Volume 13, Issue 6, June 1972
Index of content:

N‐Particle Noninteracting Green's Function
View Description Hide DescriptionA prescription is given for obtaining the Green's function for N free particles which can have different masses. The approach is systematic and straightforward. A coordinate transformation of the Fourier integral representation of the N‐particle noninteracting Green's function facilitates the integration over 3N‐1 angular variables of wavenumber space. A single radial integral can then be evaluated. The resulting Green's function representation may be of use in applying the integral form of Schrödinger's equation to calculate the ground and excited states of atoms.

Another Set of Axioms for Classical Gas Dynamics
View Description Hide DescriptionAnother set of axioms is given for a nonequilibrium classical gas composed of hard, spherical, nonattracting molecules. It is shown that the thermodynamic functions of this type of gas can be expressed as multiple integrals. It is also shown in a certain limiting case that the thermodynamic functions of this type of gas can be expressed as Wiener integrals.

Gårding Domains and Analytic Vectors for Quantum Fields
View Description Hide DescriptionIf one studies the canonical commutation relations (CCR's) of quantum field theory in the unitary Weyl form, one does not know if one can find a common dense domain for the field operators since their domain of definition depends on the test function. We consider here a general class of test function spaces including the spaces S and of Schwartz and the space of all finite linear combinations of a countable basis. It is shown that there exists an invariant Gårding domain D on which all fields are defined and strongly continuous. D consists of analytic vectors for the fields. It turns out that the test function space can be enlarged by continuity. For irreducible or factor representations it becomes even a Hilbert space. The basic idea of the proof is the same as in the Schrödinger representation for one degree of freedom and very transparent. We simply use rapidly decreasing functions in ``Q‐space'' and ``P‐space'' as smoothing factors. That this can be done in the infinite case also is due to a new and interesting measure theoretic result derived here. As an application of our results, we mention that the renormalized fields (after removing the cutoff) of the model of Glimm and Jaffe possess a Gårding domain for test functions in S or for each time.

Clebsch‐Gordan Coefficients and Special Function Identities. II. The Rotation and Lorentz Groups in 3‐Space
View Description Hide DescriptionIt is shown that the construction of concrete models of Clebsch‐Gordan decompositions for tensor products of irreducible group representations leads to a wide variety of special function identities. In this paper the representation theory of the rotation and Lorentz groups in 3‐space is used to give elegant derivations of identities involving Languerre, Gegenbauer, hypergeometric, and generalized hypergeometric functions. Some of these identities may be new in this general form.

Spectrum Generating Algebras and Symmetries in Mechanics. I
View Description Hide DescriptionCertain differential‐geometric and Lie group theoretic facts that are useful in the systematic study and search for spectrum generating algebras are presented.

Spectrum Generating Algebras and Symmetries in Mechanics. II
View Description Hide DescriptionIn case a group of symmetries of a classical mechanics system acts on a locally transitive way on the energy surface, it is shown how the time evolution of the system is related to orbits of one‐parameter subgroups of the symmetry group. In particular, this provides a group‐theoretic interpretation of certain types of regularization of collisions. Generalizations of this phenomenon to quantum mechanics are also discussed.

Expansion of the Master Equation for One‐Dimensional Random Walks with Boundary
View Description Hide DescriptionIn order to understand the behavior of coarse‐grained equations in the presence of a boundary, the following model is investigated. A homogeneous one‐dimensional random walk is bounded on one side by some boundary conditions of rather arbitrary form. The corresponding master equation is approximated by the Fokker‐Planck equation plus partial differential equations for the higher orders. The boundary condition for the Fokker‐Planck approximation is well known; but those for the higher order terms are here derived. To the second order they amount to a virtual displacement of the boundary. The case of a two‐step random walk, however, gives rise to an unexpected complication, inasmuch as nonpropagating solutions of the master equation cannot be ignored in the boundary condition, although they do not contribute to the differential equations themselves.

Topological Analysis of Self‐Generating Interactions in Lagrangian Field Theory
View Description Hide DescriptionWe determine entirely the n‐body nonlocal potentials, parameters of a given Lagrangianfield theory, in terms of a set of independent functionals of the many‐body propagators. These functionals are more suitable for the description of self‐generating interactions than the many‐body propagators themselves.

Calculation of a Certain Type of 6j‐Symbol
View Description Hide DescriptionIt is shown that, for an arbitrary finite simply reducible group, it is possible to express a certain type of 6j‐symbol in which one of the representations is one‐dimensional in terms of 3j‐symbols .

Local Existence of Solutions to the Equations When
View Description Hide DescriptionWe prove that solutions to the equations exist locally when . Previously this was only shown for the special case of .

A Class of Stationary Electromagnetic Vacuum Fields
View Description Hide DescriptionIt is shown how a new class of stationary electromagnetic vacuum fields can be generated from solutions of Laplace's equation. These fields are a stationary generalization of the static electromagnetic vacuum fields of Weyl, Majumdar, and Papapetrou, and are plausibly interpreted as exterior fields of static or steadily moving distributions of charged dust having numerically equal charge and mass densities.

Constructing Examples in Relativistic Classical Particle Mechanics
View Description Hide DescriptionA new method is proposed for constructing examples in relativistic classical particle mechanics, starting with the Poincaré group and space reflections as transformations of position and velocity variables for two free particles and deriving equations for Lorentz covariant position variables for two interacting particles. The advantages are that the equations do not couple the position variables of the two particles, and solutions yield the particle world lines directly. Solutions are obtained for a special case.

Identities Involving the Coefficients in the Asymptotic Expansion of the Outgoing Scattering Solution of the Schrödinger Equation
View Description Hide DescriptionScattering by a short‐range potential is described by the wavefunction ψ having the asymptotic form where has the series representation When only the first term in this series is taken into account, the requirement that the particle current be conserved leads to the optical theorem. By taking into account all of the remaining terms, we obtain a sequence of identities involving the coefficients f, g, …. These identities are formally analogous to the optical theorem, which is seen to correspond to the first identity in the sequence.

The Four‐Dimensionality of Space and the Einstein Tensor
View Description Hide DescriptionAll tensors of contravariant valency two, which are divergence free on one index and which are concomitants of the metric tensor, together with its first two derivatives, are constructed in the four‐dimensional case. The Einstein and metric tensors are the only possibilities.

The Weyl Basis of the Unitary Group U(k)
View Description Hide DescriptionWeyl's method for the construction of irreducible tensors of the unitary group is used to construct a basis for any irreducible representation of U(k) or GL(k) in terms of Bose creation operators. A simple way is indicated to select a complete but not over complete basis from the functions obtained. The basis obtained can be useful in nuclear or molecular calculations, as well as in some mathematical problems.

Relation between the Three‐Dimensional Fredholm Determinant and the Jost Functions
View Description Hide DescriptionIt is proved that the modified Fredholm determinant F of the three‐dimensional Lippmann‐Schwinger equation in the theory of scattering by spherically symmetric potentials is related to the Jost functions f_{l} of angular momentuml by,where K_{l} is the kernel of the lth radial Lippmann‐Schwinger equation. The relation between the multiplicity of the zeros of F and the degeneracy is discussed, and a relevant theorem for Hilbert‐Schmidt operators is proved.

Rank 1 Expansions
View Description Hide DescriptionA group expansion technique is presented which is valid for all homogeneous Riemannian symmetric coset spaces of rank 1. Examples are given.

Analytic Properties of the Free Energy for the ``Ice'' Models
View Description Hide DescriptionThe calculation of the free energy for the two‐dimensional lattice models which obey the ``ice'' condition is reviewed and summarized. The analytic properties of this function relevant to thermodynamics are obtained for the set of those models corresponding to the absence of external fields. Detailed complex temperature Reimann structures are presented for the KDP and F models.

Canonical Transformations and the Radial Oscillator and Coulomb Problems
View Description Hide DescriptionIn a previous paper a discussion was given of linear canonical transformations and their unitary representation. We wish to extend this analysis to nonlinear canonical transformations, particularly those that are relevant to physically interesting many‐body problems. As a first step in this direction we discuss the nonlinear canonical transformations associated with the radial oscillator and Coulomb problems in which the corresponding Hamiltonian has a centrifugal force of arbitrary strength. By embedding the radial oscillator problem in a higher dimensional configuration space, we obtain its dynamical group of canonical transformations as well as its unitary representation, from the Sp(2) group of linear transformations and its representation in the higher‐dimensional space. The results of the Coulomb problem can be derived from those of the oscillator with the help of the well‐known canonical transformation that maps the first problem on the second in two‐dimensional configuration space. Finally, we make use of these nonlinear canonical transformations, to derive the matrix elements of powers of r in the oscillator and Coulomb problems from a group theoretical standpoint.

Decision Procedures in Quantum Mechanics
View Description Hide DescriptionThe results of an earlier paper on finite and infinite sequences of measurements are here extended to include decision procedures. It is shown that with each decision procedure Q there is uniquely associated a probability operator measureO^{Q} , which gives the statistical properties of Q. None, some, or all of the paths of Q can be infinitely long. A result of this association is that there are two methods of measuring the probability that carrying out Q on a system in state ρ gives an outcome sequence in some set F. A remarkable aspect of this equivalence is that the purely physical operation of one method is equivalent to, or can replace, the physical operation and mathematical decision procedure of the other method.