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Volume 5, Issue 3, March 1964

A No‐Interaction Theorem in Classical Relativistic Hamiltonian Particle Dynamics
View Description Hide DescriptionIt is shown that a relativistically invariant classical mechanical Hamiltonian description of a system of three (spinless) particles admits no interaction between the particles. If a set of ten functions of the canonical variables of the three‐particle system satisfies the Poisson bracket relations characteristic of the ten generators of the inhomogeneous Lorentz group, and‐with the canonical position variables of the particles‐satisfies the Poisson bracket equations which express the familiar transformation properties of the (time‐dependent) particle positions under space translation, space rotation, and Lorentz transformation, then this set of ten functions can only describe a system of three free particles. A significant part of the proof is valid for a system containing any fixed number of particles. In this general case, a simplified form is established for the Hamiltonian and generators of Lorentz transformations, and it is shown that the generators of space translations and space rotations can be put in the standard form characteristic of free‐particle theories. The proof of the latter involves a generalization from one to many three‐vector variables of the angular momentum Helmholtz theorem of Lomont and Moses.

Quantum Electrodynamics of the Stimulated Emission of Radiation. I
View Description Hide DescriptionThe quantum electrodynamic formalism of Senitzky is developed so as to yield long‐timedifferential equations for the time‐average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two‐level molecules, and a dissipation mechanism. The cavity‐mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T _{1} and T _{2} time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules.
Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time‐average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off‐resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well‐behaved differential equations are obtained for the time‐average expectation values of field energy and molecular excess population, justified on a long‐time basis by the application of an intermittent similarity transformation. A differential equation for the time‐average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct‐product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum‐entropy inference. In conclusion, the equations of motion for the time‐average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid‐state laser beam during the emission period.

Functional Differential Calculus of Operators
View Description Hide DescriptionThe functional derivative with respect to operators of operator functionals is defined for operators which satisfy certain commutation relations of interest in quantum field theory. From this definition, a functional differential calculus is developed for functionals of tensor as well as spinor fields. It is noted that an implicit definition of the functional derivative can always be given while an explicit one seems to exist only for those operator fields which need not be restricted by supplementary operator conditions, and for which not more than one derivative occurs in the commutation relations.

Feynman Integrals and the Schrödinger Equation
View Description Hide DescriptionFeynman integrals, in the context of the Schrödinger equation with a scalar potential, are defined by means of an analytic continuation in the mass parameter from the corresponding Wiener integrals. The method yields a new interpretation of highly singular attractive potentials in quantum mechanics. For the example of the attractive 1/r ^{2} potential, this interpretation agrees with classical mechanics in the correspondence limit.

An Application of Sommerfeld's Complex‐Order Wavefunctions to an Antenna Problem
View Description Hide DescriptionUsing the orthogonality relations of Sommerfeld's complex‐order wavefunctions, the exact solution for the problem of electromagnetic radiation from a circularly symmetric slot on the conducting surface of a dielectric‐coated cone is obtained. The results are valid for the near‐zone region as well as for the far‐zone region, and they are applicable for arbitrary‐angle cones. It is noted that the technique used to solve this problem may be applied to similar types of problems involving conical structure, such as the diffraction of waves by a dielectric‐coated, spherically tipped cone.

Regge Poles and Potentials with Cores
View Description Hide DescriptionThe analytic behavior of the S matrix as a function of real momentum and complex angular momentum is examined for the case of nonrelativistic scattering by potentials with hard‐core interiors. The presence of a hard core in a potential introduces a type of symmetry which, combined with unitarity, places restrictions upon the movement of the Regge poles and upon the allowable representations of the scattering amplitude. In particular, it is found that the Watson‐Sommerfeld transform representation of the scattering amplitude is not valid. An alternate representation of the Sfunction, more practical for such cases, is postulated and examined numerically. Numerical results are presented for the case when the hard core is replaced by a finite barrier added to a repulsive Yukawa potential interior. It is found that additional trajectories appear. The mechanism for producing the threshold behavior of the phase shifts is compared to that of the case where a hard core is not present.

Generalized Adiabatic Invariance
View Description Hide DescriptionIn this paper we find the quantities that are adiabatic invariants of any desired order for a general slowly time‐dependent Hamiltonian. In a preceding paper, we chose a quantity that was initially an adiabatic invariant to first order, and sought the conditions to be imposed upon the Hamiltonian so that the quantum mechanical adiabatic theorem would be valid to mth order. [We found that this occurs when the first (m − 1) time derivatives of the Hamiltonian at the initial and final time instants are equal to zero.] Here we look for a quantity that is an adiabatic invariant to mth order for any Hamiltonian that changes slowly in time, and that does not fulfill any special condition (its first time derivatives are not zero initially and finally).

Convergence of Yukawa Theories with a Finite Number of Interacting Boson Modes
View Description Hide DescriptionConvergence of the perturbation expansion of the S matrix is demonstrated for a field theory where a quantized fermion field with regularized propagator interacts with a finite number of quantized boson modes. Conventional graphical techniques and combinatorial analysis are used to establish this result, derived earlier by Edwards, but we believe not properly clarified. A zero radius of convergence is demonstrated for the theory where a Yukawa‐type interaction takes place between one boson field interacting in only a finite number of modes with another boson field coupled to it bilinearly. The relation of this result to the convergence of infinite coupled mode theories is discussed.

The Connection between Conservation Laws and Laws of Motion in Affine Spaces
View Description Hide DescriptionIt was pointed out recently that, for any theory describing matter as a collection of mass points in a metric space and subject to a covariant conservation law for a symmetric tensor density , the geodesic law of motion as well as the form of follow from the conservation law alone, independent of any equations obeyed by the metric. This result is shown to be valid in any affine space, independent of any equations obeyed by the affine connection; conversely, the geodesic law implies a conservation law for a singular symmetric tensor density. Similarly, the existence in any affine space of a covariant conservation law for a vector density describing a collection of point charges is shown to imply the constancy of charge, and the form of ; conversely, the constancy of charge implies a conservation law for a singular vector density. Some applications of these results are presented. An Appendix contains a discussion of the laws of motion for particles with an intrinsic dipole moment.

Determinantal Method in Perturbation Theory
View Description Hide DescriptionThe determinantal method of deriving fundamental relations of a new perturbation theory, which the author has presented recently, is demonstrated. This method is simpler than the Green's function method which has been adopted in the previous paper. The Brillouin‐Wigner perturbation theory is discussed for comparison.

Energy Sharing and Equilibrium for Nonlinear Systems
View Description Hide DescriptionA study is made of a one‐dimensional system of identical particles in which the forces between neighbors are linear. The system is nonlinear because it is assumed that collisions occur between adjacent particles, which each have an effective diameter d. The energiesE_{i} in the linear normal modes are computed numerically to show that energy is freely exchanged between all the modes in the system, as predicted by the theory. Furthermore, the time averages 〈E_{i} 〉 of these energies show a strong tendency towards equipartition of energy among the modes. This is in distinct contrast to the computations of Ulam, Fermi, and Pasta, which showed that some nonlinear systems appear to be nonergodic. An equation of state and an expression for the total energy of the system as a function of thermodynamic coordinates are derived via statistical mechanics. Expected values for the pressure and temperature of the assembly may then be computed. A comparison of these with the numerical values of those variables arising from the computations shows that the nonlinear system approaches equilibrium.

Spectral Analysis of the Anisotropic Neutron Transport Kernel in Slab Geometry with Applications
View Description Hide DescriptionA spectral analysis of the transport kernel for anisotropicscattering in finite slabs is achieved by first solving a type of generalized scattering problem for a subcritical slab. Initially, the scattering problem is stated as an inhomogeneous integral transportequation with a complex‐valued source function. This is readily transformed to singular integral equations and linear constraints in which the space and angle variables enter as parameters. Dual singular equations appear in applications of Case's method to transport problems, but we cannot yet completely explain this duality. The singular equations are transformed to Fredholm equations by an extension of Muskhelishvili's standard method and by analytic continuation. It is shown that, for a wide class of scatteringfunctions, this particular Fredholm reduction yields equations which converge rapidly under iteration for all neutron productions and slab thicknesses. The ultimate solution of the singular equations contains arbitrary constants which, when evaluated by the aforementioned linear constraints, display explicitly the Fredholm determinant and the eigenfunctions of the transport kernel. An immediate consequence of this result is the criticality condition and the associated neutron distribution. Specific applications to linear anisotropic and isotropic scattering in slab geometry are discussed. In addition, it is seen that the case of isotropic scattering in spheres can be treated with this method, and, in fact, the spectral analysis of the kernel for the slab problem immediately applies to the sphere kernel.

Some Exact Radial Integrals for Dirac‐Coulomb Functions
View Description Hide DescriptionThe zero energy loss Dirac‐Coulomb integrals are evaluated using the technique of contour integration. The expressions obtained have a closed analytic form, showing that these integrals are formally similar to the corresponding classical and nonrelativistic quantum mechanical, zero energy loss integrals which also have exact elementary solutions.
Application of the zero energy loss Dirac‐Coulomb integrals occurs in inelastic electron scattering and similar problems. The investigation of the finite energy loss Dirac‐Coulomb integrals requires a study of the zero energy loss integrals as a preliminary.

Integration of the Partial Differential Equations for the Hypergeometric Functions F _{1} and F_{D} of Two and More Variables
View Description Hide DescriptionFrom Kummer's 24 solutions of the ordinary hypergeometric equation and their connections, the general solution of the partial differential equations for the Appell function F _{1} in the neighborhood of the singular points is found. Connections between the solutions are given to the extent required to continue the F _{1} function to the neighborhood of any of its singular points.
The corresponding problem for the hypergeometric function F_{D} of more than two variables is briefly indicated.

The Complete High‐Energy Behavior of Ladder Diagrams in Perturbation Theory
View Description Hide DescriptionThe Mellin‐transform method for obtaining the high‐energy behavior of Feynman integrals is modified and applied to the set of ladder diagrams. The complete set of terms of the form s ^{−1}(ln s)^{ n } is summed, and gives an equation for the trajectory function which is analogous to that obtained by Fredholm methods for a Yukawa potential. A perturbation expansion for α(t) valid for t large is given, and the threshold behavior investigated. The results confirm the reliability of the perturbation‐theory method of investigation. They also exhibit directly the connection between high‐energy behavior and the poles of the scattering amplitude.