Volume 15, Issue 1, January 1974
Index of content:

Renormalization and Ward identities using complex space‐time dimension
View Description Hide DescriptionComplex‐dimensional renormalization is defined for an arbitrary Feynman amplitude and shown to be equivalent to BPH renormalization. Using quantum electrodynamics as an example, Ward identities are proved; here Carlson's theorem extends the identities from integer to complex dimension. Both complex dimensional and analytic regularization are necessary at intermediate stages.

On a complex representation of Lorentzian random variables
View Description Hide DescriptionA Lorentzian random variable X is characterized by its mean value a and its width γ. We point out some interesting properties of the map . We illustrate the usefulness of this map by solving a particular problem in the theory of disordered materials.

An extension of Hamilton's principle to include dissipative systems
View Description Hide DescriptionThe key idea of conservative Hamiltonian systems is the fact that the closed line integral of action is an absolute invariant of the motion. Dissipation effects may be included by considering those systems for which the closed integral of action is a parameter‐dependent, conformal invariant of the motion. An application of this idea to hydrodynamics is made, and the conditions required for the validity of the Liouville theorem with respect to conformal Hamiltonian flows are examined.

Debye potentials in Riemannian spaces
View Description Hide DescriptionBy means of Debye potentials it is possible to get all solutions of source‐free Maxwell equations in vacuum from a single scalar equation. Sufficient conditions for the existence of Debye potentials in a given four‐dimensional Riemannian space have been found. Some examples of metrics are given, including plane gravitational waves, metrics with spherical symmetry, and cosmological models. The method is generalized to Maxwell fields with sources and Maxwell fields in dielectric media.

Invariant imbedding and the resolvent of Fredholm integral equations with variable parameters
View Description Hide DescriptionIn recent years Bellman and Krein have given independently a geometrical invariant imbedding of the resolvent K(t, y, x) (0 ≤ t, y ≤ x) of Fredholm integral equations with continuous kernels. In the case of symmetric kernels and constant parameters, the Cauchy systems for the symmetric resolvents have been discussed by several authors. In this paper, when the parameter depends upon the independent variable, we show how to solve exactly the Fredholm integral equation with the aid of invariant imbedding. In other words, by reducing the kernel to symmetric kernel, the Bellman‐Krein formula for the Fredholm resolvent permits us to reduce the two‐point boundary value problem to the initial value problem.

On the Lee model with dilatation analytic cutoff function
View Description Hide DescriptionThe spectrum of the Lee model Hamiltonian with Y interaction is studied; first of all we extend the work of Kato and Mugibayashi about the eigenvalues outside the essential spectrum and about the essential spectrum itself. Furthermore, it is proved that the singular‐continuous spectrum is not present; properties of resonances and eigenvalues embedded in the continuous spectrum are obtained.

Unitary representations of SO(n, 1)
View Description Hide DescriptionAll unitary representations of SO(n, 1) have been obtained in the group chain . The branching laws have been explicitly formulated. These results follow from the observation that the matrix elements of the ``noncompact'' generators of SO(n, 1) differ from the corresponding matrix elements of the same generators of SO(n + 1) by a factor of . The branching laws then follow from the unitarity condition. It is also observed that the invariants of SO(n, 1) have the same eigenvalues as the invariants of SO(n + 1). Finally we show that the normalized raising and lowering operators in SO(n + 1), obtained by Pang and Hecht in graphs, and by Wong in algebraic form, can be similarly defined and applied to SO(n, 1).

Scattering of photons by an external current
View Description Hide DescriptionConsidering the interaction of a quantized electromagnetic field with a prescribed external c‐number current, we show that, for each current, there is a representation of the field operators in a separable Hilbert space and a unitary S operator transforming the free incoming field into the free outgoing one. The representation, a generalized product representation, depends on the current so that, in general, for two different currents the corresponding representations are unitarily inequivalent. Photon states and asymptotic observables are defined for the case of an infrared divergent current.

On the approach to equilibrium in kinetic theory
View Description Hide DescriptionDefinition, existence and properties of the equilibrium states, existence and uniqueness of the approach to the equilibrium states, the Onsager reciprocity relations, and existence of hydrodynamic stage in the approach to the equilibrium states are the problems which are discussed in this paper for dynamics defined by the family of kinetic equations of the Enskog‐Vlasov type. A modification of the Chapman‐Enskog method for quantitative calculations involving the kinetic equations considered is suggested.

Cluster coefficients for the one‐dimensional Bose gas with point interactions
View Description Hide DescriptionExact expressions for the cluster coefficients b _{2} and b _{3} for the one‐dimensional Bose gas with repulsive δ‐function interactions are found. The calculation depends only on scattering information in the form of two‐ and three‐body scattering wavefunctions. Although not in agreement with a previous calculation by Servadio, our results are shown to be consistent with the work of Yang and Yang which involves the use of periodic rather than scattering boundary conditions.

Lienard‐Wiechert fields and general relativity
View Description Hide DescriptionAn analogy is established between the Lienard‐Wiechert solutions of the Maxwell equations and the Robinson‐Trautman solutions of the Einstein equations by virtue of the fact that a principal null vector field of either the Maxwell or Weyl tensor in each case satisfies the following four conditions: (1) The field is a geodesic field, (2) it has nonvanishing divergence, (3) it is shear free, and (4) it is twist (or curl) free.

Multipole moments of stationary space‐times
View Description Hide DescriptionMultipole moments are defined for stationary, asymptotically flat, source‐free solutions of Einstein's equation. There arise two sets of multipole moments, the mass moments and the angular momentum moments. These quantities emerge as tensors at a point A ``at spatial infinity.'' They may be expressed as certain combinations of the derivatives at A of the norm and twist of the timelike Killing vector. In the Newtonian limit, the moments reduce to the usual multipole moments of the Newtonian potential. Some properties of these moments are derived, and, as an example, the multipole moments of the Kerr solution are discussed.

Classical SU(3) gauge field equations
View Description Hide DescriptionAdmissible forms of the static solutions to the SU(3)gauge field equation are examined. It is shown that by a proper choice of the form of solutions which extricate the SU(3) indices, the set of nonlinear partial differential equations is reducible to nonlinear ordinary differential equations for the radial functions.

Exact calculation of the energy and heat capacity for the triangular lattice with three different coupling constants
View Description Hide DescriptionCalculation of the internal energy and heat capacity of the general anisotropic triangular Ising lattice is derived from the double integral form of the partition function. The principal result is the reduction of the elliptic integrals to a standard form for three arbitrary coupling constants. Both the standard form and the method of reduction are due to Legendre. The method of reduction is one involving two linear transformations. A straightforward reduction of elliptic integrals to standard form could not be used in this application. This is because of the functional dependence of the two linear transformations on the many combinations and permutations of the signs and relative magnitudes of the coupling energies of the lattice. A relatively simple formulation is presented in which the many combinations and permutations previously mentioned are reduced to only two distinct cases. An independent numerical solution was calculated directly from the partition function as a means of verifying the formulation presented in this paper.

The Lie algebra s o (N) and the Duffin‐Kemmer‐Petiau ring
View Description Hide DescriptionAn explicit expression is given for the unit element E of the ring generated by the Duffin‐Kemmer‐Petiau (DKP) operators β_{μ}. The relation of E to the unit operator I (unit matrix in a matrix representation) is also derived. It is pointed out that one must be careful to distinguish E from I. Bhabha's observation that one may use the irreducible representations (irreps) of the Lie algebras o (5) to obtain the irreps of the Dirac, DKP, and other algebras is given a concise and general setting in terms of a relation between the Lie algebras o (n + 1) and a family of semisimple operator rings. We emphasize that for the case n + 1 = 5 this means that there is an underlying relationship between the physical DKP and Diracalgebras and wave equations.

Zeroes of the partition function for the Ising model in the complex temperature plane
View Description Hide DescriptionFor special boundary conditions, the zeroes of the partition function of the square Ising model are shown to lie on Fisher's two circles in the complex exp(− 2βJ) plane. For some more general boundary conditions, the zeroes distribute asymptotically on these circles.

Statistical mechanics for velocity dependent interactions
View Description Hide DescriptionThe quantum corrections to the phase space distribution function are obtained for general velocity dependent interactions. It is noted that a study of the thermodynamic properties of bulk nuclear matter may settle the question of velocity dependence of nucleon‐nucleon interaction.

General expressions for the position and spin operators of relativistic systems
View Description Hide DescriptionSome general assumptions based on the physical properties of elementary and of composite quantized systems lead in a natural way to general expressions for the position and spin operators of massive particles. All previously proposed position and spin operators, defined in the enveloping algebra of the Poincaré group, appear as special cases. By algebraic arguments a general local position operator is obtained which is proved to coincide under special conditions with the Pryce‐Newton‐Wigner position operator.

Asymptotic solution of neutron transport problems for small mean free paths
View Description Hide DescriptionA method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψ^{ i }, a boundary layer part ψ^{ b }, and an initial layer part ψ^{0}. Then ψ^{ i } is sought as a power series in ε, while ψ^{ b } decays exponentially with distance from a boundary or interface at a rate proportional to ε^{−1}. Similarly ψ^{0} decays at a rate proportional to ε^{−1} with time after the initial time. For a near critical reactor, the leading term in ψ^{ i } is determined by a diffusion equation. The leading term in ψ^{ b } is determined by a half‐space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ^{0} and ψ^{ b } to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.

Dimensional regularization for zero‐mass particles in quantum field theory
View Description Hide DescriptionWe show by a suitable redefinition of momentum‐space integrals that the technique of dimensional regularization can be extended consistently to include nonnormally ordered theories describing zero‐mass particles.