Volume 9, Issue 6, June 1968
Index of content:

Global Existence of Spin Structures for Gravitational Fields
View Description Hide DescriptionThe notion of a spin structure for a gravitational field is defined, and it is shown that a spin structure exists if and only if the second Stiefel‐Whitney class of the space‐time M vanishes. The number of different spin structures is then equal to the number of elements in H ^{1}(M, Z _{2}).

Lie‐Group and Lie‐Algebra Inhomogenizations
View Description Hide DescriptionA systematic formulation of the concept of inhomogenization is given both for Lie groups and for Lie algebras, and the connection between the two structures is clarified in terms of the notion of semidirect product. Special emphasis is devoted to the classification of the inhomogenizations of semisimple Lie algebras. As an application, a lemma due to O'Raifeartaigh is generalized to a wide class of inhomogeneous structures.

Inhomogenizations and Complex Representations
View Description Hide DescriptionReal Lie‐algebra inhomogenizations are considered for complex defining representations and some emphasis is devoted to the case when the defining representation is irreducible. A theorem is given concerning the conditions under which nonequivalent complex representations give rise to isomorphic inhomogenizations and a classification is made of the complex inhomogenizations of semisimple Lie algebras.

Ising‐Model Spin Correlations
View Description Hide DescriptionIn this paper we study a class of planar Ising lattices which have two or more directions along which the pair‐correlation function admits a Toeplitz determinant representation. On the basis of the recently developed theory of Toeplitz determinants, we discuss the asymptotic behavior of these correlations at the critical point.

Exact Quantization Conditions
View Description Hide DescriptionThe method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one‐dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.

Some Total Invariants of Asymptotically Flat Space‐Times
View Description Hide DescriptionThe total energy, momentum, supermomentum, and angular momentum of asymptotically flat space‐times are calculated in terms of coordinate and conformally invariant expressions by taking the limit in an invariant way of the asymptotic symmetry linkages through a sequence of finite closed two‐spaces which converge to a sphere at null infinity. The resulting expressions consist of integrals over the sphere at null infinity of cordinate and conformally invariant quantities. In the case of energy and momentum these integrals may be reduced to expressions previously proposed by Penrose.

Electric‐Field Penetration into a Plasma with a Fractionally Accommodating Boundary
View Description Hide DescriptionLandau's field penetration study is extended to a plasma with a boundary that reflects a fraction σ of the incident electrons specularly and the remainder diffusely. Exact solutions for specular and diffuse reflection, and series solutions for fractional accommodation are obtained. At great depths in the plasma the field is found to exhibit negligible σ dependence for ω near ω_{ p } and weak dependence through a factor (1 + σ) at other frequencies.

Generalized Ornstein‐Zernike Approach to Critical Phenomena
View Description Hide DescriptionA generalization of the Ornstein‐Zernike integral equation is derived and suggestions are made about a possible application to an improved theory of critical phenomena. A fundamental maximum principle of statistical mechanics is used to place the generalized equation in the context of phase transitions and critical points. The equation is a relationship between a generalized correlation matrix by means of which the average fluctuation product of any two sum functions may be expressed and a generalized direct‐correlation matrix which is the second functional derivative of the functional in the maximum principle. The existence of a critical eigenvector of the direct‐correlation matrix is proposed and three physical meanings of this vector are given. An explicit formula for the direct‐correlation matrix is given and is used to derive two asymptotic properties. This formula exhibits an unexpected relationship between the generalized Ornstein‐Zernike equation and the Percus‐Yevick equation.

Convergence of the Born Series
View Description Hide DescriptionThree types of Born series which can be associated with a transition amplitude are discussed, and the criteria for the convergence of the different series are compared. With regard to the divergence of the Born series, the ordering, matrix‐element series diverges implies vector series diverges implies operator series diverges, is obtained for the natural vector and operator Born series that can be abstracted from the expression for the transition amplitude. The conclusion that the divergence of the operator Born series does not ensure that the Born series of physical matrix‐elements divergences is applied to an example of three‐body rearrangement scattering.

Are Bloch Bands at Finite Field Adiabatically Connected to Those at Zero Field?
View Description Hide DescriptionIt has been shown previously that if a potential consists of a superposition of a periodic part and a uniform electric field, then for a particle moving in this field there are Bloch bands closed in time. The present paper addresses itself to the question whether these bands may be identified with the field‐free bands. The most natural thing is to expect that the bands are slightly field dependent, but converge toward the field‐free bands as E goes to zero. Bands for which this is true are said to be adiabatically connected to corresponding bands at zero field. In Sec. 2 of the paper, two model cases are given for which this adiabatic connection pertains. Section 3 is the central part of the paper and provides the conclusion that the answer to the question in the title is almost always negative. In this proof the positive cases serve an essential function. It is shown that the parameters of the periodic potential must obey at least one supplementary condition to allow adiabatic connection, and that the collected cases precisely obey this condition. Adiabatic connection is thus generally not possible. Section 4 provides an explicitly soluble case which does not allow adiabatic connection. An infinite number of field values E converging toward zero are found at each of which the two bands under consideration switch identity (hyperbolic rather than linear connection at energy crossings). The connection postulated in the effective‐mass approximation must therefore be of a nonadiabatic nature. It probably involves the ``sudden'' approximation of quantum theory.

Angular Momentum and the Kerr Metric
View Description Hide DescriptionFor Kerr's rotating metric, it is shown that −ma is the angular momentum of the body where m is the mass and a is the rotation parameter. This is true even for large m and a.

New Derivation of the Integro‐Differential Equations for Chandrasekhar's X and Y Functions
View Description Hide DescriptionThe X and Y functions of radiative transfer satisfy a system of integro‐differential equations which form the basis of an effective numerical treatment. These integro‐differential equations are derived from the integral equation for the source function and the differential equation for the resolvent.

Diffuse Transmission of Light from a Central Source through an Inhomogeneous Spherical Shell with Isotropic Scattering
View Description Hide DescriptionA partial‐differential‐integral equation is derived in this paper for the angular distribution of the radiation which is diffusely transmitted through an inhomogeneous, isotropically scattering, spherical shell when there is a constant net flux of radiation normally incident on the inner surface. An equation is also derived for the strength of the diffusely reflected radiation when the shell is illuminated at each point on the outer surface by constant isotropic incident radiation.
The equations obtained appear to lend themselves well to numerical solution. Astrophysically, the situation corresponds to determining the brightness of a spherical planetary nebula. As far as is known, the equations are new and exact.

Electromagnetic Two‐Body Problem for Particles with Spin
View Description Hide DescriptionA two‐body system is considered for classical particles with charge, spin, and magnetic moment. The particles move in circular orbits, with spins orthogonal to the orbital plane, and they interact through time‐symmetric electromagnetic fields. Rigorous relativistic equations of motion and rigorous expressions for the total energy and angular momentum of the system, including contributions from the field, are obtained.

Possible Interpretation of Quantum Mechanics
View Description Hide DescriptionIt is shown that to Schrödinger's equation, one may associate a Markoff process in the Smoluchowski approximation with an external force acting on the system which is a measure of its interaction with the vacuum. This interaction is, in this scheme, responsible for the stochastic character of the motion. The physical interpretation of the usual momentum and energy operators emerges in a natural way from the theory. Thus, we are immediately led to Heisenberg's uncertainty relations. However, this interpretation of quantum mechanics is valid only within the limits of validity of Smoluchowski's equation. As a simple example, we treat the one‐dimensional harmonic oscillator.

Simple Generalization of Schrödinger's Equation
View Description Hide DescriptionA possible generalization of quantum mechanics is examined by showing that the motion in phase space of a classical Brownian particle may be described by a complex probability amplitude depending on the phase‐space coordinates and the time, and obeying a Schrödinger‐like equation. However formal this result may seem, the usual dynamical operators may be defined whose physical meaning stems directly from the theory. An outstanding feature of the formalism is that ordinary quantum mechanics in configuration space may be recovered in a limiting process whereby the velocity variable, defined now through a statistical distribution, is eliminated. Therefore, it plays the role of a hidden variable. This result supports recent reinterpretations of von Neumann's theorem on the nonexistence of such variables in quantum mechanics and serves as a counterexample of the usual interpretation of his theorem.

Local Field Theory and Isospin Invariance. I. Free‐Field Theory of Spinless Bosons
View Description Hide DescriptionThe properties of free‐field theories of spinless bosons are investigated. The self‐conjugate boson field theories of isospin ½, , , ⋯ are shown to be nonlocal: the energy, isospin, and number densities fail to commute for spacelike separations. The difficulty is traced to the necessary occurrence of non‐independent field variables. For the anomalous case the dependent variables are nonlocally related to the independent ones; elimination of the former makes superficially local quantities depend nonlocally on the independent fields. The connection with canonical field theory is investigated. The p and q coordinates have anomalous commutation relations in the nonlocal case. Although the Hamiltonian appears normal in every case, the anomalous theories are characterized by zero (integrated) Lagrangian.Antiparticle conjugation is investigated in detail with attention to phase questions. For anomalous theories two types of conjugation are found, one of which nonlocally relates the field to a superposition of its adjoint. A unitary transformation is constructed which converts one type to the other. Finally, CPT transformation properties of normal and abnormal theories are derived and compared.

Nonequilibrium Statistical Mechanics of Open Systems
View Description Hide DescriptionA theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed. This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary systemS in interaction with a ``large'' reservoir R. The dynamics of S are analyzed on the basis of a precise quantum‐mechanical treatment of the microscopic equations of motion for the combined systemS + R. On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many‐particle structure of R, its initial state, and its coupling to S. This master equation corresponds to a self‐contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state. This dynamical law is a generalization of the quantum‐mechanical Liouville equation to a form appropriate to open systems.

Matrix Elements of Operators of Class I Representations of SL(n, C)
View Description Hide DescriptionLet D be an irreducible representation of the group SL(n, C) by operators T_{g} in a Hilbert spaceH. We introduce a canonical basis of vectors h ∈ H, and show how to evaluate the ``D‐functions'' (h _{1}, T_{g}h _{2}). It has been suggested that one such function has the physical interpretation of the electric‐charge form factor of the pion. This function is evaluated in detail.

Path‐Integral Calculation of the Quantum‐Statistical Density Matrix for Attractive Coulomb Forces
View Description Hide DescriptionA method for the calculation of the two‐particle statistical density matrix for attractive Coulomb forces is described. The path‐integral expression for the density matrix is reduced to a modified path integral which involves summation over only one‐dimensional paths. This expression is then approximated by an iteration procedure using direct numerical quadratures. The results obtained are related directly to the quantum‐mechanical radial‐distribution function for a plasma at small ion‐electron separations.