Volume 4, Issue 6, June 1963
Index of content:

Propagator Ghosts and Regge Poles
View Description Hide DescriptionIn the approximation of considering only vertex contributions to the propagator, we determine sufficient conditions on the two‐particle scattering amplitude which does not generate ghosts in the propagator. The Regge type of behavior does not satisfy these conditions, and will give rise to ghosts unless cancellations occur coming from the higher inelastic contributions to the vertex‐function unitarity condition, which have been neglected in our calculation. The double‐pole behavior discussed by Frye is found to be the only two‐particle scattering‐amplitude behavior which will have no propagator ghosts, under certain general conditions.

Divergence Conditions and the Gravitational Field Equations
View Description Hide DescriptionIt is shown that in addition to the four divergence laws T ^{αβ} _{β} = 0, stress tensors satisfying the Einstein field equations must also satisfy the additional divergence conditions . Conversely, under quite general assumptions these ten conditions on the source, T _{αβ}, dictate the metric—matter connection proposed by Einstein.

Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry
View Description Hide DescriptionFermi coordinates, where the metric is rectangular and has vanishing first derivatives at each point of a curve, are constructed in a particular way about a geodesic. This determines an expansion of the metric in powers of proper distance normal to the geodesic, of which the second‐order terms are explicitly computed here in terms of the curvature tensor at the corresponding point on the base geodesic. These terms determine the lowest‐order effects of a gravitational field which can be measured locally by a freely falling observer. An example is provided in the Schwarzschild metric. This discussion of Fermi Normal Coordinate provides numerous examples of the use of the modern, coordinate‐free concept of a vector and of computations which are simplified by introducing a vector instead of its components. The ideas of contravariant vector and Lie Bracket, as well as the equation of geodesic deviation, are reviewed before being applied.

Distortion in the Metric of a Small Center of Gravitational Attraction due to its Proximity to a Very Large Mass
View Description Hide DescriptionThe problem of two bodies in general relativity in an especially restricted mode is analyzed. One body is of small mass, and, under the influence of gravitational attraction, moves toward a much larger mass whose field produces ``tidal'' deformations in the geometry of the smaller one. To evaluate these deformations, we treat the Schwarzschild metric of the particle by a perturbation analysis similar to that performed by Regge and Wheeler. The boundary conditions for this analysis are obtained from the metric of the background field expressed in a novel set of comoving coordinates, here called Fermi normal coordinates. These coordinates have been described in the previous paper. In this paper we use the Schwarzchild metric given there in comoving coordinates as an asymptotic approximation for the full solution which is evaluated here.
A perturbation analysis using the ``tidal deformation'' (background curvature) as an expansion parameter is performed on the metric of a small Schwarzschild particle. The solution so obtained satisfies Einstein's equations for empty space for small deviations from a nonflat metric. This solution also reduces to the Fermi metric in the appropriate limit, namely, when the ``distance'' from the geodesic is large compared to the radius of the small mass but small compared to the separation of the two masses. Thus only the quadratic terms in the distance are important. This solution is then used to find the deformations in shape of the throat of the wormhole and to locate this region of symmetry (the throat) by finding a coordinate transformation which leaves the metric invariant.

Parameter Differentiation of Quantum‐Mechanical Linear Operators
View Description Hide DescriptionQuantum‐mechanical linear operators have some real‐number parameters in them. The differential coefficients of functions, including eigenvalues and eigenvectors, of such an operator with respect to a parameter are calculable from the differential coefficient of the operator with respect to the parameter. The formulas for this purpose are presented together with their proofs.

Galilei Group and Nonrelativistic Quantum Mechanics
View Description Hide DescriptionThis paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up‐to‐a‐factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero‐mass case (though in fact there are no physical zero‐mass systems in nonrelativistic mechanics). Finally, we study the two‐particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

Principle of General Q Covariance
View Description Hide DescriptionIn this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space, the lattice of subspaces has a symmetry group which is isomorphic to the group of all co‐unitary transformations in . In contrast to the complex space (ordinary Hilbert space), this group is connected, while for it consists of two disconnected pieces.
The subgroup of transformations in which associates with every quaternion q of magnitude 1, the correspondence ψ → qψq ^{−1} for all (called Q conjugations), is isomorphic to the three‐dimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point.
The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q‐covariant derivative and Q curvature are natural consequences thereof.
There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion η(x) which may be a continuous function of the space—time coordinates. It corresponds to the i in the Schrödinger equation of ordinary quantum mechanics. We consider η(x) as a fundamental field, much like the tensorg _{μν} in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of η and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.

Reduction of S‐Matrix Elements
View Description Hide DescriptionWe have obtained expressions for the commutators of in‐ and out‐fields with the generalized retarded operators of Burgoyne and Ruelle. These are then used to show that the matrix elementwhere p ≡ (p _{1}, p _{2}, ···, p_{n} ) is arbitrary, can be reduced in terms of retarded functions without using the assumption of stability of one‐particle states.

Theorem on Energy Shifts Due to a Potential in a Large Box
View Description Hide DescriptionA theorem conjectured by Cohen, which gives a concise form to the Rayleigh—Schrödinger perturbation series for energy shifts due to a potential in a large box, is proved.

Generalized Dielectric Function for Classical Many‐Body Systems
View Description Hide DescriptionA dielectric function is introduced for classical systems of many interacting particles in terms of the N‐particle distribution function in phase space. The system is considered as a whole using the Liouville equation instead of the canonical equations or the B–B–G–K–Y hierarchy. To facilitate the calculation, the diagrammatic technique of Prigogine and Balescu is introduced in terms of which a systematic asymptotic analysis is carried out for two simple cases: (1) a system of weak interaction and (2) a simple model of plasma. The treatment is restricted to systems in thermal equilibrium.

Lorentz—Lorenz and Sellmeier Formulas in Irregular Gases
View Description Hide DescriptionThe integral equation expressing the condition of dynamical equilibrium of dipoles in an applied electromagnetic field is solved to determine the index of refraction for the ensemble‐average wave in a media in which the polarizability varies irregularly. The solution closely follows that employed in the usual derivation of the Ewald—Oseen theorem for isotropic media. A Lorentz—Lorenz formula, modified by the appearance of an additional term, is determined. The additional term is real when the scale length of the irregularities is small compared to the wavelength. For larger irregularities, however, the correction leads to a complex index of refraction, expressing the attenuation due to scattering. Applying the concept of the depolarizing effect of electron—ion collisions, these results are extended to the case of an ionized gas. In this latter instance, a similar modification of the Sellmeier formula for the index of refraction is determined. Explicit formulas are given for the case of irregularities with scale large compared to a wavelength.

Zeros of Hankel Functions and Poles of Scattering Amplitudes
View Description Hide DescriptionThe complex zeros ν_{ n }(z), n = 1, 2, ··· of H _{ν} ^{(1)}(z), dH _{ν} ^{(1)}(z)/dz and are investigated. These zeros determine the poles in the scattering amplitudes resulting from scattering of various kinds of waves by spheres and cylinders. Formulas for ν_{ n }(z) are obtained for both large and small values of z and for large values of n. In addition, for H _{ν} ^{(1)}(z) and dH _{ν} ^{(1)}(z)/dz,numerical solutions are found for real z in the interval 0.01 ≤ z ≤ 7 and n = 1, 2, 3, 4, 5. The resulting loci of ν_{ n }(z) in the complex ν plane are presented. These loci are the trajectories of the so‐called Regge poles for scattering by spheres and cylinders.

Charged Spheroid in Cylinder
View Description Hide DescriptionThe problem of a charged conducting spheroid within a coaxial conducting cylinder is solved by a slight variation of the method in J. Appl. Phys. 31, 553 (1960). Errors in the terminal digits of Table I in that paper have been corrected and the table extended. The charge density on the spheroid, the potential between it and the cylinder, and its capacitance are given for ratios of the spheroid equatorial radius to cylinder radius of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95. Tables give numerical results for five cases: the disk, the oblate spheroid with 2 to 1 axial ratio, the sphere, and the prolate spheroid with axial ratio 1 to 2. The thin prolate spheroid requires special treatment.

Quadratic Invariants of Surface Deformations and the Strain Energy of Thin Elastic Shells
View Description Hide DescriptionA conceptual theory of thin, elastic shells is developed. A general form for the strain energy is postulated in terms of parameters which characterize, in the sense of Euclid, the geometry of the shell. The strain energy permits a formal definition of generalized stresses and then a statement of equilibrium. The physical interpretation of the stresses is obtained from the boundary conditions. The energy coefficients, which are not derivable within the scope of this theory, are related to material constants by using classical results derived from the three‐dimensional theory of elasticity. The result is a complete, self‐contained theory of shells which can be used for prediction. The methods used depend on a combination of Cartesian tensors and exterior differential forms.

Statistical Behavior of Linear Systems with Randomly Varying Parameters
View Description Hide DescriptionWe consider an nth‐order system of ordinary linear differential equations whose coefficients are random functions of the time. In particular, we discuss the case where each of these coefficients is a random noise. A differential equation for the probability distribution of the solutions of the random D. E. is derived and from this the moments can be calculated. Special attention is given to the case of Gaussian noise but the treatment is applicable to any type of noise. Finally, various conditions for stability are discussed.