Index of content:
Volume 6, Issue 12, December 1965

On the Representations of the Semisimple Lie Groups. V. Some Explicit Wigner Operators for SU _{3}
View Description Hide DescriptionA general method for the calculation of the [21 … 10] Wigner operators is presented and used to obtain specific results for the SU _{3} group. General expressions for the SU _{3} reduced Wigner operators are given for tensor operators transforming like the representations [100], [110], and [210].

A Relation between the Hydrogen Atom and Multidimensional Harmonic Oscillators
View Description Hide DescriptionA simple connection between the radial Schrödinger equation for the bound states of a hydrogen atom with angular momentuml and that of an isotropic harmonic oscillator of even dimension from 2 to 4l + 4 is noted. The case of highest dimensionality 4l + 4 is shown to lead in a very simple way to the energy levels and degeneracies of the hydrogen atom, once the energy levels of a one‐dimensional harmonic oscillator are known.

Quantum Electrodynamics of the Stimulated Emission of Radiation. II
View Description Hide DescriptionThe equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second‐order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self‐consistent expressions for commutators such as 〈[P _{ n }(t _{1}), P _{ m }(t _{2})]〉_{ d }, P _{ n } being the operator‐amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self‐consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose‐Einstein formula for thermal energy in each mode at time t _{2} = t _{1}. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space‐time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator‐amplitudes of a parametric amplifier are derived in the creation‐annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss‐averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.

Cartesian Tensors in Spaces with an Indefinite Metric with Applications to Relativity
View Description Hide DescriptionA new formalism for dealing with Cartesian tensor analysis in flat spaces with an indefinite metric with the same ease as in Euclidean spaces is introduced. It avoids the necessity of distinguishing covariant and contravariant indices and the consequent use of the metric tensor in raising and lowering indices; neither does it require the introduction of imaginary coordinates and components of tensors. It is based on the use of a modified Einstein summation convention combined with a modified differentiation with respect to tensor components in such a way that all formal manipulations are essentially identical with those employed in Cartesian tensor analysis in Euclidean spaces. The further introduction of a modified matrix multiplication and a modified definition of a determinant serves to round out the formal analogy with the Euclidean space. The convenience, simplicity, and typographical economy of the new formalism is illustrated by examples drawn from special relativity. The formalism can be trivially generalized to complex linear vector spaces with an indefinite metric.

Thermodynamic Evolution Equation for a Quantum Statistical Gas
View Description Hide DescriptionIt is shown by a perturbation diagram method that partial Green's functions satisfy closed evolution equations exact in the thermodynamic limit except for the neglect of the contribution of initial particle correlations. This contribution is negligible for a system at a time t ≫ τ_{c} (average collision duration) when the main interaction processes are localized in space and time. In particular this makes unnecessary the often imposed assumption of the absence of initial particle correlations for the derivation of the usual Boltzmann equation or its generalizations.

Momentum‐Space Analyticity Properties of the Bergman‐Weil Integral for the Three Point Function
View Description Hide DescriptionThe momentum‐space analyticity domain of the Bergman‐Weil integral representation of the vertex function or three‐point function is investigated using the assumption that thresholds can be introduced simply as lower limits of integration for the mass variables in the representation. It is shown that this assumption leads to a regularity domain which is larger than the domain following from general physical assumptions of Lorentz invariance, local commutativity and reasonable mass spectrum. To simplify the discussion we also assume that the vertex function is regular when all three complex variables lie in the same half‐plane. Standard techniques for the evaluation of Feynman diagrams have proved to be inconvenient for this investigation and we have developed new methods taking explicit advantage of the fact that we only have two external vectors and, hence, can work in a two‐dimensional Lorentz space. Further, the vanishing of the masses for certain internal lines has also been exploited. The techniques we have used here might be of interest also in other connections.

Green's Functions and Superfluid Hydrodynamics
View Description Hide DescriptionThe two‐fluid hydrodynamics of Landau and Tisza is related to the thermodynamicGreen's function formulation of the many‐body problem. A specific approximation for the self‐energies yields directly the superfluidhydrodynamics in the limit of slow space and time variation. The approximation chosen is the simplest possible form for the self energies which includes the effects of collisions and which also satisfies the differential conservation laws for the mass, momentum, and energy. Of course these self energies are not adequate for a realistic description of ^{4}He. Nonetheless the structure of this model may provide some insight into the exact theory. For example, in the model employed here the self‐energies and Green's functions obey several integral identities which are utilized in the derivation of the two‐fluid hydrodynamics, and which yield useful expressions for some thermodynamic quantities. It is speculated that the identities and the consequent expressions for the various thermodynamic quantities may have a wider range of validity than their derivation.

A New Phase‐Space Distribution Function in the Statistical Theory of the Electromagnetic Field
View Description Hide DescriptionIn a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase‐space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvaluesz of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase‐space distribution function is discussed.

An Exactly Soluble Model Showing Ferromagnetism
View Description Hide DescriptionA Hamiltonian of a one‐dimensional Heisenberg model with ferromagnetic interaction is expressed in terms of fermion operators, and renormalized linked cluster expansion has been carried out. It is shown that the approximation leads to ferromagnetic behaviors similar to the molecular field approximation. A model for which the second‐ and higher‐order terms vanish is presented and regarded as an example of quantum‐mechanical systems that shows ferromagnetism rigorously. It is also noted that the model is very much like the Husimi‐Temperley model or the van der Waals gas with respect to a long‐range character of interaction and the vanishing of higher‐order terms.

Expansion of a Function of Noncommuting Operators
View Description Hide DescriptionThe Taylor series expansion of a function f(A + B) of noncommuting operators A and B is written in several ways. Recursion relations, generating functions, and some explicit formulas for operator coefficients of this expansion are given.

On Expanding the Exponential
View Description Hide DescriptionSystematic methods for generating various expansions of the function exp (a + λb)t, of noncommuting operators a and b, are presented. The usual perturbation expansion in powers of λ, and several other formulas found in the earlier literature occur as special cases. Brief remarks about relative merits and physical applications are made.

Note on Orthogonal Polynomials which are ``Invariant in Form'' to Rotations of Axes
View Description Hide DescriptionThe precedure of Bhatia and Wolf for constructing orthogonal sets whose elements are ``invariant in form'' with respect to rotations of axes, is extended to include sets which are defined over the entire two‐dimensional plane. Use was made of the Gram‐Schmidt process to derive general expressions for generating elements of many unique and complete sets corresponding to different circularly symmetric weight functions for two cases, where in the first case the elements are only functions of the real variables x and y while in the other case they are also functions of the real variable r = (x ^{2} + y ^{2})^{½}. These general expressions were used to obtain two new, unique and complete sets corresponding to Gaussian and exponential weight functions, respectively. The radial polynomials for these two sets were found to be closely related to the Laguerre polynomials. The generatingfunctions for these radial polynomials are also given.

Half‐Space Neutron Transport with Linearly Anisotropic Scattering
View Description Hide DescriptionThe method developed by Case is used to solve four time‐independent, one‐speed problems for neutron transport in a homogeneous medium where the scattering function is linear in the cosine of the scattering angle. The solutions to the albedo, Milne, Green's function, and constant isotropic source problems for a half‐space are facilitated by the use of half‐range bi‐orthogonality relations between the eigenfunctions of the homogeneous transport equation. Expressions are also derived for the emerging angular densities and the densities and net currents on the surface of the half‐space.

An Expansion Method for Treating Singular Perturbation Problems
View Description Hide DescriptionCochran's method for treating singular perturbation problems is shown to give, in some cases, expansions which are not uniformly valid. This method is modified and extended to give uniformly valid expansions. The new method is applied to the problem of heat transfer in a duct to give a solution in agreement with that obtained by the WKBJ method.

Generalized Solutions for Massless Free Fields and Consequent Generalized Conservation Laws
View Description Hide DescriptionGeneralized solutions to the equations for a massless free field with arbitrary spin are written down. It is shown that they lead immediately to generalizations of all the usual conservation laws.

Reduction of a One‐Loop Feynman Diagram with n Vertices in m‐Dimensional Lorentz Space
View Description Hide DescriptionAn explicit formula is given for the evaluation of a diagram with one loop and n vertices in m‐dimensional Lorentz space (n > m.). The result is given as a sum of terms each corresponding to a one‐loop diagram with m vertices and with a coefficient which can be obtained from rules analogous to the rules of residue calculus.

The Froissart‐Gribov Continuation and Reggeon Unitarity Conditions. I
View Description Hide DescriptionAn identity is derived which is useful in the discussion of unitary integrals. It is used to discuss single Regge pole insertions in the Froissart‐Gribov continuation and with the help of perturbation‐theory models a derivation is given of reggeon unitarity conditions. The general form suggested by Gribov, Pomeranchuk, and Ter‐Martirosyan is confirmed. Finally cancellation mechanisms are discussed and their relation to the mechanism generating Regge poles is emphasized.

Critical Störmer Conditions in Quadrupole and Double Ring‐Current Fields
View Description Hide DescriptionA theoretical study has been made of the behavior of the critical Störmer pass points for general axially symmetric magnetic configurations. A topological method has been derived to predict the occurrence of the critical Störmer conditions for charged‐particle exclusion. This analytic technique, when applied to geomagnetically interesting fields, should be a useful aid to the understanding of experimental data. The method is applied here to three magnetic geometries: double ring currents with parallel dipole moments and with antiparallel dipole moments, and the axial magnetic quadrupole. The topology of the regions representing allowed motion is treated systematically and the behavior of the critical pass points is illustrated in typical Störmer plots. For the quadrupole and the antiparallel ring system critical pass points are found to occur only out of the equatorial plane. For the parallel ring system, critical points can occur in or out of the equatorial plane. For certain special conditions as many as three simultaneous critical pass points are found, and two simultaneous points occur for a wide range of parameters.

Isotopic Space, Complex Conjugation, and U _{3} Symmetry for Particles
View Description Hide DescriptionThe inherent U _{4} symmetry over spinor spaces of the regular decomposition indicated previously is discussed from the viewpoint of a single spinor space. It is noted that this symmetry does not yet include isotopic spin. Isotopic space is introduced by considering that the space‐time vectors and its vector Clifford algebra are imbedded in the algebra of a six‐dimensional Euclidean space. Three of the latter's dimensions are identified with ordinary space, and three are identified with isotopic space. The ``time vector'' corresponds to the pseudoscalar element in isospace, and the imaginary unit is expressed as a linear combination of the bivectors in isospace. The relation among complex conjugation, time inversion, and space inversion is thereby clarified. Some comments on the application of these ideas to particle symmetry discussions are made. In particular one obtains as the first extension of the SU _{2} symmetry of isotopic spin, a U _{3} symmetry for particles.

Spin‐Matrix Polynomials and the Rotation Operator for Arbitrary Spin
View Description Hide DescriptionA technique for the expansion of an arbitrary analytic function of a spin matrix is developed in terms of a complete set of polynomials based on the characteristic equations of the spin matrices. The expansion coefficients are determined by use of the ascending difference operator from the calculus of finite differences. The expansions are valid not only for functions of a spin matrix but also for functions of a complex variable. In the latter case the eigenvalues of the spin matrices are the zeros of the polynomials. In either case the expansion coefficients are the same although of course in the case of a spin matrix the series terminates. As an example the rotation operator for arbitrary spin is developed via its functional analog in terms of these polynomials, and the region of convergence of the series to the function is investigated.