Volume 8, Issue 10, October 1967
Index of content:

Unitary Irreducible Representations of SU(2, 2). I
View Description Hide DescriptionThis paper, which is the first of three, is concerned with the general properties of the noncompact group SU(2, 2), the Lie algebra of which is isomorphic to the Diracalgebra. In the course of our study of the unitary representations, we first obtained all the finite dimensional irreducible representations. The associated Young diagrams are shown to have simple properties; the degenerate Young diagrams always denote degenerate representations. Through a theorem of Harish‐Chandra, which relates the finite representations to the unitary representations in the discrete series, we are able to obtain explicitly all the unitary infinite‐dimensional irreducible representations in this series, both degenerate and non‐degenerate. The notion of multiplicity for nondegenerate representations is introduced and discussed in connection with a new operator F _{3}, which is required for a complete labeling of states.

Representation Theory of Local Current Algebras
View Description Hide DescriptionWe prove the following theorem: Every finite‐dimensional irreducible representation of the local current algebra,where the are the structure constants of a semisimple Lie algebraL, has the form,where is a finite‐dimensional representation of L, and for i ≠ j, x_{i} ; commutes with for all ρ and σ.

Interpolation Formulas in the Angular Momentum Plane
View Description Hide DescriptionWhen the potential is an analytic function of a rational power of r, analytical properties of the wave‐functions enable one to derive interpolation formulas in the angular‐momentum plane. In these formulas, products of the wavefunctions which correspond to two different potentials are expanded in terms of their values at an infinite discrete set of points on the real axis. These expansions hold uniformly in z. Properties of the expansion coefficients are investigated. They allow the author to construct generating functions, and to obtain several interpolation formulas for functions of mathematical physics. Results are applied to the inverse scattering problem at fixed energy. They provide further information on an ``angular‐momentum dispersion relation'' obeyed by the Jost function. A semiclassical approximation is given for the interpolation coefficients, and its meaning is investigated. Finally, an elementary derivation of the interpolation formulas is sketched and suggests a curious symbolic writing of the wavefunctions.

General Space‐Momentum Commutation Relation
View Description Hide DescriptionThis paper derives the general commutator [q^{m}, p^{n} ] for all integers m and n, positive or negative. Two new identities for the binomial coefficients are derived using the general commutator. A simple example of the use of the general commutation relation is given in the expansion of (Ap + Bq)^{ n } (A, B, constants) in normal form for integer n.

Phase‐Integral Approximation in Momentum Space and the Bound States of an Atom
View Description Hide DescriptionThe phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrödinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman‐type functional integral which was recently proposed by Garrod for the energy‐momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound‐state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.

Compatibility Requirements in BBGKY Expansion
View Description Hide DescriptionThe BBGKY equations are written in nondimensionalized form leaving two parameters α and β explicit. The coefficient α is a measure of the ``strength'' of the potential of interaction, while β is a measure of the ``range'' of the interaction potential. The s‐particle distribution function is expanded about vanishing correlation with ε entering as a parameter of smallness. Nine cases are considered, depending on the order of magnitude of α and β (viz., ε, 1, ε^{−1}). Lowest order terms, in each of the cases considered, give a kinetic equation, together with subsidiary conditions. In the ``rigid sphere'' limit, the Vlasov equation appears together with a second‐order differential constraint. The Vlasov equation alone appears in the Rosenbluth‐Rostoker limit. A critical survey is made of the compatibility problem which is inherent to expansions of the BBGKY sequence. For the most part, the problem reduces to the compatibility of ``type‐l'' constraint equations. These constraint conditions cause any given order set of equations to be over determined.

Analytic Properties of Certain Radial Matrix Elements
View Description Hide DescriptionThe analytic properties of the Laplace transform of products of two confluent hypergeometric functions as function of certain parameters are investigated. The radial matrix element of multipole type that contain Coulomb‐Dirac wavefunctions or special cases thereof are considered. These are investigated as functions of the momenta p of the particles. Branch cuts and the behavior along the branch cuts in the complex p plane are given. The formulas presented permit accurate calculation of the matrix elements also, when the parameters have values arbitrarily close to singular points. The prime concern, however, is to facilitate second‐ and higher‐order perturbation calculations.

Evaluation of a Partial‐Wave Unitarity Integral
View Description Hide DescriptionThe integral , where ρ(s) = (1/s)[(s ‐ s _{+})(s ‐ s _{−})]^{½}, 0 ≤ s _{−} < s _{+}, is evaluated for an arbitrary function f(s) which is meromorphic in the complex s plane and has the behavior at infinity . Some applications to the N/D method of elementary particle physics are discussed.

Upper Bounds on Holomorphy Envelopes for Wightman Functions
View Description Hide DescriptionAs a consequence of Lorentz invariance, reasonable mass spectrum and local commutativity, the holomorphy envelopes for the analytic functions arising from vacuum expectation values of products of n field operators are related in such a way that furnishes upper bounds (in the coordinates) for all H(S_{ N }) with N > n.

Invariants of Nearly Periodic Hamiltonian Systems
View Description Hide DescriptionA new and simple method of finding an invariant J of a nearly periodic dynamical system is presented. The Hamiltonian is written as H = p _{1} + εΩ(q_{i}p_{i} ), where Ω is periodic in q _{1} and ε « 1. The first four terms of the invariant series are found explicitly in terms of Ω using Poisson bracket and averaging operators. This invariant is related to the adiabatic invariant and to various constants of motion discussed in celestial mechanics, such as Whittaker's adelphic integral. J is shown to be an asymptotic constant by using the rigorous methods of Kruskal to calculate the adiabatic invariant K; it is found that K/τ = H ‐ εJ, where τ is the period in q _{1}. The adelphic integral has different functional forms depending on the presence of resonant denominators, but is shown to be always a function of H and J. The present method provides a single functional form which is applicable even when Ω is only almost periodic in q _{1}. It is also much simpler than the methods of adiabatic invariant theory.

Results on Certain Non‐Hermitian Hamiltonians
View Description Hide DescriptionWe present a few results on the spectral properties of a class of physically reasonable non‐Hermitian Hamiltonians. These theorems relate the spectral properties of a non‐self‐adjoint operator (of the aforementioned class) in terms of that of a self‐adjoint operator. These theorems can be specialized to yield conditions under which the perturbed eigenvalues (of the above class of operators) vary continuously from the eigenvalues of the unperturbed operators. If the Schrödinger equation has to be solved numerically, a knowledge of the spectral properties of the non‐Hermitian Hamiltonian would insure when the eigensolutions exist.

Further Exact Solutions of the Dirac Equation
View Description Hide DescriptionThe four exactly solvable problems for the Dirac equation previously found by the author are discussed in detail. The presentation is mainly within the two‐component relativistic description of a spin ½ particle. Both bound state and scattering solutions are considered.

Realization of Lie Algebras by Analytic Functions of Generators of a Given Lie Algebra
View Description Hide DescriptionIn this paper we discuss the problem of the Poisson bracket realization of various Lie algebras in terms of analytic functions of the generators of a given Lie algebra. We pose and solve the problem of realizing the general O(4), O(3, 1), and E(3)algebras in terms of analytic functions of the generators of a prescribed realization of an E(3)algebra. A similar problem is solved for the symmetric tensor realizations of SU(3) and SL(3,R). Related questions are discussed for O(n + 1), O(n, 1), E(n), SU(n), and SL(n, R). We study in some detail the finite canonical transformations realized by the generators of the various groups. The relation of these results to the reconstruction problem is briefly discussed.

Systems of First‐Order Linear Ordinary Differential Equations as Canonical or Euler‐Lagrange Equations
View Description Hide DescriptionA simple but complete theory is given of the necessary and sufficient conditions which must be satisfied in order that a system of first‐order linear ordinary differential equations, whose coefficients are functions of the independent variable, be the canonical or Euler‐Lagrange equations of a variational principle.

n‐Representability Problem for Reduced Density Matrices
View Description Hide DescriptionIn this paper we prove some theorems about the n‐representability problem for reduced density operators. The first theorem(Theorem 6) sharpens a theorem proved by Garrod and Percus. Let be the set of all n‐representable p‐density operators. Then a density operator D^{p} belongs to (the bar indicates the closure with respect to a certain topology) if and only if Tr (D^{p}B^{p} ) ≥ 0 for all bounded self‐adjoint p‐particle operators B^{p} , such that their n‐expansionis a positive operator in n‐particle space. Moreover, it is shown that is the closed convex hull of the exposed points of of finite one‐rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8).
Consider the set γ^{ p } of subspaces of the n‐particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n‐particle operator which is the n expansion of some p‐particle operator. Choose from every element of γ^{ p } one (and only one) vector (function) and form the corresponding reduced p‐particle operator. is the closed convex hull of all these p‐density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.

Phase Transition of a Bethe Lattice Gas of Hard Molecules
View Description Hide DescriptionA pseudo‐lattice, or homogeneous Husimi tree, is simpler statistically than a true lattice in two or three dimensions, since the pseudo‐lattice contains only low‐order cycles. The prototype is the ``Bethe lattice,'' or Cayley tree, containing no cycles at all. Reported here are the results of studies of the lattice statistics of the Bethe lattice of coordination number three, using the language of the hard molecule lattice gas; the size of an adsorbed molecule prevents simultaneous occupancy of the same site or any of the three nearest‐neighbor sites. Using methods related to those used in enumerating graphs, a recursion relation is obtained which must be satisfied by the grand partition function. It is shown that the resulting equation of state is not the quasi‐chemical equation expected because of the absence of cycles. There is no phase transition of lower than third order and in all likelihood none at all. The quasi‐chemical equation of state is obtained only if ``surface'' effects are eliminated; even then the solution is valid only for activities z < 4, at which point the ``interior'' system undergoes a second‐order transition to an ordered state with a finite discontinuity in compressibility.

Electro‐Optical Effects. I
View Description Hide DescriptionThe theory of Toupin and Rivlin for the propagation of electromagnetic waves in a centro‐symmetric isotropic material to which static electric and magnetic fields are applied is specialized to the case when only a static electric field is applied. The reflection‐refraction problem for an electromagnetic wave normally incident on the interface between free space and the material is studied for arbitrary direction of the applied electric field. It is found that, in general, there are two refracted rays and that these are not generally normal to the interface.

Formal Properties of the Solution to the Radiationless Relativistic Kepler Problem
View Description Hide DescriptionFormal properties of the solution to the classical radiationless relativistic Kepler problem are discussed with particular attention to the strong coupling limit, a case rarely considered. It is found that fall to the origin can occur for potentials less strongly attractive than required nonrelativistically. One conclusion is that the relativistic centrifugal barrier is less ``effective'' than the nonrelativistic kind. It is possible to describe these effects in terms of a dimensionless parameter resembling a classical ``coupling constant.'' The quantum‐mechanical case is also briefly treated.

Complementary Variational Principles and Their Application to Neutron Transport Problems
View Description Hide DescriptionSeveral variational principles are developed which give upper and lower bounds for the linear functional (S, ψ), where ψ is the solution of the inhomogeneous equationHψ = S with H a self‐adjoint, positive‐definite, linear operator. Some of the principles bound this functional only with respect to small or local variations, whereas others give bounds for arbitrary variations. Several of our results coincide with those of other authors widely scattered throughout the literature, and we show that these principles have a common origin. Other results given are new. Examples of the use of these principles are taken from the field of neutron transport theory, and we use both the linear Boltzmann or transportequation and the diffusionequation. One interesting result is that certain ``exact'' values of the extrapolated endpoint for the Milne problem which have been reported in the literature fall, due to numerical inaccuracies, outside the bounds computed here.

Systems of Observables in Axiomatic Quantum Mechanics
View Description Hide DescriptionSystems of observables are considered in an axiomatic framework for quantum mechanics which generalizes the usual Hilbert space formulation. Familiar concepts such as complete systems of observables and superselection rules are generalized and it is shown that many of the Hilbert space theorems carry over to this abstract formalism. Also functions of observables are considered and some theorems due to John von Neumann are generalized.