Volume 18, Issue 8, August 1977
Index of content:

Quantum axiomatics, representation theorem, and communicability of observations in physics
View Description Hide DescriptionWe show how a fundamental assumption in the Dirac formulation of quantum mechanics, namely that the states of a physical system at a particular time are mathematically represented by unit vectors in Hilbert space, can be deduced from certain aspects of our experimental procedures and of the observed outcome of quantum mechanical experiments. Our assumptions have clear empirical meaning and the results hold true for any dimensionality of the system, without anomalies in low dimensions which exist in the two well‐known axiomatic approaches to quantum mechanics. The propositional logic approach of Birkhoff and von Neumann does not work for quantum systems of dimension less than four and requires an assumption which does not have an empirical basis. Jordan algebra axioms, on the other hand, also lead to anomalies in low dimensions and, moreover, are formal and cannot be directly physically interpreted. In our work it was possible to avoid these shortcomings.

On the field of the quantum mechanical Hilbert space
View Description Hide DescriptionThe well‐known limitations on the field of the quantum mechanical vector space, which, within the framework of the propositional logic of Birkhoff and von Neumann, can be obtained only for systems of dimension greater than or equal to four, are obtained here for all dimensions. The propositional logic fails to provide any information about the quantum systems of dimension less than four and, moreover, one does not know the empirical meaning of one of its basic assumptions. The Jordan algebra approach, on the other hand, which provides the same limitations on the field, also suffers from anomalies in low dimensions and is altogether formal rather than physical. In the present work, which is based on a recent study of the topological properties of quantum states, there are no low‐dimensional anomalies and the assumptions have clear empirical meaning.

Symmetries of the stationary Einstein–Maxwell field equations. I
View Description Hide DescriptionThe Einstein equations for stationary axially symmetric gravitational fields are written in several extremely simple forms. Using a tensor generalization of the Ernst potential, we give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations and gage transformations, and (iii) the infinite parameter group K of Geroch which combines both. We then show how the same thing can be done to the Einstein–Maxwell equations. The enlarged internal group H′ now includes the Harrison transformations, and is isomorphic to SU(2,1). The enlarged group K′ contains even more parameters, and generates even more potentials and conservation laws.

Symmetries of the stationary Einstein–Maxwell field equations. II
View Description Hide DescriptionFrom Einstein–Maxwell fields which are stationary and axially symmetric, we show how to construct an infinite hierarchy of potentials. The potentials form a representation of K′, the infinite‐parameter symmetry group of the Einstein–Maxwell equations. For flat space, the hierarchy is calculated explicitly.

Stability of solutions of the compressible Navier–Stokes equations
View Description Hide DescriptionThe purpose of the paper is to consider the effect of the viscous terms of the compressible Navier–Stokes equations in situations where significant variations of the flow variables occur only over many mean‐free paths. Under these conditions it is shown that the magnitude of the viscous terms is comparable to the addition in the inviscid equations of a forcing term which is small relative to appropriately normalized initial data. The appropriate normalization is discussed. Stability, in terms of a stability parameter, is defined relative to such perturbations. A large value of the stability parameter implies that the solution is ill‐conditioned. Application of the results proceeds in the following manner. Let (t,x,y) be a given solution to the Navier–Stokes equations. Let V (t,x,y) be the corresponding solution to the inviscid equations. Let M be the maximum deviation that occurs between and V. From , one obtains an ε which represents an upper bound on the magnitude of the viscous terms. If M is not significant, then it is clear, without any further analysis, that the viscous terms are not significant. If M is significant, one proceeds to obtain a lower bound on the stability parameter λ, namely λ⩾M/ε. If ε is small, it is now possible to conclude that the solution is ill‐conditioned. (If ε is large, the analysis remains valid, but no useful information is obtained.) Two particular applications are made. The first considers the known solutions to shock wave structure equations (particularly the case of the weak shock). The second considers solutions to the incompressible equations. It is shown that in many situations the use of the time‐dependent incompressible Navier–Stokes equations is unjustified.

The Fermi method of quantizing the electromagnetic field as a model for quantum field theory
View Description Hide DescriptionIn a previous paper we demonstrated that the Fermi method for quantizing the electromagnetic field had a rigorous C*‐algebra version. Here we investigate some properties of our formalism and show (a) that the Fermi method provides an example of spontaneous symmetry breaking in a quantum field theory, (b) that it raises some interesting questions about automorphisms of Weyl systems, and (c) that it provides a prototype for higher‐spin zero‐mass field theories.

Wavepacket scattering in potential theory
View Description Hide DescriptionA contour integration technique is developed which enforces the initial conditions for wavepacket‐potential scattering. The expansion coefficients for the exact energy eigenstate expansion are automatically expressed in terms of the plane wave expansion coefficients of the initial wavepacket thereby simplifying what is usually a tedious, mathematical process. The method is applicable regardless of the initial spatial separation of the wavepacket from the scattering center.

Causally symmetric spacetimes
View Description Hide DescriptionCausally symmetric spacetimes are spacetimes with J ^{+}(S) isometric to J ^{−}(S) for some set S. We discuss certain properties of these spacetimes, showing for example that if S is a maximal Cauchy surface with matter everywhere on S, then the spacetime has singularities in both J ^{+}(S) and J ^{−}(S). We also consider totally vicious spacetimes, a class of causally symmetric spacetimes for which I ^{+}(p) =I ^{−}(p) =M for any point p in M. Two different notions of stability in general relativity are discussed, using various types of causally symmetric spacetimes as starting points for perturbations.

Asymptotic behavior of integral and integrodifferential equations
View Description Hide DescriptionLong time behavior of integral and integrodifferential equations is studied. Some of them are generalizations of the models for the transport of charged particles in a random magnetic field; the solution of the homogeneous integrodifferential equation has an algebraic‐logarithmic decay for long times, whereas the solution of the inhomogeneous equation has a slower logarithmic decay.

Group‐theoretical foundations of classical and quantum mechanics. I. Observables associated with Lie algebras
View Description Hide DescriptionThis paper is a first attempt to explore the relationship between classical and quantum mechanics from a group‐theoretical point of view. We deal here with the algebraic aspects of the sets of classical and quantum observables in the framework of the algebraic structures associated with finite‐dimensional Lie algebras. In particular, we investigate the canonical structure of the quotient fields predicted by the Gel’fand–Kirillov and Vergne conjectures in order to study the types of observables that emerge from a given Lie algebra.

Nonlocal interactions: The generalized Levinson theorem and the structure of the spectrum
View Description Hide DescriptionFredholm theory is applied to the Lippmann–Schwinger equation for nonlocal potentials without spherical symmetry. For a specified large set of trace‐class interactions it is proved that when, for real k≠0, the Fredholm determinant vanishes, k ^{2} is the energy of a bound state. The point k=0 is examined and the analog of the distinction between zero‐energy bound states and zero‐energy resonances for local central potentials is found. A generalized Levinson theorem is proved.

Geometrodynamics with tensor sources. IV
View Description Hide DescriptionWe develop the Hamiltonian hypersurface dynamics of the gravitational field derivatively coupled to general tensor sources. The closing of constraints follows from the independence of the hypersurface action on the path in the space of embeddings. The derivative coupling breaks the DeWitt supermetric in Riem (m).

Product integrals and the Schrödinger equation
View Description Hide DescriptionA brief introduction to product integration is given. The theory developed is used to give a simple and rigorous analysis of the asymptotic behavior (r→∞) of positive‐energy solutions of the radial Schrödinger equation. Absence of positive‐energy bound states is proved for various classes of potentials. It is shown that E=1 is the o n l y positive energy for which the Wigner–von Neumann potential can have a positive‐energy bound state. The results proved imply (as will be shown in a later publication) existence of the Mo/ller wave matrices for the potential V (R) = (sinr)/r and various related potentials. A brief discussion is given to justify the WKB approximation which gives the wavefunction asymptotically for large positive values of the energy E.

On the theory of time‐dependent linear canonical transformations as applied to Hamiltonians of the harmonic oscillator type
View Description Hide DescriptionWriting the canonical variables (q^{ T }, p^{ T }) as (ω^{ T }), we develop a method for transforming the time‐dependent Hamiltonian H=A _{μν}(t) ω^{μ}ω^{ν}+B _{μ}ω^{ν} +C (t) to the time‐independent form = (1/2) δ_{μν} ^{μ} ^{ν} using the linear transformation ^{μ}=s ^{μ} _{ν}(t) ω^{ν} +r ^{μ}(t). Differential equations are obtained for the parameters s ^{μ} _{ν} and r ^{μ}. The transformed Hamiltonian enables the construction of an invariant I and an invariant matrix [I ^{μν}]. These invariants apply to both the classical and quantum mechanical problems. The invariant I has the dynamical symmetry group SU(n), and this characterizes all systems with Hamiltonians of the form of H.

Derivation of quantum mechanics from stochastic electrodynamics
View Description Hide DescriptionFrom the equation of motion for a radiating charged particle embedded in the zero‐point radiation field we construct a stochastic Liouville equation which serves to derive, by a smoothing process, a Fokker–Planck‐type equation with infinite memory. We show that an exact alternative form of this phase‐space equation is the Schrödinger equation in configuration space, with radiative corrections. In the asymptotic, radiationless limit (when the radiative corrections became negligible), the phase‐space density reduces to Wigner’s distribution, thus confirming Weyl’s rule of correspondence. We briefly discuss several other implications of stochastic electrodynamics which are relevant for quantum theory in general.

Impedance, zero energy wavefunction, and bound states
View Description Hide DescriptionWe show that the presence, or absence, of bound states in the three‐dimensional Schrödinger equation directly depends on the existence of zeros for a function which is a zero energy solution of the equation and which has the meaning of an impedance in a related equation. Several inequalities that are sufficient to prevent the existence of bound states are obtained from this remark. Some of them are new and bridge the gap between previous results.

Varieties of symmetry breaking in a class of gauge theories
View Description Hide DescriptionIn unified gauge theories, the Higgs particles can interact in various ways. The problem of finding the symmetry‐breaking directions can become very complicated in nontrivial cases, where the scalar fields have many interactions. A method is presented which predicts, in a simple way, the possible types of spontaneous symmetry breaking in a theory symmetric under the group U(N _{1}) ⊗⋅⋅⋅⊗U(N _{ j }) ⊗SU(M _{1}) ⊗⋅⋅⋅⊗SU(M _{ k }). Within its framework it is possible to obtain results by drawing a new kind of graph. It is found that in such models, various phases (and hence phase transitions) are possible. There are distinct hierarchies in the symmetry breaking strengths and they are related.

Spontaneous symmetry breaking of U_{1}(M) ⊗U_{2}(M) ⊗U_{3}(M) with colored Higgs particles
View Description Hide DescriptionIn a previous article a general method for finding the symmetry‐breaking directions in a theory which is symmetric under a group U(N _{1}) ⊗⋅⋅⋅⊗U(N _{ j }) ⊗SU(M _{1}) ⊗⋅⋅⋅⊗SU(M _{ k }) was presented. In order to explain and exhibit its utility, we derive in this paper the possible directions of spontaneous symmetry breaking in a U_{1}(M) ⊗U_{2}(M) ⊗U_{3}(M) theory with ’’colored’’ Higgs particles. As an example we discuss the symmetry breaking in the Pati–Salam model which is found to be a legitimate one. A simple explanation of the stability of the quarks inside a ’’particle‐phase’’ is exhibited.

Correlation functions in the spherical and mean spherical models
View Description Hide DescriptionA transformation is obtained relating spherical and mean and spherical averages. The kernel of the transformation is the probability density of N ^{−1}Σ^{ N } _{ i=1} x _{ i } ^{2} in the mean spherical model. The transformation is inverted to obtain a simple method for computing spherical averages from mean spherical averages. Averages in the two ensembles are identical except in zero field below the critical temperature.

Solitons on moving space curves
View Description Hide DescriptionIt is shown that the motion of certain types of helical space curves may be related to the sine–Gordon equation and to the Hirota equation (and consequently to the nonlinear Schrödinger equation and to the modified Korteweg–de Vries equation). The intrinsic equations that govern the motion of space curves are shown to provide the various linear equations that have been introduced to solve these evolution equations by inverse scattering methods.