Volume 10, Issue 4, April 1969
Index of content:

S‐Matrix Singularity Structure in the Physical Region. II. Unitarity Integrals
View Description Hide DescriptionThe general techniques developed in an earlier paper [J. Math. Phys. 10, 494 (1969)] are applied to evaluate the singularities and discontinuities of unitarity integrals. The results are conveniently expressed in terms of what we call mechanism (or M) diagrams.

S‐Matrix Singularity Structure in the Physical Region. III. General Discussion of Simple Landau Singularities
View Description Hide DescriptionDrawing on the understanding of unitarity integrals acquired in previous papers [J. Math. Phys. 10, 494, 545 (1969)], we show that unitarity demands that connected amplitudes be singular on the positive‐α arcs of all ``simple'' Landau curves in their physical regions. Further, we show that the amplitudes are nonsingular on mixed‐α arcs, while on the positive‐α arcs their discontinuities are given by the Cutkosky rule. This confirms arguments from perturbation theory and demonstrates how a weak analyticity assumption can generate in an exact way a singularity scheme relating to causality.

Lower Bounds to the Ground‐State Energy of Systems Containing Identical Particles
View Description Hide DescriptionLower bounds to the ground‐state energy of a system composed of N identical particles interacting by two‐body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground‐state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.

Group‐Theoretic Approach to the New Conserved Quantities in General Relativity
View Description Hide DescriptionThe Newman‐Penrose formalism for obtaining the recent conserved quantities in general relativity is discussed and a group‐theoretic interpretation is given to it. This is done by relating each triad of the orthonormal vectors on the sphere to an orthogonal matrix g. As a result, the spin‐weighted quantities η become functions on the group of three‐dimensional rotation, η = η(g), where g ∈ O _{3}. An explicit form for the matrix g is given and a prescription for rewriting η(g) as functions of the spherical coordinates is also given. We show that a quantity of spin weight s can be expanded as a series in the matrix elements of the irreducible representation of O _{3}, where s is fixed. Infinite‐ and finite‐dimensional representations of the group SU _{2} are then realized in the spaces of η's and . It is shown that the infinite‐dimensional representation is not irreducible; its decomposition into irreducible parts leads to the expansion of η in the , the latter providing invariant subspaces in which irreducible representations act.

Electron in a Given Time‐Dependent Electromagnetic Field
View Description Hide DescriptionThe theory of a quantized Dirac field interacting with a classical electromagnetic field is considered. The resulting q‐number problem is reduced to a closely related c‐number problem. The theory is then shown to be without divergences. The interpolating field is shown to exist and is local. Also, the S matrix is shown to be unitary.

Invariant Imbedding and Case Eigenfunctions
View Description Hide DescriptionA new approach to the solution of transport problems, based on the ideas introduced into transport theory by Ambarzumian, Chandrasekhar, and Case, is discussed. To simplify the discussion, the restriction to plane geometry and one‐speed isotropic scattering is made. However, the method can be applied in any geometry with any scattering model, so long as a complete set of infinite‐medium eigenfunctions is known. First, the solution for the surface distributions is sought. (In a number of applications this is all that is required.) By using the infinite‐medium eigenfunctions, a system of singular integral equations together with the uniqueness conditions is derived for the surface distributions in a simple and straight‐forward way. This system is the basis of the theory. It can be reduced to a system of Fredholm integral equations or to a system of nonlinear integral equations, suitable for numerical computations. Once the surface distributions are known, the complete solution can be found by quadrature by using the fullrange completeness and orthogonality properties of the infinite‐medium eigenfunctions. The method is compared with the standard methods of invariant imbedding, singular eigenfunctions, and a new procedure recently developed by Case.

Realization of Chiral Symmetry in a Curved Isospin Space
View Description Hide DescriptionThe nonlinear realizations of the chiral group are studied from a geometric point of view. The three‐dimensional nonlinear realization, associated with the pion field, is considered as a group of coordinate transformations in a three‐dimensional isospin space of constant curvature, leaving invariant the line element. Spinor realizations in general coordinates are constructed by combined coordinate‐spin‐space transformations in analogy to Pauli's method for spinors in general relativity. The description of vector mesons and possible chiral‐invariant Lagrangians, which yield the various nonlinear models in specific frames of general coordinates, are discussed.

Solution of the Bethe‐Salpeter Equation with a Square‐Well Potential
View Description Hide DescriptionThe ladder approximation Bethe‐Salpeter equation for (i) a bound spin‐0 boson‐boson system of zero total mass and (ii) a bound spin‐½ fermion‐antifermion system of zero total mass is solved for a four‐dimensional square‐well potential.

Method for the Evaluation of Certain Time‐Dependent Thermal Averages
View Description Hide DescriptionA simple method is presented for the evaluation of time‐dependent thermal averages relating to the Debye‐Waller form.

Unitary Representations of the Poincaré Group Contained in a Class of Representations of the Conformal Group
View Description Hide DescriptionA unitary irreducible class of representations of the conformal group is constructed and reduced with respect to the Poincaré group to see which unitary irreducible representations of the Poincaré group it contains. In particular, it is shown that this class of representations of the conformal group does not contain the continuous‐spin representations of the Poincaré group. It is concluded that the representations of the conformal group cannot be used to eliminate the continuous‐spin representations.

Powers of the D Functions
View Description Hide DescriptionWhile powers of distributions in general do not exist, [D _{+}]^{ n } and [D _{−}]^{ n } are the exceptions among all D functions; they do exist and are given explicitly. A ``modified'' power [D _{Γ}]_{ n } based on the representation of D _{Γ} as analytic functionals can be defined. It exists for all homogeneous D _{Γ}, i.e., for D _{±}, D, and D _{1}. The relation of [D _{±}]_{ n } to [D _{±}]^{ n } is given, and explicit expressions are found in x space and in p space. The Källén‐Lehmann‐Umezawa‐Kamefuchi representation of these distributions is derived. The extent to which these considerations are applicable to the Δ_{Γ}(x, m), m > 0, is discussed.

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor
View Description Hide DescriptionA Riemannian space V_{n} is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξ^{ i } for which , where is the Riemann curvature tensor and £_{ξ} denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V _{4} with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form , involving the Bel‐Robinson tensorT^{ijkm} . Also it is found that the identity of Komar, , which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£_{ξ} K is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.

Renormalization of a Finite Matrix Hamiltonian
View Description Hide DescriptionWe investigate the eigenvalues of a finite matrix Hamiltonian H = H _{0} + g _{0} V, where H _{0} is diagonal with eigenvalues 1, 2, …, N, and where all the elements of V are equal to 1. We are interested in the case N → ∞. The radius of convergence of the perturbation series is (ln N)^{−1}, but nevertheless the exact eigenvalues of H tend to well‐defined limits when N → ∞. It is shown that if we define and if we let g _{0} → 0 as N → ∞ in such a way that g is constant, then it is possible to obtain a perturbation series with the ``renormalized'' coupling constant g, provided that suitable counter terms are introduced. We also investigate a different model (where V_{mn} = mn) and show that no such renormalization is possible there.

Massive Vector Meson Interacting with the Gravitational Field. I. General Formalism
View Description Hide DescriptionStarting from the covariant field equations for a vector meson, the energy‐momentum tensor entering in Einstein's field equations is derived. It is shown that its most general algebraic form involves two vector fields and two scalars. Specifying the formalism to the special cases for which the fields are either parallel or perpendicular to each other, it is found that the vector field cannot be described in terms of a perfect fluid involving only density and pressure, but includes an additional term involving the stresses. The conservation laws are given, which, in addition to the ones of relativistic hydrodynamics, also include the ones describing the streaming of the vector field.

Asymptotic Fields in Some Models of Quantum Field Theory. II
View Description Hide DescriptionA quantum field with nonlocal but translation‐invariant interaction is considered. We prove that, with a proper smoothness condition on the interaction, the asymptotic limits of the annihilation‐creation operators exists. The asymptotic limits are then used to prove that the state space decomposes as a tensor product of an incoming (outgoing) Fock space and a zero‐particle space.

Ground‐State Many‐Boson Problem with Repulsive Potentials
View Description Hide DescriptionWe extend to the N‐body case previous techniques for solving three‐body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one‐dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according tofrom the Schrödinger equation we find a basic equation for a typical ψ_{ ij }. As a test of this equation, we apply it for N = 4 and the case of δ‐function interactions, and solve it numerically to find close agreement between the energy per particle of a four‐body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that E_{g} (∞), which is the ground‐state energy per particle of the system with an infinite number of particles, is related to E_{g} (N), which is that energy for the N‐body system of the same density, by E_{g} (N) = E_{g} (∞)[1 − (1/N)] for weak repulsion and by E_{g} (N) = E_{g} (∞)[1 − (1/N ^{2})] for very strong repulsion. We begin a similar comparison for the three‐dimensional hard‐sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L _{2} such that the density is the same as for an N‐body problem in volume . We find that we get the analog of the one‐dimensional result, for which case E_{g} (2) = E_{g} (∞)/2 for N = 2. That is, we find E_{g} (2) = πa ^{2} h/mv, which is just one‐half of the many‐body theory result. We also calculate the two‐body correlation function and find good agreement with the many‐body one, as calculated by Lee, Huang, and Yang.

Some Analytic Properties of Scattering Amplitudes for Long‐Range Forces
View Description Hide DescriptionWe examine the properties of the partial‐wave amplitude a(l, k) and the full amplitude A(k, cos θ) for scattering by a long‐range potential made up of a Coulomb part 2α/r and a short‐range part V(r). The properties of a(l, k) as an analytic function of l and k are shown to be quite similar to those of the usual short‐range amplitude, except in the neighborhood of the threshold k = 0, which point we examine in detail. The full amplitude is treated as a function of cos θ for fixed physical momentum k; using the Sommerfeld‐Watson transformation, we show that A(k, cos θ) is analytic in the cut plane of cos θ.

Functional Methods in Statistical Mechanics. I. Classical Theory
View Description Hide DescriptionA statistical‐mechanical theory of fields is developed. Since a field has an infinite number of degrees of freedom, it is natural and convenient to use functional methods for its description. The most general statistical‐mechanical state for a field is represented by a distribution functional which satisfies a functional differential equation analogous to the Liouville equation. The functional Fourier transform (characteristic functional) is introduced and its properties are studied. Multitime functionals and various reduced distribution functions are also discussed. The formalism is applied to the free electromagnetic fields as well as to a system of charged particles (plasma) interacting via the electromagnetic field.

Functional Integrals Representing Distribution Functions in Statistical Mechanics
View Description Hide DescriptionWe show how to obtain formal solutions of the chain of equations for distribution functions in classical statistical mechanics. These solutions are in the form of complex functional integrals. They are not unique, which fact is a fundamental property of the equations, and the different solutions are recognized by different integration paths in the complex function space. The different manners of integration correspond to different phases, of which some can be identified with the possible physical states. The treatment of the integrals in some cases is also discussed. They are closely related to generalizations of the molecular field approach to the problem. It is also shown that the functional integrals can be written as averages over an external field and that essentially the same form is valid in the quantum‐mechanical case.

SU _{3} Algebra in the Mixed‐Shell Space (0p, 0d, 1s). I
View Description Hide DescriptionThe operators of Elliott are extended in such a way that they also describe an SU _{3}algebra in the mixed‐shell space , where is given as the direct sum of two spaces and spanned by the single‐particle wavefunctions in the (0p) and (0d, 1s) shells of a harmonic‐oscillator potential. The representation of SU _{3} in this space is investigated in detail by the aid of the weight diagram in a way analogous to that of Banerjee and Levinson. The basis is expressed in an explicit manner using the one‐particle wavefunctions in the usual shell model. The states arising from two‐ and three‐particle systems are classified according to the irreducible representations in this extended space.