Volume 6, Issue 5, May 1965
Index of content:

Relativistic Formulation of the Lifetime Matrix in the Potential Theory of Collision
View Description Hide DescriptionThe lifetime matrices for collision processes as described by the Klein‐Gordon and Dirac equations are formulated. The results obtained are exactly the same as in the case of a collision described by the Schrödinger equation. This gives us a consistent operator theory for the collision lifetime in the case of potential scattering.

A Set of Harmonic Functions for the Group SU(3)
View Description Hide DescriptionWe construct a set of harmonic functions carrying all the irreducible representations of the group SU(3). Some features of these functions are discussed in detail.

Operators that Lower or Raise the Irreducible Vector Spaces of U _{ n−1} Contained in an Irreducible Vector Space of U_{n}
View Description Hide DescriptionWe define operators that lower or raise the irreducible vector spaces of a semisimple subgroup of a semisimple Lie group contained in an irreducible vector space of the group. We determine the lowering and raising operators for the canonical subgroup U _{ n−1} of the unitary group U_{n} . With the help of these operators, which are polynomial functions of the generators of U_{n} , and the corresponding operators for the subgroups in the canonical chain U_{n} U _{ n−1} … U _{2} U _{1} we can obtain, in this chain, the full set of normalized basis vectors of an irreducible vector space of U_{n} from any given normalized basis vector of the vector space. In particular we can obtain, using only the lowering operators, the set of basis vectors from the basis vector of highest weight of the vector space. This result is of importance in applications to many‐body problems and in the determination of the Wigner coefficients of U_{n} . In future papers we plan to determine the lowering and raising operators for the orthogonal and symplectic groups.

Symbolic Calculus of the Wiener Process and Wiener‐Hermite Functionals
View Description Hide DescriptionA new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener‐Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener‐Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener‐Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener‐Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly‐Gaussian random function.

Wiener‐Hermite Expansion in Model Turbulence in the Late Decay Stage
View Description Hide DescriptionThe Wiener‐Hermite functional expansion, which is the expansion of a random function about a Gaussian function, is here substituted into the Burgers one‐dimensional modelequation of turbulence. The result is a hierarchy of equations which (along with initial conditions) determine the kernel functions which play the role of expansion coefficients in the series. Initial conditions are postulated, based on physical reasoning, criteria of simplicity, and the assumption that the series is to represent the late decay stage (in which the Gaussian correction is small and also decreasing with time). These are shown to justify an iterative solution to the equations. The first correction to the Gaussian approximation is calculated. This is then tested by evaluating the correction to the flatness factor, which for an exactly Gaussian function has the value 3, but which has been found by experiment (in real three‐dimensional fluids, of course) to have a value which deviates from the Gaussian value increasingly rapidly with the order of the derivative. We utilize this effect as a test of the inherent ability of the Wiener‐Hermite expansion to bring to realization the physical properties implicit in the Navier‐Stokes or Burgers equations. The various contributions to the flatness‐factor deviation, when computed, do show a potential capability of providing a theoretical basis for the effect.

Algebraic Tabulation of Clebsch‐Gordan Coefficients of SU _{3} for the Product (λ, μ) (1, 1) of Representations of SU _{3}
View Description Hide DescriptionAn algebraic tabulation is made of the Clebsch‐Gordan (CG) coefficients of SU _{3} which occur in the reduction into irreducible representations of the direct product (λ, μ) (1, 1) of irreducible representations of SU _{3}. Full explanation is made of the method of handling the complications associated with the possible double occurrence of the representation (λ, μ) itself in the direct product. The phase convention employed is an explicitly stated generalization of the well‐known Condon and Shortley phase convention for SU _{2}. The relationship of the CG coefficients associated with the direct product (1, 1) (λ, μ) to those coefficients already mentioned is also exhibited.

On ``Diagonal'' Coherent‐State Representations for Quantum‐Mechanical Density Matrices
View Description Hide DescriptionIt is proved that every density matrix is the limit, in the sense of weak operator convergence, of a sequence of operators each of which may be represented as an integral over projection operators onto coherent states (in the sense of Glauber) with a square‐integrable weight function. This result is a special case of one that holds for all operators with trace and for overcomplete families of states other than just the coherent states. We prove our more general result, at no cost of complexity, within the more general framework of continuous‐representation theory. The significance of our results for representing traces of operators is indicated.

S‐Operator Theory. I. Formulation
View Description Hide DescriptionIt is postulated that the S operator is dynamically independent of the wavefunctions of the asymptotic states. From this postulate a functional differential equation for the S operator is developed. Its solutions include both the renormalized and unrenormalized Feynman‐Dyson S operators; appropriate boundary conditions distinguish the two solutions. Interpolating quantum fields are defined in terms of the S operator and are only of secondary importance in this theory. Calculations in this theory are not appreciably more difficult than the corresponding calculations with the unrenormalized S operator.

Tensor Methods and a Unified Representation Theory of SU _{3}
View Description Hide DescriptionStarting with irreducible tensors, we develop an explicit construction of orthonormal basic states for an arbitrary unitary irreducible representation (λ, μ) of the group SU _{3}. A knowledge of the simple properties of the irreducible tensors can then be exploited to obtain a variety of results, which ordinarily require more abstract algebraic methods for their derivation. As illustrative applications, we (i) derive Biedenharn's expressions for the matrix elements of the generators of SU _{3}, (ii) compute the matrix elements of octet‐type operators for the case (λ, μ) → (λ, μ), and (iii) develop an explicit unitary transformation connecting the isospin and the U‐spin states in any arbitrary irreducible representation.

Multipole Matrix Elements of the Translation Operator
View Description Hide DescriptionFormulas are given for the expansion of multipole fields of arbitrary tensorial character into multipole fields about a shifted origin. The expansion coefficients are given as matrix elements of the translation operator. In analogy to the matrix elements of the rotation operator, we introduce for these matrix elements a standard form which represents a parallel displacement of the coordinate system along the z axis. Any arbitrary translation of the coordinate system then consists of a consecutive application of a rotation, a standard translation, and a rotation. Since the multipole fields form a complete set any arbitrary function can in principle be expressed in a shifted coordinate system by means of the given formulas. All mathematical derivations are given.

Variational Principle for Saturated Magnetoelastic Insulators
View Description Hide DescriptionIn a previous paper, a system of nonlinear differential equations and boundary conditions governing the macroscopic behavior of arbitrarily anisotropic nonconducting magnetically saturated media undergoing large deformations, was derived. The derivation utilized the classical procedure of defining field vectors and determining the equations relating them by applying energy and momentum conservationtheorems. The macroscopic effect of the quantum mechanical exchange interaction was included as was dissipation and the associated thermodynamics. The magnetic field was assumed to be quasistationary. In this paper a variational principle is presented, which is shown to yield the aforementioned system of equations and boundary conditions in the absence of dissipation and heat flow.

Lagrangian Theory for the Second‐Rank Tensor Field
View Description Hide DescriptionThe second‐rank tensor field φ^{μν} is decomposed into its various subspaces under the Lorentz group and the appropriate projection operators are exhibited explicitly. The most general local, Hermitian, free‐field Lagrangian which can be formed from this field is written down, and the corresponding equations of motion and subsidiary conditions are derived by means of a variational principle. Finally some possible applications of this theory are discussed (in particular spin‐2 boson theory), and all the possible couplings of this field to a Dirac particle are listed in full.

Exact Bootstrap Solutions in Some Static Models of Meson‐Baryon Scattering
View Description Hide DescriptionWe study the exact bootstrap solutions to four well‐known models of meson‐baryon scattering in the nonrecoil, one‐meson approximation. The models are the neutral scalar theory, the charged scalar theory, the symmetric scalar theory, and the neutral pseudoscalar theory. A bootstrap solution is defined to be a solution satisfying Levinson's theorem of potential scattering. It is found that the existence of a bootstrap solution depends crucially on the high‐energy conditions, which enter the problem through a cutoff function and through subtractions in the dispersion relations. In all the models considered there is no bootstrap solution with no subtraction. With one subtraction there exists more than one bootstrap solution. However, the requirements that (a) the meson‐baryon coupling constant should be different from zero, and (b) there should be no inelastic threshold below the elastic threshold, render the bootstrap solution unique. Positions of bound states and their coupling constants depend on two arbitrary parameters, which may be taken to be the cutoff momentum and the subtraction constant.

Compactification of Minkowski Space and SU(3) Symmetry
View Description Hide DescriptionSeveral ideas from the geometric side of Lie grouptheory are presented that may be relevant to the search for groups containing the internal and Poincaré group symmetries.

Problem of Subtractions in Potential Scattering
View Description Hide DescriptionWe study the problem of subtractions in potential scattering theory. We show that the single spectral functions can be obtained by analytic continuation of the Mellin transform of the double spectral function. The N/D method could thus be avoided, in principle.

Note on Non‐Landau Singularities
View Description Hide DescriptionSome non‐Landau singularities are discussed using the formalism of Fotiadi, Froissart, Lascoux, and Pham. The simple cases of self‐energy and vertex diagrams are treated, as well as the sixth‐order scattering ladder diagram.