Volume 6, Issue 4, April 1965
Index of content:

Microcausality in Asymptotic Quantum Field Theory
View Description Hide DescriptionAsymptotic quantum field theory can be formulated without the axiom of microcausality, (strong local commutativity). It is shown here that in certain cases microcausality is nevertheless a consequence of the remaining assumptions of the theory. These cases are the ones for which solutions of the basic equations can be obtained in perturbation expansion, i.e., the scalar, spin ½, and neutral vector field. For the charged vector boson, for example, the microscopic causality condition cannot be proved from the basic postulates. In the provable cases the functional derivative in the Bogoliubov causality condition can be defined explicitly, and this condition can then be derived from the postulates of asymptotic quantum field theory. In the other cases, Bogoliubov causality cannot be defined explicitly.

High‐Order Limit of Para‐Bose and Para‐Fermi Fields
View Description Hide DescriptionPara‐Bose and para‐Fermi quantization are schemes of second quantization which generalize the usual Bose and Fermi schemes. The different cases of para‐Bose (para‐Fermi) quantization are labeled by a positive integer p, which is called the ``order.'' For p = 1, the usual Bose (Fermi) quantization is recovered. The high‐p limit of para‐Bose (para‐Fermi) quantization is studied, and it is shown that, to a large extent, the high‐p limit of a para‐Bose (para‐Fermi) field is a Fermi (Bose) field. The nature of the limit is studied in detail, and a paradox relating to the connection of spin and statistics is resolved.

Statistical Mechanics of Assemblies of Coupled Oscillators
View Description Hide DescriptionIt is shown that a system of coupled harmonic oscillators can be made a model of a heat bath. Thus a particle coupled harmonically to the bath and by an arbitrary force to a fixed center will (in an appropriate limit) exhibit Brownian motion. Both classical and quantum mechanical treatments are given.

Proof that Scattering Implies Production in Quantum Field Theory
View Description Hide DescriptionA proof of the necessity of production processes in quantum field theory is carried out in the axiomatic framework. It is shown that a field theory that is assumed to have a nontrivial scattering amplitude violates crossing symmetry, if production processes are absent. The proof is based on the rigorous analytic properties of the scattering amplitude, particularly the analyticity in the invariant‐scattering variables s, t, and u. In the case of a scalar theory with pairing symmetry, the scattering amplitude is known to be analytic within the domain stu < 7168m ^{6}, except for the usual cuts. Under the working assumption that production processes are null, it is shown that this domain can be enlarged by applying the elastic unitarity conditions beyond the (usual) elastic region. The domain is enlarged sufficiently to include the first Landau singularity of the absorptive part of the scattering amplitude. This singularity is not symmetric in s and t within the extended domain, and this is incompatible with the crossing symmetry of the scattering amplitude. In order to avoid a contradiction, the discontinuity across this Landau singularity must be null. It follows that the scattering amplitude must itself be null.
In the course of the proof it is shown that the conclusion is valid for a scattering amplitude satisfying the requirements of an S‐matrix theory embodied in the Mandelstam representation.

Classification of Three‐Particle States According to SU _{3}
View Description Hide DescriptionIt is shown that the set of states for three noninteracting particles can be put into a one‐to‐one correspondence with the set of irreducible representations of SU _{3}. This classification leads to three‐particle angular momentum states which treat all particles on an equal footing. The states exhibit the maximum localization compatible with a given total energy, momentum, and angular momentum. Only nonrelativistic particles are treated.

Diagram Renormalization, Variational Principles, and the Infinite‐Dimensional Ising Model
View Description Hide DescriptionThe Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi‐invariants, which play the role of the one‐particle density or the one‐particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite‐dimensional (infinite‐range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi‐invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.

Note on Positronium
View Description Hide DescriptionBy means of an inequality providing an upper bound for the norm of integral operators, it is shown that the Bethe‐Salpeter equation for bound states of the electron‐positron system (in the ladder approximation) admits solutions associated with a discrete spectrum of binding energies. It is found that in the weak coupling limit the spectrumB(α) approaches asymptotically the Coulomb spectrum in the sense,where α is the fine‐structure constant and m the electron rest mass.

Algebraic Construction of the Basis for the Irreducible Representations of Rotation Groups and for the Homogeneous Lorentz Group
View Description Hide DescriptionThe basic functions for a class of the irreducible representations of the rotation groups in n dimensions (R_{n} ) are explicitly constructed by an algebraic method in which the basic functions are taken to be homogeneous polynomials in the variables of E_{n} . The solutions correspond to the hyperspherical harmonics of the mathematical literature and are of interest for problems exhibiting invariance under a certain R_{n} . The method is also applied to derive a basis for the infinite‐dimensional irreducible representations of the homogeneous Lorentz group if we then look for homogeneous functions in the variables of the corresponding pseudo‐Euclidean space.

On a Special Class of Type‐I Gravitational Fields
View Description Hide DescriptionThis paper contains an investigation of the algebraic structure and the analytic properties of a class of normal hyperbolic Riemannian 4‐spaces restricted by the following condition: There exists a timelike unit vector u^{a} such that the Riemann tensor satisfies *R_{abcd}u^{b}u^{d} = 0. This condition is shown to be equivalent to the statement that the conform tensor is Petrov type I with real eigenvalues,u^{a} being a principal vector and an eigenvector of the Ricci tensor. This means that there is no flux of nongravitational energy relative to an observer travelling with 4‐velocity u^{a} .
The eigen null directions (Debever vectors) of the conform tensor lie in a timelike hyperplane spanned by u^{a} and the two eigenvectors of ε_{ ac } ≡ −C_{abcd}u^{b}u^{d} belonging to the eigenvalues with largest absolute value. The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer u^{a} is an eigendirection of ε_{ ab }.
The complete set of Bianchi identities is examined. It yields an expression for the covariant eigentime derivative of ε_{ ab } and an algebraic relation linking the rotation and shear of u^{a} to the curvature tensor of the Riemannian space.
The general results are applied to special Einstein spaces (R_{ab} = 0) admitting a congruence of timelike curves without shear and rotation. We get a new simplified proof of the theorem that in the case of nondegeneracy such spaces are static, the curves of the congruence being paths of an isometric motion.

New Homogeneous Solutions of Einstein's Field Equations with Incoherent Matter Obtained by a Spinor Technique
View Description Hide DescriptionEinstein's field equations with incoherent matter are solved for the case of homogeneous spacetime, i.e., for metrics allowing a four parametric simply transitive group of motions. Two families of new solutions are obtained by use of a spinor technique. As a special result a proof emerges for Gödel's theorem, which states that there exist only two homogeneous solutions of Einstein's field equations with incoherent matter and rigid rotation, namely the Gödel cosmos and the Einstein static universe.

Uniform Asymptotic Estimates for Wave Packets in the Quantum Theory of Scattering
View Description Hide DescriptionThe theory of the scattering of particles by a potential field V(x) according to the classical wave mechanics of Schrödinger is considered. Various conditions on V(x) are known which imply that the wave packets ψ(x, t) describing scattered particles are asymptotically equal in L _{2}(R^{n} ) to wave packets ψ_{0}(x, t) for free particles when t → ∞ or − ∞. In this paper, additional conditions on V(x) and the initial values of the wave packets are given which imply that ψ(x, t) and ψ_{0}(x, t) have square‐integrable (or continuous) partial derivatives of a prescribed order whose difference tends to zero in L _{2}(R^{n} ) (or uniformly in R^{n} ) when t → ∞ or − ∞.

Excitation Operators and Intrinsic Hamiltonians
View Description Hide DescriptionAn operator A ^{†} that satisfies [H, A ^{†}] = h/ωA ^{†} converts a stationary‐state eigenfunction of the Hamiltonian H into another eigenfunction with energyeigenvalue increased by h/ω. Such operators describe collective excitations of many‐particle systems, and their properties can be used to construct an intrinsic Hamiltonian that is dynamically independent of the collective degrees of freedom, without introducing subsidiary conditions. The procedure developed by Lipkin, valid when h/ω is real and positive, is extended to make possible the construction of an intrinsic Hamiltonian when h/ω vanishes and A ^{†} is Hermitian, and also when h/ω is complex. The nuclear cranking model is shown to be a special case of the proposed general method for vanishing h/ω in which effective moments of inertia occur as eigenvalues of linear equations. Several examples are worked out in detail, all dealing with an interacting phonon‐electron system in the random‐phase approximation. Results derived are the explicit screened Coulomb interaction resulting from electronic plasma excitations, a verification of the renormalized phonon frequency spectrum and phonon‐electron interaction derived in the adiabatic approximation, and the resulting screened Coulomb and phonon‐induced electronic interactions obtained when plasma and phonon excitations are treated simultaneously.

Approach to Stability of a Plasma
View Description Hide DescriptionThe stabilization of a spatially homogeneous plasma by the collision term of the new kinetic equation of Balescu for a weakly unstable plasma is shown for a simplified model. The one‐particle velocity distribution consists of two symmetrically displaced Lorentzdistribution functions in which the displacement velocity and velocity spread are functions of time. For this distribution function, the collision term decreases the displacement velocity and increases the velocity spread until the plasma is quite stable. Physically this process can be viewed as conversion of relative energy to kinetic energy. This is demonstrated to be equivalent to the passage of the zero of the plus dielectric constant from the upper half complex plane, characterizing an unstable plasma, to the lower half complex plane, characterizing a stable plasma.

Upper Bound for the Double Spectral Function in Potential Scattering
View Description Hide DescriptionFor potentials V(z) holomorphic in Re z > 0 and bounded bywe show that the double spectral ``function'' ρ(s, t) is a continuous function of s and t in s > 0, t > 0, and we obtain an upper bound for it. This upper bound shows clearly that the double integral of the Mandelstam representation in fact exists and defines an analytic function of s and t in two cut planes. We indicate how to generalize these results to the case when ρ(s, t) is no longer a function but a distribution.

Uniqueness of Steady‐State Solutions to the Fokker‐Planck Equation
View Description Hide DescriptionThe uniqueness of steady‐state probability densities in certain continuous Markov processes is proven when the Fokker‐Planck or Kolmogorov equation satisfied by the transition probability is of a variety called ``steady.'' The tendency of other probability densities to approach such steady‐state densities is formally demonstrated.

Property of the Vertex Function in Potential Theory
View Description Hide DescriptionThe vertex function of an s‐wave bound state in potential theory is discussed in terms of a non‐relativistic Bethe‐Salpeter formalism. It is shown that the Born series defining the off‐shell amplitude, Γ(k ^{2}, P _{0}), has the property that , where D(k ^{2}, λ) is the Jost function for potential strength λ, and where the bare (Γ_{0}) and physical (Γ_{ c }) couplings are related by Γ_{0} = Γ _{c}Z _{1}, with Z _{1}, the vertex renormalization constant equal to D(−α^{2}, λ). The particular proof presented derives from a detailed comparison in momentum space of alternative, but equivalent, coordinate space definitions of the Jost function originating with Jost and Pais.

Vertex Modification for Coulombic Interactions by Exact Summation of Ladder Diagrams
View Description Hide DescriptionWe have been able to sum a class of ladder diagrams for Coulombic interactions to infinite order exactly and are able therefore to replace a ``bare'' interaction with a self‐consistent effective interaction or equivalently we are able to modify the vertex operator in a manner which may be of particular significance for systems of low particle density in analogy with what can be done for nuclear matter and liquid helium. The method is only valid for nearly free particles, the exact energy being negative but the zero‐order description being that of free particles. Effective attractive interactions between two electrons or interaction with an external Coulombic field constitute perturbations which fall within the range of validity of the scheme. Despite the limitation of the specific results, the method is probably of more general use and interest and both the approach and specific results are described in this note.

Internal Symmetry and the Poincaré Group
View Description Hide DescriptionTheorems concerning the interrelation between the Poincaré group and the symmetry group of elementary‐particle interactions have been proved under weaker conditions than that of McGlinn.

Necessary Condition on the Radial Distribution Function
View Description Hide DescriptionTrial functions g(r) may be used as radial distribution functions to study the ground‐state properties of a uniform extended quantum fluid (liquid He, nuclear matter). A result obtained by Wigner and Seitz in the study of the charged electron gas imposes an integral inequality on any assumed g(r). The relation is . This inequality places an effective constraint on the location, magnitude, and width of the nearest‐neighbor peak.

High‐Energy Behavior at Fixed Angle for the Five‐Point Function in Perturbation Theory
View Description Hide DescriptionThe validity of the d‐line method used by Halliday in studying the high‐energy behavior at fixed angle of the four‐point function is examined in more detail. This is carried out within the context of determining the high‐energy behavior at fixed angle for the five‐point function. The leading asymptotic behavior of the sum of all planar five‐point graphs in a φ^{3}theory at fixed angle is s ^{−2}.