Volume 11, Issue 7, July 1970
Index of content:

Mandelstam Representation in Potential Scattering
View Description Hide DescriptionA simpler and more general proof of the Mandelstam representation based on Hartog's theorem is presented within the framework of potential scattering; the potential is chosen to be a superposition of the Yukawa type with no further restrictions.

De Sitter Symmetric Field Theory. II. Tensorial and Spinorial Realizations
View Description Hide DescriptionPlane‐wave solutions with definite helicity are obtained for the field equations (S _{μν} p _{ν} + mγ_{μ})ψ = p _{μ}ψ. These equations are then rewritten in spinorial form. The relations between spinors, 4‐ and 5‐dimensional tensors and spin‐tensors, and the components of ψ are derived.

De Sitter Symmetric Field Theory. III. Quantization in the Tung‐Weinberg Basis
View Description Hide DescriptionThe de Sitter symmetric fields, satisfying the field equations (S _{μν} p _{ν} + κγ_{μ})ψ = p _{μ}ψ, are quantized in the Tung‐Weinberg basis. The properties of the associated Green's and causal functions are studied. Transformations of the Hilbert space under CPT and inhomogeneous Lorentz transformations are given. A simple closed expression is obtained for the Shaw‐Weinberg matrix tensor.

Peratization of a Class of Exponentially Singular Potentials
View Description Hide DescriptionFirst, second, and third peratizations are performed for evaluation of the scattering length for repulsive potentials of the form V(r) = gr^{−m} exp (λ/r) with m an integer ≥4. For m = 4 a nontrivial result is found in second and higher peratization which has the same weak‐coupling limit as the exact answer and agrees with the result of Calogero and Cassandro. For m > 4 one finds a zero result in both first, second, and third peratization, suggesting the inadequacy of the procedure as generally applied. The procedure of separate peratization of numerator and denominator of a ratio which is employed in this work is discussed at length. The optimum n method is introduced in the course of this discussion as a powerful and convenient method of evaluating the scattering length without summing the appropriate series. It also clarifies the limitations of the peratization approximation. A modification of the peratization procedure is suggested which has promise of being a more successful approximation scheme.

Peratization of Logarithmically Singular Potentials
View Description Hide DescriptionFirst and second peratizations are performed for a wide class of repulsive potentials with singular behavior near r = 0 of the form L(r)r^{−m} (m > 3), where L(r) is ``logarithmically singular'' at the origin. More precisely, when L(r) is of the class of ``slowly varying functions,'' i.e., functions such that rL′(r)/L(r) → 0 as r → 0, first peratization gives a (generally infinite) constant multiple of the result for the r^{−m} potential. Second peratization for a wide class of slowly varying function offers no substantive improvement.

One‐Dimensional Phase Transitions in the Spherical Lattice Gas from Inverse Power Law Potentials
View Description Hide DescriptionThe spherical model of the lattice gas of Gersch and Berlin is studied by making an integral approximation to the Fourier coefficients of the interaction potential. A phase transition is found in one dimension for potentials −gr ^{−α} with 1 < α < 2. A necessary condition which a potential must satisfy to allow a 1‐dimensional phase transition is given. General expressions are derived for specific heatC_{v} and isothermal compressibility K_{T} . Discontinuities are found in the thermodynamic properties at the transition for α less than times the number of dimensions. Asymptotic expansions are given for C_{v} and K_{T} near the critical point.

Approximate Solutions of the Unitarity Equation
View Description Hide DescriptionThe possible use of the classical Newton approximation method, as extended by Kantorovich to Banach spaces, is discussed in connection with the problem of solving the unitarity equation for the phase of a scattering amplitude, in terms of the corresponding differential cross section.

Maxwell's Equations Having a Gradient as Source
View Description Hide DescriptionWe study Maxwell's equations having a source given by a gradient. The source function f obeys the empty‐space wave equation as a consequence of current conservation. We find that this system can be written as a mass‐zero Dirac equation. Our relationship between tensor and spinor equations is an improvement over other published results, so far as simplicity is concerned.

Clebsch‐Gordan Series and the Clebsch‐Gordan Coefficients of O(2, 1) and SU(1, 1)
View Description Hide DescriptionThe Clebsch‐Gordan series of the O(2, 1) group and its covering group SU(1, 1) are derived for all cases except that of the supplemental series. The continuable Clebsch‐Gordan coefficients (or equivalently, the Wigner coefficients) are explicitly expressed in terms of the generalized hypergeometric function _{3} F _{2}. The spectra in the decomposition of the product of the two principal series are discussed. The applications to the UIR of O(2, 2) are also studied.

Self‐Interaction of Gravitational Radiation
View Description Hide DescriptionA nonsingular, sourceless, first‐order pulse of gravitational radiation imploding from infinity to a focus and then exploding back out to infinity is examined to second order. It is found that, contrary to what might be expected, the nonsingular second‐order field contains no radiation. In space‐time regions outside of the pulse, the second‐order field is nonvanishing only in the region with retarded times earlier than the passing of the exploding pulse and advanced times later than the passing of the imploding pulse. In this region, the second‐order field is that of a Schwarzschild mass, plus a nonradiative quadrupole, plus a nonradiative sixteen pole.

Some Simple Lattice‐Spin Systems
View Description Hide DescriptionA simple mathematical scheme is derived here. With this scheme, three problems of lattice‐spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz‐Ising model in a zero magnetic field with nearest, as well as next‐nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest‐neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1‐dimensional Heisenberg‐Dirac model, similar to the partially solved ``Ising‐Heisenberg'' model of Lieb‐Mattis‐Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given aswhere A_{q}, B_{q} , and E_{q} are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.

Criteria of Accuracy of Approximate Wavefunctions
View Description Hide DescriptionThe accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ  ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.

Foundations for Quantum Mechanics
View Description Hide DescriptionA discussion is given of the structure of a physical theory and an ``ideal form'' for such a theory is proposed. The essential feature is that all concepts should be defined in operational terms. Quantum (and classical) mechanics is then formulated in this way (the formulation being, however, restricted to the kinematical theory). This requires the introduction of the concept of a mixed test, related to a pure test (or ``question'') just as a mixed state is related to a pure state. In the new formulation, the primitive concepts are not states and observables but certain operationally accessible mixed states and tests called physical. The notion of a C*‐system is introduced; each such system is characterized by a certain C*‐algebra. The structure of a general C*‐system is then studied, all concepts being defined in terms of physical states and tests. It is shown first how pure states and tests can be so defined. The quantum analog of the phase space of classical mechanics is then constructed and on it is built a mathematical structure, called a q‐topology, which is a quantum analog of the topology of classical phase space. Mathematically, a q‐topology is related to a noncommutative C*‐algebra as an ordinary topology is related to a commutative C*‐algebra. Some properties of the q‐topology of a C*‐system are given. An appendix contains some physically motivated examples illustrating the theory.

Finite Integral Equations for Green's Functions for :φ^{4}: Coupling. I
View Description Hide DescriptionA fully renormalized set of integral equations is derived for the Green's functions of a theory with :φ^{4}: coupling.

Photoelectron Shot Noise
View Description Hide DescriptionThe instants of time emission of photoelectrons generated by a detector immersed in an optical field constitute a point compound Poisson process. A complete definition of such a process is introduced to calculate some average values of the distribution. The shot noise due to this point process is also considered and we study the difference between the ``deterministic'' and the ``random'' shot noises. They are completely defined by the set of their characteristic functions. We consider also the asymptotic properties of the shot noise and we show that for large mean density of the point process the fluctuations are not described by a Gaussian, but by a Gaussian compound random function. Thus the central limit theorem is not strictly valid. An experimental setup to obtain these fluctuations is described and some statistical properties of the asymptotic shot noise are presented.

Local Gauge Fields in General Relativity
View Description Hide DescriptionThe carrier space for internal degrees of freedom is assumed to be the 6‐dimensional space spanned by generatorsL _{ iκλ } of local Lorentz transformations defined at each event of a Riemann space. Gauge potentials are introduced in this internal space which are like the Yang‐Mills potentials except that they are related to parameters of the event connection by a map which involves the L _{ iκλ } as connecting quantities. The map is similar to the one which relates the Dirac spin space and spin connection to the Riemannian geometry of event space. An algebraic uniqueness condition is shown which is necessary and sufficient for the map to be one‐to‐one in a neighborhood of a solution of the mapping equations: The metric field is undetermined by a uniform scale transformation. If the gauge potentials have an internal holonomy group which is a subgroup of the Lorentz group and satisfy the free Yang‐Mills equations, and if the uniqueness condition is satisfied, then the committed by the inverse map automatically satisfy the free Einstein field equations, indicating that only the gravitational field is included in this mathematical framework. In an attempt to include nongravitational properties of matter, we have then considered internal holonomy groups larger than the Lorentz group. The part of the gauge potentials causing the enlargement is shown to form a current which may be interpreted as a source of the gravitational field.

Anholonomic Bases in the Study of the Spin Connection and Gauge Potentials in a Riemann Space
View Description Hide DescriptionIf parameters of a spin connection have a holonomy group which is the Lorentz group or a subgroup thereof and if a certain algebraic restriction involving the Riemann tensor, say , is satisfied, then there is a one‐to‐one relation between the spin connection ^{s}Γ_{μa b } and parameters of a Riemannian event connection Γ_{μκλ }, in a neighborhood of an assumed solution of the equations which establish the correspondence [see H. G. Loos, Ann. Phys. 25, 91 (1963)]. We have, by the use of anholonomic bases in the Riemann space, simplified the equations which determine the map and have shown that if the algebraic condition is satisfied, a one‐to‐one correspondence holds also in the large. The same method, involving anholonomic bases, has also been used to study the map between gauge potentials acting in the internal space formed by generators of local Lorentz transformations and parameters of connection of a Riemannian event space. An analogous theorem is proved which asserts that if is satisfied, then the correspondence between the gauge potential and the parameters of the event connection is one‐to‐one.

Solution of a Three‐Body Scattering Problem in One Dimension
View Description Hide DescriptionThe scattering problem of three equal particles interacting via repulsive inverse‐cube forces is solved in one dimension both in the classical and the quantal cases. The quantal S matrix is similar to that produced by repulsive δ‐function interaction of infinite strength.

Expansions in Spherical Harmonics. V. Solution of Inverse Legendre Transforms
View Description Hide DescriptionExplicit formulas are derived for inverse Legendre transforms, i.e., for the solutions of the integral equationfor arbitrary integer L and general functions f(r). It is shown that the functions f(r), defined by 2‐sided transforms (b = −1), follow uniquely from the 1‐sided ones (b = 0), but an arbitrary function of parity opposite to L (supplementary function) may be added to G_{L} in the 2‐sided case. For L ≥ 2, an arbitrary polynomial of the same parity as L, but of degree lower than L (complementary function), may be added to G_{L} . The complementary and supplementary functions do not affect the values of the integrals for the radial dependence in the expansion of a function of a vector sum in spherical harmonics. A power term in f(r) leads to a power term in G_{L} , except for those powers which occur in the complementary function for which G_{L} involves logarithms. Inverse transforms are also obtained for a restricted number of negative powers and some recurrence relations are derived.

Tensor Harmonics in Canonical Form for Gravitational Radiation and Other Applications
View Description Hide DescriptionAn analysis is made of the relation between the tensor harmonics given by Regge and Wheeler in 1957 and those given by Jon Mathews in 1962. This makes it possible to use the Regge‐Wheeler harmonics, which are given in terms of derivatives of scalar spherical harmonics, for calculations while using Mathews' form of the harmonics [linear combinations of the elements of the product basis formed from a basis for scalar functions on the 2‐sphere and a basis for symmetric tensors such that the product basis is split into sets which transform under the irreducible representations of SO(3)] to elucidate the properties of tensor harmonics. Thus, a convenient orthonormal set of harmonics is given which is useful in studying, for example, gravitational radiation.