Volume 26, Issue 8, August 1985
Index of content:

Mixed supertableaux of the superunitary groups. II. SU(n‖m)
View Description Hide DescriptionThe Young supertableaux of the superunitary groups are studied. We give the conditions on the size of legal supertableaux, the type of atypicity for tensor representations which are irreducible, making the link between the Kac–Dynkin parameters and those of the supertableau. A discussion is given on nonfully reducible representations and generalized atypical supertableaux.

Quantum‐mechanical representations of the group of diffeomorphisms and local current algebra describing tightly bound composite particles
View Description Hide DescriptionA semidirect product of Schwartz’ space functions on R^{3} and the group of diffeomorphisms of R^{3} can describe quantum‐mechanical systems. We interpret a class of continuous unitary representations of this group, characterized by multipole moments, as describing tightly bound composite particles.

On the Wigner coefficients of the generalized Lorentz groups in the parabolic basis
View Description Hide DescriptionWigner coefficients for the class 1 representations of the generalized Lorentz groups SO(n+1,1) in the parabolic basis corresponding to the group reduction SO(n+1,1)⊃E(n)⊃T(n) are calculated. They are in general expressible in terms of Appell’s F _{4} hypergeometric functions of two variables. However, in the case of n=1,2 they can also be expressed in terms of ordinary hypergeometric functions _{2} F _{1}.

Generalized FS×W versus Dyson’s classification of irreducible representations
View Description Hide DescriptionWe study the characterization of the 13 cases obtained in the classification of the irreducible linear–antilinear representations of semigroups in a finite‐dimensional vector space X over an algebraically closed field K with a conjugation j (generalized Frobenius–Schur–Wigner, or FS×W, classification). It has already been shown that each case can be characterized by various equivalent properties, some of which can be endowed with a physical interpretation. We show here that, whenever K is the complex field C, each case can be characterized by the structure (in the sense specified by the Weyl theorem on the structure of the matrix algebras and their commutators) of a pair of suitable operator algebras over the real field R. This characterization coincides with the one given by Dyson for each case of his classification of symmetry groups. Thus, the latter classification is recovered under more general assumptions and in a generalized framework, and its one‐to‐one correspondence with the generalized FS×W classification is shown. In the process, we obtain a classification of the algebras over R generated by complex semigroup representations of the aforesaid kind, and inquire into the connections between some properties of the representations in X over C and the structure (in the Weyl sense) of the representations in the space X _{ R } over R obtained by decomplexification of the space X.

Explicit form of the Haar measure of U(n) and differential operators
View Description Hide DescriptionThe Haar measure of the group U(n) is explicitly introduced using the Euler‐like parameters, and the differential operators of the first‐ and second‐parameter groups are given. The D‐matrix elements are defined through the Gel’fand and Tsetlin basis and the orthogonality and the completeness relations for the d‐matrix elements are given explicitly.

Formula for invariant integrations on SU(n)
View Description Hide DescriptionThe explicit formula for evaluating integrals of a product of the matrix elements of SU(n) is given in terms of Kronecker delta symbols.

New expressions for the eigenvalues of the invariant operators of the general linear and the orthosymplectic Lie superalgebras
View Description Hide DescriptionWe obtain expansions for the eigenvaluesC _{ p } of the invariant operators (Casimir operators) of the general linear, and the orthosymplectic Lie superalgebras in terms of p r o d u c t s of suitably defined graded power sums P _{ k }. The resulting expressions are closed and provide unified formulas for computing the C _{ p }’s for those superalgebras as well as their corresponding Lie algebras. The formulas are remarkably simple to suggest that the power sums used in this text could play a more basic role in the understanding of the pattern of the expansion coefficients. Explicit illustrations are given for the various series for p≤8.

The symmetric and antisymmetric structure constants for SU(6)
View Description Hide DescriptionThe 124 completely antisymmetric f _{ i j k } and 221 completely symmetric d _{ i j k } (nonzero) structure constants for a simple representation of SU(6) are tabulated. The basis matrices λ_{ i } used to generate the structure constants are also given.

Canonical map approach to channeling stability in crystals
View Description Hide DescriptionWe state and prove rigorous mathematical results on the orbital stability of certain rectilinear trajectories of sufficiently energetic particles subjected to appropriate periodic potentials. This is done in the context of nontrivial classical Hamiltonian models, nonrelativistic and relativistic, in two space dimensions. The main steps involved in the proofs are the derivation of the asymptotic form of certain canonical maps in the plane in the limit of large particle energies and the application of a version of Moser’s twist theorem. When suitably specialized, these results establish rigorously for the first time that the pertinent straight‐line channeling trajectories of fast particles in two‐dimensional rigid crystal lattices have this stability property under reasonable conditions on the crystal potential.

Second‐order corrected Hadamard formulas
View Description Hide DescriptionThe second‐order correction to the Hadamard formulas for the Green’s function, harmonic measures, and period matrix of a two‐dimensional domain is obtained in the context of the domain‐variational theory.

On a particular solution of the equation of Ernst with n fields, parametrized by an arbitrary harmonic function
View Description Hide DescriptionWe determine a particular solution, dependent upon an arbitrary harmonic function in cylindrical coordinates, of the system of npartial differential equations, which characterizes both the axially symmetric field solution of the Einstein (n−1)–Maxwell equations and one class of axially symmetric static self‐dual SU(n+1) Yang–Mills fields.

Some remarks on torsion in Kaluza–Klein unification
View Description Hide DescriptionThe role of manifolds endowed with a parallelizing torsion in Kaluza–Klein theories is examined. In particular the spin connection on such manifolds is demonstrated to be a single pure gauge almost everywhere on the manifold. This follows from the Frobenius integration theorem. As a consequence of this result the computation of the representation of massless fermions follows immediately and trivially. The spectrum of massless fermions on manifolds with a parallelizing torsion is contrasted with the analogous spectrum on manifolds with nontrivial topological configurations. While the remarks are primarily of pedagogic value much of the relevant mathematics is made intuitive.

Concepts of conditional expectations in quantum theory
View Description Hide DescriptionA concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a p o s t e r i o r i states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.

On the one‐ and two‐dimensional Toda lattices and the Painlevé property
View Description Hide DescriptionThe Toda lattice and the two‐dimensional Toda lattice (2‐DTL) are shown to possess a type of ‘‘Painlevé property’’ that is based on the use of separate ‘‘singular manifolds’’ for each dependent variable. The isospectral problem for the 2‐DTL found by both Mikhailov and by Fordy and Gibbons can be simply and logically derived from this analysis. Some remarks are made about the connection between our work and independent work of Kametaka and Airhault on the relationship between the Toda lattice and the second Painlevé transcendent.

Canonical transformations theory for presymplectic systems
View Description Hide DescriptionWe develop a theory of canonical transformations for presymplectic systems, reducing this concept to that of canonical transformations for regular coisotropic canonical systems. In this way we can also link these with the usual canonical transformations for the symplectic reduced phase space. Furthermore, the concept of a generating function arises in a natural way as well as that of gauge group.

How to construct integrable Fokker–Planck and electromagnetic Hamiltonians from ordinary integrable Hamiltonians
View Description Hide DescriptionIntegrable Hamiltonians with velocity‐dependent potentials, including those of Fokker–Planck‐type H=1/2(p ^{2} _{ x }+p ^{2} _{ y })+K _{ x } p _{ x }+K _{ y } p _{ y }, are constructed from integrable Hamiltonians of type H=1/2(p ^{2} _{ x }+p ^{2} _{ y })+V(x,y) using certain canonical and noncanonical transformations. Some of the Hamiltonians obtained this way are integrable only for zero energy. Candidates for the Φ potential, which is of interest for Fokker–Planck models, are constructed in several cases.

Lax‐pair operators for squared‐sum and squared‐difference eigenfunctions
View Description Hide DescriptionAn interrelationship between various representations of the inverse scattering transformation is established by examining eigenfunctions of Lax‐pair operators of the sine–Gordon equation and the modified Korteweg–de Vries equation. In particular, it is shown explicitly that there exist Lax‐pair operators for the squared‐sum and squared‐difference eigenfunctions of the Ablowitz–Kaup–Newell–Segur inverse scattering transformation.

On the convergence of the Rytov approximation for the reduced wave equation
View Description Hide DescriptionThe reduced wave equation Δu+k ^{2} n ^{2}(x)u=0 is treated where n(x) fluctuates about unity in a compact domain D, and is equal to unity in the region exterior to D. In the Rytov approximation the total field u(x) generated by an incident field u ^{ i }(x) has the form u(x)∼u ^{ i }(x)exp φ(x) where φ(x)u ^{ i }(x)=(1/4π)∫_{ D } (e ^{ i k‖x−y‖}/‖x−y‖)k ^{2}(n ^{2}−1) ×u ^{ i }(y)dτ_{ y }. It is shown that under suitable conditions on n(x) and restrictions on u ^{ i }(x), the approximation is a leading term of a convergent expansion holding for all x in D. This is in contrast to previous theory, which treated the Rytov approximation as an asymptotic expansion valid in the forward‐scattered direction.

Bilinear phase‐plane distribution functions and positivity
View Description Hide DescriptionThere is a theorem of Wigner that states that phase‐plane distribution functions involving the state bilinearly and having correct marginals must take negative values for certain states. The purpose of this paper is to support the statement that these phase‐plane distribution functions are for hardly any state everywhere non‐negative. In particular, it is shown that for certain generalized Wigner distribution functions there are no smooth states (except the Gaussians for the Wigner distribution function itself) whose distribution function is everywhere non‐negative. This class of Wigner‐type distribution functions contains the Margenau–Hill distribution. Furthermore, the argument used in the proof of Wigner’s theorem is augmented to show that under mild conditions one can find for any two states f, g with non‐negative distribution functions a linear combination h of f and g whose distribution function takes negative values, unless f and g are proportional.

Proof of the Levinson theorem by the Sturm–Liouville theorem
View Description Hide DescriptionThe Levinson theorem is proved by the Sturm–Liouville theorem in this paper. For the potential ∫^{1} _{0} r‖V(r)‖d r <∞,V(r)→b/r ^{2} when r→∞, the modified Levinson theorem is derived as n _{ l }=(1/π)δ_{ l }(0) +(a−l)/2− 1/2 sin^{2}{δ_{ l }(0)+[(a−l)/2]π}, if a(a+1)≡b+l(l+1)> 3/4 or a=0. Two examples which violate the Levinson theorem and satisfy the modified Levinson theorem are discussed.