is established for the six Euclidean generators (plus the Hamiltonian) of a Euclidean‐invariant theory, and for the ten Poincaré generators of a Lorentz‐invariant theory. Only general properties and certain operator domain conditions are essential to our argument. The WCP holds whether the representation of π and φ is irreducible or reducible; in the latter case, the WCP holds even if the vectors f, g〉 do not span the Hilbert space, or even if the generator is not a function solely of π and φ. Thus, the WCP is an exceedingly general and completely representation‐independent connection between a classical theory and its quantum generators which is especially useful in the formulation of nontrivial, Euclidean‐invariant quantum field theories.

Internal Dynamics of Particlelike Solutions to Nonlinear Field Theories
View Description Hide DescriptionWe set up a Rayleigh‐Ritz procedure for determining the approximate classical and quantum internal motion of a singularity‐free solution to a nonlinear field theory. The dynamical approximation procedure is applied to the particlelike solutions of a nonlinear modelscalar field theory and to the particlelike solutions of a class of nonlinear model scalar‐spinor theories. Dynamically stable, metastable, periodic, and unstable particlelike solutions all follow for certain ranges of the physical parameters from the classical theory, but only corresponding metastable and unstable associated quantum stationary states are obtained for the nonlinear modeltheories considered here. The functional dependence of the decay constant for the metastable quantum stationary states is such that very long‐lived, practically stable, states are admissible. Group‐theoretic techniques for the systematic derivation of rigorous particlelike solutions are also described and illustrated with examples.

Proof that Successive Derivatives of Boltzmann's H Function for a Discrete Velocity Gas Alternate in Sign
View Description Hide DescriptionIt has been conjectured by McKean that the particular property of Boltzmann'sH function which singles it out from a wide class of functionals of the Boltzmann solution may be that its successive derivatives alternate in sign. We consider here the proof of this alternating property for a discrete velocity gas. For the linearized‐model Boltzmann equation, the proof is trivial. For the full (i.e., nonlinear) model Boltzmann equation, the proof is shown to be equivalent to demonstrating the positivity of a particular polynomial. The proof of this property is then demonstrated. It is also shown that H ^{(n)}, like H ^{(1)}, is zero only for the equilibrium distribution.

Energy Requirement for Nonlinear Density Fluctuations in a Vlasov Plasma
View Description Hide DescriptionExact solutions of an isoperimetric problem are presented which lead to a lower bound on the kinetic energy of the particles in a Vlasov plasma, in terms of the mean‐density fluctuation:,where E is the kinetic energy per unit volume, n the mean number density,k the Boltzmann constant, T the temperature, and Δn the deviation of density. This energy requirement associated with Δn must be taken into account when considering the energetic aspect of the growth of instabilities. The present work, which relies entirely on mathematical analysis, confirms the main results of an earlier nonrigorous calculation.

Results on the Analyticity of Many‐Body Scattering Amplitudes in Perturbation Theory
View Description Hide DescriptionThe problem of analytic continuation of the many‐body scattering amplitude associated with a perturbation‐theory diagram under the rotation of the final momenta from real to complex momenta, k → (1 + iθ)k, is studied. It is shown that the contour of integration over internal momenta can be distorted avoiding singularities of the integrand, as θ varies for small enough θ. If the diagram is connected enough, the potentials are Yukawa‐type, Re E > 0, and Im E < 0. The rotation angle can be picked independently of Im E.

Daughter Regge Trajectory in a Field Theory Model
View Description Hide DescriptionAll contributions of ladder diagrams in a λφ^{3}theory with unequal masses which are asymptotically of order t ^{−2} ln^{ m } t are summed to give a family of secondary Regge trajectories. The analysis is carried out with the Mellin transformed scattering amplitude. The daughter Regge trajectory predicted by Freedman and Wang is identified. For equal internal masses on the sides of the ladder, the daughter pole has the following properties: (i) The trajectory function α(s) is constant when calculated to lowest order in the coupling constant; that is, the pole is fixed. (ii) The pole moves in higher order, but the trajectory function has no two‐particle cut. (iii) The residue of the pole has no two‐particle cut, and vanishes to all orders for equal external masses. The amplitude with unequal internal masses is considered in lowest order, and it is shown that three‐particle scattering should continue to dominate the daughter trajectory function although there is a two‐particle cut. The daughter pole moves towards the physical region less rapidly than the leading pole, and it develops a smaller imaginary part at the two‐particle threshold. It is concluded that a detailed statement about the motion of the daughter pole requires an accurate treatment of three‐particle scattering. As a by‐product of this work an earlier error in a treatment of the mixing of Regge poles and cuts is corrected.

Selection Rules and the Decomposition of the Kronecker Square of Irreducible Representations
View Description Hide DescriptionThe irreducible representations occurring in the decomposition of the Kronecker squares of irreducible representations of finite and continuous groups are shown to be readily separable into symmetric and antisymmetric parts using Littlewood's method of plethysm. Particular applications of the rotation and symplectic groups, together with selection rules for isoscalar factors, are given.

Exact Occupation Statistics for One‐Dimensional Arrays of Dumbbells
View Description Hide DescriptionExact relationships are developed which describe the occupation statistics for one‐dimensional arrays of dumbbells. It is shown that in the limit, as the number of compartments per array tends to infinity, these relationships reduce to those calculated, using the Bethe approximation when the number of nearest neighbors is two. A partition function, which includes the influence of the configurational correlation inherent in a one‐dimensional array of dumbbells, is also derived and discussed.