Volume 13, Issue 11, November 1972
Index of content:

Operational Statistics. I. Basic Concepts
View Description Hide DescriptionA generalized notion of a ``sample space'' is developed which allows for the simultaneous representation of the outcomes of a set of related ``random experiments.'' Affiliated with each such generalized sample space is a so‐called ``logic,'' the elements of which are propositions that can be confirmed or refuted by observing the outcomes of the random experiments. Stochastic models for the experimental situation represented by a given sample space are introduced, and it is shown that such stochastic models induce generalized probability measures on the logic of this sample space.

Some Theorems on Atomicity in Axiomatic Quantum Mechanics
View Description Hide DescriptionGiven a complete orthocomplemented lattice L and a set S of nonnegative real functions on L, sufficient conditions are established that S should fulfill in order that L be atomic. The conditions are investigated under which L may be represented by the lattice of all closed subspaces of a separable Hilbert space. (As is well known, the atomicity of L plays an important role here.) Some unsolved problems are pointed out. In axiomatic quantum mechanics, the lattice L may represent the set of propositions whereas the set of functionsS represents the set of physical states. The conditions imposed on the pair (L,S) then have a simple and plausible physical interpretation; an important condition imposed on (L,S) is the existence of the ``maximal'' (i.e., maximally determined) states which appear in the theory as limit constructions.

Approximate Functional Integral Methods in Statistical Mechanics. I. Moment Expansions
View Description Hide DescriptionIn this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)) . The lowest‐order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum‐statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck‐Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration‐space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum‐mechanical perturbation theory, in order to answer the perennial question, does the path‐integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman‐Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path‐integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.

All Stationary Vacuum Metrics with Shearing Geodesic Eigenrays
View Description Hide DescriptionThe general solution of the field equations of stationary vacuum gravitational fields possessing geodesic eigenrays with nonvanishing shear is obtained. Nontrivial solutions exist only if the eigenrays do not rotate. The resulting metrics fall into two classes: either there is a functional dependence among the field quantities (this class belongs to the Papapetrou solutions), or the quantity γ^{0}, which in the shear‐free case has been interpreted as the central mass, is uniquely determined. This latter class consists of two space‐times. The curvature invariants vanish in the r→∞ limit for both solutions; however, the metrics exhibit singular behavior in this limit.

Scalar‐Tensor Theory of Gravitation in a Lyra Manifold
View Description Hide DescriptionA closed‐form exact solution to the field equations of a scalar‐tensor theory, formally similar to the Brans‐Dicke theory, is obtained. It is shown that the present theory predicts the same effects, within observational limits, as the Einstein theory.

Gauge Invariant Decomposition of Yang‐Mills Potentials
View Description Hide DescriptionThe Yang‐Mills (YM) potentials are decomposed into an isovector part and a part which transforms nonhomogeneously under local gauge transformations. Two decompositions are shown; one of them is based on a gauge‐invariant version of the transversality condition, and the other arises from a gauge‐invariant modification of the Lorentz condition. The latter is Lorentz as well as gauge invariant. The gauge invariance of the decompositions is obtained at the expense of locality since the separate parts of the decomposed potential are functionals of the full YM potential. The transverse‐longitudinal decomposition is used to throw the YM sourceless field equations into a gauge‐invariant Hamiltonian form. Static fields in the Hamiltonian formulation are discussed. The decompositions are used to construct massive, gauge‐invariant but nonlocal Lagrangians. A Lorentz and gauge‐invariant nonlocal interaction of the YM field with a spinor‐isospinor field is formed. The transverse‐longitudinal decomposition is used to investigate the geometric structure of a configuration space Ω of YM potentials. The nonexistence of submanifolds of Ω orthogonal to the gauge‐invariant manifoldsX ∈ Ω is proved in contradistinction to the electromagnetic case. A Green's functional for the Yang‐Mills field is represented explicitly by an infinite power series of functionals and is shown to be self‐adjoint.

Null Plane Restrictions of Current Commutators
View Description Hide DescriptionWe consider some mathematical aspects of the problems of defining restrictions of quantum fields and commutators of such fields to null planes. We give precise meanings to these restrictions and discuss how these lead to unambiguous derivations of the usual formal results. We discuss and relate various definitions of null plane charges and derive some of their properties such as vacuum annihilation. We define and exhibit finite null plane restrictions for causal solutions of the Klein‐Gordon equation. Commutator functions defined by integral representations with various spectral functions are then considered. Light cone operator product expansions are used to calculate some null plane current commutators. In this way we can give precise derivations of Fubini sum rules and electroproduction structure function asymptotic behavior.

Quantization of a General Dynamical System by Feynman's Path Integration Formulation
View Description Hide DescriptionThe Schrödinger equation is obtained by Feynman's path integration method of quantization for a general dynamical system. The meaning of the results is discussed.

An ``H‐Theorem'' for Multiplicative Stochastic Processes
View Description Hide DescriptionIn a recent paper the author showed how multiplicative stochastic processes lead to a potentially comprehensive theory for nonequilibrium phenomena. In this paper an ``Htheorem'' is proved from results obtained using multiplicative stochastic processes.

Algebraic Equations for Bethe‐Salpeter and Coulomb Green's Functions
View Description Hide DescriptionThe equation for the Bethe‐Salpeter Green's function in the case of two scalar quarksinteracting via the exchange of a scalar particle of zero mass is transformed to an algebraic equation with the help of the dynamical group SO(5, 2). From this equation a one‐parameter integral representation of the Green's function is obtained in the case of maximal binding, and from this representation the Green's function is calculated in terms of a hypergeometric function. The equation for the f‐dimensional nonrelativistic Coulomb Green's function is also transformed to an algebraic equation.

Absence of Long‐Range Order in Thin Films
View Description Hide DescriptionThin films are described as idealized systems having finite extent in one direction but infinite extent in the other two. For systems of particles interacting through smooth potentials (e.g., no hard cores), it is shown that an equilibrium state for a homogeneous thin film is necessarily invariant under any continuous internal symmetry group generated by a conserved density. For short‐range interactions it is also shown that equilibrium states are necessarily translation invariant. The absence of long‐range order follows from its relation to broken symmetry. The only properties of the state required for the proof are local normality, spatial translation invariance, and the Kubo‐Martin‐Schwinger boundary condition. The argument employs the Bogoliubov inequality and the techniques of the algebraic approach to statistical mechanics. For inhomogeneous systems, the same argument shows that all order parameters defined by anomalous averages must vanish. Identical results can be obtained for systems with infinite extent in one direction only.

Exponential Fourier Transforms for Coupled Harmonic Oscillator Chains
View Description Hide DescriptionThe exponential Fourier transform is used to study the dynamics of semi‐infinite and infinite chains of interacting harmonic oscillators. In addition to the harmonic coupling between nearest neighbors, each oscillator is subjected to frictional and other external time‐dependent forces. In contrast with previous studies on such systems, the initial conditions (at t = 0) are not specified, and the motion of all the oscillators is expressed in terms of the given applied forces only. The analytic structure of the transforms as well as some properties of the propagators are studied for all possible values of physical constants including the limiting values for uncoupled oscillators. The inverse transforms not readily available from tables are obtained by carrying out the integrations explicitly.

Some Asymptotic Expressions for Prolate Spheroidal Functions and for the Eigenvalues of Differential and Integral Equations of Which They Are Solutions
View Description Hide DescriptionThe asymptotic behavior of the prolate spheroidal functions of order zero f_{n} (x, c), where n is the number of zeros of the function in the interval − 1 ≤ x ≤ 1, is studied for large values of the parameter c and all values of n. The method used involves solving the differential equation which defines the functions by using a classical approximation. The corresponding eigenvalues χ_{ n } are given by an implicit equation and the norm of the functions is calculated. The functions f_{n} (x, c) are also solutions of an integral equation and associated with eigenvalues λ_{ n }(c). Asymptotic expressions of [1 − λ_{ n }(c)] are derived by using the values obtained for the norm of f_{n} (x, c). All these results generalize and interpolate partial results obtained by Slepian and others in two special cases, namely, n finite and n ≃ c.

Generalized Semidirect Product in Group Extensions
View Description Hide DescriptionThe case when there exists a homomorphism σ of a group G into Aut(K) of a non‐Abelian group K[σ having at most one image in every coset of Aut(K) with respect to I(K)] is investigated. It is shown that any extension E ∈ ext_{σ}(G, K) can be obtained as a generalized semidirect product , where H belongs to ext_{σ}(G, C) (the group C being the center of K), the semidirect product of K and H is based on τ which equals (n being the homomorphism of H onto G), and C′ is the antidiagonal of . The GSP is a natural generalization of the central extensions, it is applicable to most groups in theoretical physics, and it has a suitable form for the derivation of the irreducible representations of E.

Relation between the Statistical Representations of Real and Associated Complex Fields in Optical Coherence Theory
View Description Hide DescriptionIn the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristicfunctionals that in order to determine the joint probability distributions of the complex field at N space‐time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well‐known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.

Existence and Uniqueness Questions for the Unitarity Equation
View Description Hide DescriptionIn early articles on this subject, it was asserted that the transformation which arises from the generalized optical theorem for inverse scattering of scalar waves is completely continuous in certain spaces. In this article we show this is not correct: This transformation is not compact in these spaces. We also obtain an improved uniqueness result for the case where there is no spherical symmetry. The article concludes with a discussion of the proper setting of the unitarity equation (generalized optical theorem) in the larger context of the inverse scattering problem.

Note on Nonlinear Representations
View Description Hide DescriptionIn the theory of non‐linear representations of a continuous group G with respect to a closed subgroup H, the peculiar transformation behavior of the reduction matrix L _{φ} is found to be identical with the transformation behavior of a set of coset representative elements of G with respect to H. The limitations of the extended definition of the boost by Salam and Strathdee are discussed.

A Study of Relaxing Waves
View Description Hide DescriptionThe equation is studied when an initial finite pulse u(x, 0) is given. For the linearized equation, general solutions in terms of Laplace transforms are obtained and more explicit expression for exponential kernels is given. An iteration expansion scheme is established for general kernels. For positive kernels, it is found that the stability condition for the solution is . Then the large time solution as well as the solution representing the main disturbance is obtained. For the nonlinear equation, the condition for the shock formation is obtained for the special case f(t) = μe ^{−μt }, or when the nonlinearity is weak.

Polarized Elastic Materials with Electronic Spin‐A Relativistic Approach
View Description Hide DescriptionIn this article, we derive the balance laws and constitutive equations of polarized elasticsolids with electronic spin by use of a relativistic (special) variational principle. The theory is fully dynamical and nonlinear. It is shown that this approach encompasses several previous works in micromagnetism (magnetoelastic interactions) and elasticdielectrictheory.

Relativistic Continua with Directors
View Description Hide DescriptionIn this article, we give an axiomatic introduction of directors in the space‐time continuum of Minkowski. A variational principle is presented that leads to the complete set of field equations, boundary conditions, and jump relations of the Kafadar‐Eringen theory of relativistic polar media. The constitutive equations follows from the variational formulation.