Volume 20, Issue 8, August 1979
Index of content:

On the coupling of self‐conjugate systems with S̄L̄ (3̄,R) symmetry
View Description Hide DescriptionThe coupling of unitary self‐conjugate S̄L̄ (3̄,R) multiplicity‐free irreps is explicitly calculated. It is proven that pair‐wise coupling of a finite number of these self‐conjugate irreps never contains a continuum S̄L̄ (3̄,R) irrep which is multiplicity‐free. The significance of S̄L̄ (3̄,R) symmetry in nuclear and hadronic systems is discussed, including implications for the coupling of quarkels.

Comment on the Wigner 9‐j symbol
View Description Hide DescriptionA number of errors in the algebraic formulas of Rotenberg e t a l’s T h e 3j a n d 6‐j S y m b o l s have been corrected. Attention is drawn to a few other simple new algebraic relationships.

Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups
View Description Hide DescriptionWe give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

Evaluation of SU(6) ⊇ SU(3) ⊗ SU(2) Wigner coefficients
View Description Hide DescriptionA technique is described which allows the calculation of any SU(6) Wigner coefficient (isoscalar factor) from known one‐body SU(6) coefficients. The technique is applied to calculate the SU(6) coefficients for q ^{3}×q ^{3} leading to SU(3) singlet, octet, and decuplet states of six quarks. A sum rule of SU(3) 9−λμ symbols is given.

Singular spin harmonics
View Description Hide DescriptionThe definition of the spin‐weighted spherical harmonics, _{ s } Y _{ l m }, is extended so that all integral spin weights, s, are allowed. We then discuss a set of spin‐weighted functions, _{ s } Z _{ l m }, which are analogous to the spherical harmonics of the second kind.

An application of Lax’s estimates to the determination of critical crystal thickness in nonlinear laser optics
View Description Hide DescriptionLax’s estimates concerning the development of singularities of solutions of the Cauchy problem for 2×2 quasilinear hyperbolic systems are applied to the Maxwellsystem of nonlinear optics, which describes the propagation of a plane polarized laser wave through a uniaxial piezoelectric crystal, in a direction orthogonal to the optic axis, and neglecting absorption and dispersion. The ’’critical thickness’’ of the crystal, i.e., the distance after which (shock) singularities necessarily occur, is evaluated within an error of order less than 10^{−3}, and is found to be proportional to the ratio of laser wavelength to peak amplitude times the nonlinear optical coefficient. The critical thickness appears to be ordinarily of the order of a few centimeters.

Solutions of the nonlinear 3‐wave equations in three spatial dimensions
View Description Hide DescriptionRecently Ablowitz and Haberman have shown that, in three spatial dimensions, the nonlinear 3‐wave evolution equation results from the compatibility condition between two well‐defined first‐order linear differential 3×3 systems having common solutions. We construct inversionlike integral equations (I.E.) associated with both these two linear differential systems so that the solutions of the I.E. embody their compatibility conditions. The scalar kernels of these I.E. depend upon three independent variables in such a way that there exist degenerate kernels confined in the three‐dimensional coordinate space. Consequently we exhibit, for the nonlinear 3‐wave evolution equations, an infinite number of solutions which, at fixed time, are confined in the three‐dimensional coordinate space.

Rigid body motions, space curves, prolongation structures, fiber bundles, and solitons
View Description Hide DescriptionThe dynamics of a nonlinear string of constant length represented by a helical space curve may be studied through a consideration of the motion of an arbitrary rigid body along it. The resulting set of compatibility equations is shown to result in the class of nonlinear evolution equations solvable through the two component inverse scattering phenomenology. A class of pseudopotentials and prolongation structures follow naturally due to the intrinsic group structure of the phenomenon. This leads to an identification of the underlying fiber bundle structure and connection forms. Thus a unified picture emerges for a class of soliton possessing evolution equations.

Remarks on a theorem by G. Epifanio, ’’On the matrix representation of unbounded operators’’ [J. Math. Phys. 1 7, 1688 (1976)]
View Description Hide DescriptionWe give a generalization of a theorem concerning the change of basis for the matrix representation of unbounded operators defined in a scalar product space. We introduce for the proof a suitable structure which can be useful when one has to make operations with operators defined between different scalar product spaces.

Transformation between the normal and antinormal expansions of boson operators
View Description Hide DescriptionWe show that the expansion coefficients for the normal and antinormal forms of a boson operator are related by a simple transform.

Markov fields in noncommutative probability theory on W* algebras
View Description Hide DescriptionMany important results in ordinary probability theory do not have extensions in noncommutative probability theory. Here we consider Markov fields, and we prove that new Markov fields may be generated from old ones by means of multiplicative measurable operators. There is an analog of this result in ordinary probability theory involving multiplicative functionals. It is envisaged that the result established here would be of use in the study of interacting Fermion quantum fields.

A note on the Lorentz transformation
View Description Hide DescriptionUsing Lie theory of one‐parameter transformation group, we show that the (linear) Lorentz transformation can be embedded into a class of nonlinear transformations.

The inverse problem for random sources
View Description Hide DescriptionThe problem of deducing the statistical structure of a localized random source ρ (r) of the reduced wave equation from measurements of the field external to the source is addressed for the case when the measurements yield the autocorrelation function of the field at all pairs of points exterior to the source volume and the quantity to be determined is the source’s autocorrelation function R _{ρ}(r_{1},r_{2}) =〈ρ* (r_{1}) ρ (r_{2}) 〉.This problem is shown to be equivalent to that of determining R _{ρ} from the autocorrelation function of the field’s radiation pattern and is found, in general, not to admit a unique solution due to the possible existence of nonradiating sources within the source volume. Notable exceptions are the class of delta correlated (incoherent) sources whose intensity profiles are shown to be uniquely determined from the data and the class of quasihomogeneous sources whose coherenceproperties can be determined if their intensity profiles are known and vice versa.

Theory of strings and membranes in an external field. II. The string
View Description Hide DescriptionWe apply the general formulation of the theory of extended systems developed previously to the case of a relativistic string in an external field. This problem is relevant for the construction of models of strongly interacting particles. When the external field is static and uniform, explicit solutions are presented and classified according to their symmetry properties.

A Lie group framework for composite particles and mass spectrum
View Description Hide DescriptionWe propose a concept of internal structure and a relativistically covariant method of unifying the external and internal structures, leading to a dynamical Lie algebra without superfluous generators. In this framework we study in more detail a Lie algebra unifying external space with an internal 3‐space, and several representations which describe models of composite particles and give rise to various mass formulas capable of describing the hadronspectrum; we make use of both unitary irreducible global representations and partially integrable, Schur‐irreducible, symmetric local representations.

A uniqueness result in the Segal–Weinless approach to linear Bose fields
View Description Hide DescriptionWe prove a theorem, which, while it fits naturally into the Segal–Weinless approach to quantization seems to have been overlooked in the literature: Let (D,σ) be a symplectic space, and T (t) a one parameter group of symplectics on (D,σ). Let (H, 2Im〈⋅ ‖ ⋅〉) be a complex Hilbert space considered as a real symplectic space, and U(t) a one‐parameter unitary group on H with strictly positive energy. Suppose there is a linear symplectic map K from D to H with dense range, intertwining T (t) and U(t). Then K is unique up to unitary equivalence.

On Blume’s integration of Schrödinger’s equation for a quantum system subject to random pulses
View Description Hide DescriptionIn this note we dispose of Blume’s objections to a recent article by Szyl. We show that he is forgetting about causality and that he makes obvious mistakes taking limits.

Settling the question of the high‐energy behavior of phase shifts produced by repulsive, strongly singular, inverse‐power potentials
View Description Hide DescriptionFor repulsive, strongly singular, inverse‐power potentials it is rigorously shown that the JWKB expression for the nonrelativistic phase shift tends to exactness in the high‐energy limit. The hitherto open question as to the correct expression for the leading term in the high‐energy expansion of the phase shift for these potentials is thus definitely settled, and it is further confirmed that even the next term in the expansion yielded by the JWKB expression is significant.

Classical, cross‐section generating solutions of field equations
View Description Hide DescriptionThe symmetry properties of classical, cross‐section generating solutions of field equations, sectons, are investigated. It is shown that under general conditions on the interaction Lagrangian of the field theory the symmetry group of the solutions can only be 0(1,1) ×0(2) for finite momenta. Such solutions generate inclusive cross sections with Feynman scaling.

Scattering of a beam of particles by a potential
View Description Hide DescriptionWe discuss the potential scattering into a cone C in position space of a beam of particles with localized momenta distributed over a set K, and the derivation from time‐dependent scattering theory of the formula ∫_{ K } dk̃∫_{ C } dΩ σ_{k̃}(Ω) [where σ_{k̃}(Ω) is the scattering cross section] as a measure of the probability of this scattering. We make use in our discussion of a result of Dollard giving the probability of scattering into a cone for a single wavepacket.