Volume 16, Issue 6, June 1975
Index of content:

The notion of essential locality for nonlocalizable fields
View Description Hide DescriptionThe classification of extension of the field commutator outside the light cone suggested by Constantinescu and Taylor is analyzed and shown to be to a large extent mathematically equivalent to the notion of essential locality, introduced in a recent paper by the present authors. Simple model fields are constructed which disprove the interpretation given by Constantinescu and Taylor. Essential locality is shown to hold for the two‐point function of every scalar Hermitian field, including the massless case. It is, moreover, shown to be weaker than locality and independent of the other Wightman axioms. Unfortunately, essential locality turns out to be unstable under limits. In order to indicate the possibility that there are essentially local fields which do not fall into Jaffe’s class and the commutator of which is concentrated in the closed light cone, Jaffe’s concept of strict localizability is generalized. As a by‐product it is indicated that local fields (in the generalized sense) may have extreme high energy behavior.

The separating topology for the Lorentz group L
View Description Hide DescriptionSome properties of the Lorentz groupL are presented if it is endowed with a topology induced by one of the topologies for the Minkowski space M, proposed by E. C. Zeeman.

Physical applications of multiplicative stochastic processes. III. Nonequilibrium entropy
View Description Hide DescriptionIt is argued that the expression − K _{ B } Trace [〈ρ (t) 〉 ln〈ρ (t) 〉], which appears in a stochastic treatment of the dynamics of the density matrix is indeed the nonequilibrium entropy. The reasoning involves consideration of the time evolution of the free energy for a relaxing magnetic moment in a fluctuating magnetic environment. It is shown that the Htheorem, and the monotonic decrease of the free energy, as described by Pauli’s master equation can be generalized to the full density matrix, at least for the case of magnetic relaxation, which requires the presence of off‐diagonal density matrix elements.

Solution of potential problems near the corner of a conductor
View Description Hide DescriptionThe Green’s function for a space defined (in cylindrical coordinates) by the intersection of two half‐planes S _{1} (φ=0) and S _{2} (φ=ϑ where 0 < ϑ ⩽ 2π) is found by a technique due to Sommerfeld. The Green’s function (or its normal derivative) is required to vanish on the surface S _{1} + S _{2} as well as at infinity. When ϑ = mπ/k where k and m are integers, the solution can be written in terms of the Green’s functionu _{ m } for a Riemann space of m windings (in φ). For m = 1 and 2, u _{ m } can be expressed in terms of elementary functions. For m = 3, we find u _{ m } to be given in terms of complete elliptic integrals. Application to some simple electrostatic and magnetostatic problems is made, particularly for ϑ = 3π/2.

Cross sections in quantum mechanics
View Description Hide DescriptionThe definition of scattering cross sections requires an averaging over wavepackets with random impact parameters ρ; this leads to an integral of the scattering probability over all ρ in a plane perpendicular to the incident beam. We show that, for scattering off a potential which is O (1/r ^{β}) as r→∞, the scattering probability is O (1/ρ^{2β−4}) as ρ→∞. Thus for any β ≳ 3, the integral over impact parameters is well‐defined and convergent.

Harmonic oscillator Green’s function from a BCH formula
View Description Hide DescriptionAn integral operator expression is formulated for the n‐dimensional harmonic oscillator by exploiting the U (n) symmetry group of the oscillator Hamiltonian. The operator expression is disentangled using a Baker–Campbell–Hausdorff formula appropriate to the dynamical group S p (4;R) of the Green’s function. The BCH formula is computed in a faithful matrix representation of S p (4;R). It is sufficient to compute the disentangling theorem for the more restricted dynamical group S O (2,2).

The de Donder coordinate condition and minimal class 1 space–time
View Description Hide DescriptionBy means of immersion techniques a set of ’’adapted coordinates’’ are introduced as preferred coordinates for class 1 space–time. It is proved that the necessary and sufficient condition for the adapted coordinates to be harmonic coordinates is that class 1 space–time be a minimal variety. Some interesting features of the embedding approach to curved space–time are also shown in terms adapted coordinates.

Evaluation of lattice sums. IV. A five‐dimensional sum
View Description Hide DescriptionAn approach for evaluating lattice sums is presented, which requires the use of basic hypergeometric functions. The sum J (x _{1} x _{2} + x _{2} x _{3} + x _{3} x _{4} + x _{4} x _{1} + x _{2} x _{5})^{−s } is given as an example.

Eigenvalues of the invariant operators of the orthogonal and symplectic groups
View Description Hide DescriptionEigenvalues of the invariant operators of the orthogonal and symplectic groups have been obtained in closed form. All semisimple Lie groups, the unitary, orthogonal, and symplectic groups, are treated in a systematic way by modifying Perelomov and Popov’s method. The eigenvalues of the invariant operators for the orthogonal and symplectic groups are then calculated with reference to the unitary group.

A Goursat problem for the fourth moment equation
View Description Hide DescriptionThe solution of a Goursat problem for the pseudoparabolic equation satisfied by the fourth statistical moment of an initially plane wave propagating in a random medium is presented, using an integro–differential equation technique. Two‐dimensional propagations are considered.

Two applications of the Racah coefficients of the Poincaré group
View Description Hide DescriptionRacah coefficients of the Poincaré group are defined and compared with the rotation group. It is then shown how they arise in a natural manner in the inelastic unitarity equations and in crossing multiparticle amplitudes.

General chiral S U _{2} × S U _{2} pion Lagrangian and the generators of O (4,1) group
View Description Hide DescriptionIt is shown that the general chiralS U _{2} × S U _{2} invariant pionLagrangian in the form given by Gürsey can be used to obtain a general and convenient parametrization of an infinite set of generators of the noncompact O (4,1) group. Various special cases of the general form of the generators are given, and one particular form is shown to coincide with the generators of the O (4,1) group used in the literature.

Discrete Coulomb gas in one dimension: Correlation functions
View Description Hide DescriptionIn this short note we calculate the correlation and cluster functions of the discrete Coulomb gas in one dimension considered recently by Gaudin. We consider also the discrete versions of the Gaussian ensembles previously studied extensively.

Two‐level radiative systems and perturbation theory
View Description Hide DescriptionWe obtain expressions for the radiative level shifts of a two‐level system in terms of (i) recurrence relations, (ii) ratios of determinants, (iii) continued fractions, and (iv) a Lidstone expansion. These expressions are shown to be very useful for numerical computations. It is pointed out that perturbation series in powers of the coupling constant are not the most appropriate way of representing the solutions of the problem, but if they are to be used, different series should be employed depending on the relative value of the frequency of the field to the frequency of the two‐level system. The significance of these results in the general theory of perturbation is discussed.

The general relativistic fields of a charged rotating source
View Description Hide DescriptionThe gravitational and electromagnetic fields of a charged, rotating source are obtained by an elementary algebraic method.

Petrie matrices and generalized inverses
View Description Hide DescriptionThe connection between Petrie matrices and a special group of generalized inverses deriving from an incidence matrix is established. These matrices are encountered in the theory of the excluded volume effect in polymers, but have wide applicability to other problems. As an example of one such application we present a greatly simplified derivation of Lagrange’s theorem, which relates the inertial tensor of a mechanical system to the distances between all pairs of particles.

On the stability of equilibrium states of finite classical systems
View Description Hide DescriptionThe state of a system is characterized, in statistical mechanics, by a measure ω on Γ, the phase space of the system (i.e., by an ensemble). To represent an equilibrium state, the measure must be stationary under the time evolution induced by the systems Hamiltonian H (x), x‐Γ. An example of such a measure is ω (d x) = f (H) d x;d x is the Liouville (Lebesgue) measure and f (H (x)) is the ensemble density. For ’’nonergodic’’ systems there are also other stationary measures with ensemble densities, e.g., for integrable dynamical systems the density can be a function of any of the constants of the motion. We show, however, that the requirement that the equilibrium measure have a certain type of ’’stability’’ singles out, in the typical case, densities which depend only on H.

Optical theorems for three‐to‐three processes
View Description Hide DescriptionOptical theorems for three‐to‐three processes are derived from S‐matrix principles. These theorems express all single, double, and multiple discontinuities across all combinations of normal threshold cuts in terms of physical scattering amplitudes. The 2^{16} functions corresponding to all combinations of sides of the 16 normal threshold cuts are determined by analyticity requirements and the generalized Steinmann relations. These two conditions guarantee that these functions can be identified with the corresponding functions in the Regge discontinuity formulas of Weis. This identification provides for a possible enlargement of the scope of Regge–Mueller‐type analyses of high‐energy processes.

Self‐interacting, boson, quantum, field theory and the thermodynamic limit in d dimensions
View Description Hide DescriptionBy use of a finite volume, lattice approximation, we set up an approximation to the analytic continuation of a polynomial, self‐interacting boson quantum field theory from Minkowski space to Euclidean space. The infinite volume limit for various boundary conditions is shown to exist and to be asymptotic to the perturbation expansion in the coupling constant g at g=0. For g:φ^{4}:_{ d }theory we prove mass renormalizability and show how, by use of Nelson’s reconstruction theorem, the corresponding Minkowski space quantum field theory can be obtained. We discuss, at least for d?4, how statistical mechanical techniques, used to analyze the Ising model in the critical region just above the critical temperature, can be used to compute the properties of quantum field theory.

Minimal coupling and complex line bundles
View Description Hide DescriptionThe concept of minimal coupling, which leads to the Schrödinger equation of a particle in an external electromagnetic field, is reformulated within the theory of complex line bundles. The possible generalizations are discussed, and the case of the magnetic monopole is investigated with the help of the new formalism.