Volume 25, Issue 10, October 1984
Index of content:

A note on general involutional transformations
View Description Hide DescriptionFor two matrices A and B of order n×n which satisfy x ^{ n }−1=0, it is shown that the transformation matrix T _{ A B } which connects A and B via A T _{ A B }=T _{ A B } B becomes general involutional when A B=ωB A, ω being a primitive nth root of unity.

Simultaneity and reality of U(n) and SU(n) 3j m and 6j symbols
View Description Hide DescriptionA simple proof is given of the simultaneous existence of a real set of 3j m and 6j symbols for the unitary groups U(n) and SU(n); including the case of mixed tensor representations. We observe that simultaneity is incompatible with the conventional permutation symmetries for U(n) with n>3. The relevance of these results to the Schur–Weyl duality of U(n) with the symmetric groups S _{ l } is discussed.

Collectivity and geometry. III. The three‐dimensional case in the Sp(6)⊇Sp(2)×O(3) chain for closed shells
View Description Hide DescriptionAs was indicated in previous papers, general Hamiltonians for systems of n particles in three‐dimensional space can be formulated in the enveloping algebra of the symplectic group Sp(6n). This group admits, among others, the subgroup Sp(6)×O(n) and, as has been noticed by many authors, collective Hamiltonians can be formulated in the enveloping algebra of Sp(6), so that their eigenstates can be characterized by a definite irreducible representation (irrep) of this group. The mathematical problem is then to determine the matrix elements of the generators of Sp(6) in a basis characterized by irreps of this group as well as of appropriate subgroups. In the present series of papers the subgroups chosen where Sp(2)×O(3) as the Casimir operator of Sp(2) when n → ∞ is formally related to the Bohr–Mottelson vibrational Hamiltonian (BMVH), while O(3) gives the angular momentum of the state. We give an algorithm for determining these states that closely parallels the procedure followed for the BMVH. Programs are being developed to convert our algorithm into a computational tool for determining collective excitations in nuclei.

Application of double Gel’fand polynomials to the symmetric group and spin–isospin wave functions of cluster systems
View Description Hide DescriptionThe theory of double Gel’fand polynomials is applied to irreducible representations of the symmetric and SU_{4} groups with the aim to treat spin–isospin wave functions of nuclear cluster systems. Multiplicity‐free recoupling coefficients of the symmetric group are connected with special types of Clebsch–Gordan coefficients of the unitary group. The standard phase conventions of the Yamanouchi basis and of the multiplicity‐free recoupling coefficients are proved to be derivable from natural phase conventions of double Gel’fand polynomials and these special Clebsch–Gordan coefficients. By extending the concept of double Gel’fand polynomials, useful expansion formulas are derived with respect to the determinant associated with a matrix tensor product. A simple example of their application is given for normalization kernels of two‐body systems composed of s‐shell clusters and for SU_{4} Clebsch–Gordan coefficients in the spin–isospin representation needed therein.

General charge conjugation operators in simple Lie groups
View Description Hide DescriptionA description of particular elements (‘‘charge conjugation operators’’) found in any compact simple Lie groupK is presented. Such elements R _{ i } transform a physical state (weight vector of a basis of a representation space) into others with opposite ‘‘charge (ith component of the weight), sometime changing also the sign of the state. It is demonstrated that exploitation of these elements and the finite subgroup N of K generated by them offer new powerful methods for computing with representations of the Lie group. Their application to construction of bases in representation spaces is considered in detail. It represents a completely new direction to the problem.

Enveloping algebra annihilators and projection techniques for finite‐dimensional cyclic modules of a semisimple Lie algebra
View Description Hide DescriptionSome results on the structure of finite‐dimensional cyclic modules for a semisimple Lie algebra are presented. Cyclic modules arise naturally in constructing symmetry adapted states of a system using projection. Projecting out states with definite symmetry from an arbitrary state ψ is related to the properties of the cyclic module generated by ψ. An important example of a cyclic module is the tensor product of two irreducible modules V(λ)⊗V(μ) which is cyclically generated by the vector v ^{λ} _{−}⊗v ^{μ} _{+}, where v ^{λ} _{−}(resp., v ^{μ} _{+}) is the minimal (resp., maximal) weight vector of V(λ) [resp., V(μ)]. For this particular case we determine the explicit form of the annihilator, in the universal enveloping algebra, of the cyclic vector v ^{λ} _{−}⊗v ^{μ} _{+}. It is hoped that this result may add new insight into the Clebsch–Gordan multiplicity problem. As an application of this result projection operators are constructed which project, from an arbitrary vector of weight λ, a maximal weight vector of weight λ.

Poisson bracket realizations of Lie algebras and subrepresentations of (ad^{⊗k })_{ s }
View Description Hide DescriptionA procedure which associates Poisson bracket realizations of a Lie algebraL to subrepresentations of the extension (ad^{⊗k })_{ s } of the adjoint action to the algebra of polynomials defined on the dual space L* is pointed out. The procedure is applied, for k=2, to the real forms of the semisimple Lie algebras of types D _{3} and B _{2}∼C _{2}, in particular to the algebras so(4,2), so(4,1), and so(3,2)∼sp(4,R). The results obtained for the algebra sp(4,R) have led to an algebraic foundation for the constraints satisfied by the dynamical variables for the classical limit of the generalized helium problem.

Clebsch–Gordan coefficients for E _{6} and SO(10) unification models
View Description Hide DescriptionWe illustrate here a new method for computing Clebsch–Gordan coefficients (CGC) for E _{6} by computing CGC for the product 27⊗27 of the irreducible representation (100000) of E _{6} with itself. These CGC are calculated thrice: once in a weight vector basis independent of any semisimple subgroup, then in a basis which refers to SO(10)⊆E _{6}, and finally in a basis referring to SU(5)⊆SO(10)⊆E _{6}.

On the Fourier series representations of path integrals
View Description Hide DescriptionThe direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ^{2}−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N‐segment polygonal paths of the type contributing significantly to the integral, and their N‐term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T ^{−} ^{1}∫^{ T } _{0} d t x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω^{2} x ^{2}, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω^{2} values for which the integrals are finite or infinite in the imaginary time cases.

The expression for the triple vector product solid harmonic
View Description Hide DescriptionThe solid harmonic y _{ L M }[(r _{1}Λr _{2})Λr _{3}] was expressed in terms of the spherical harmonics Y _{ L 1 M 1 }(r̂_{1}), Y _{ L 2 M 2 }(r̂_{2}), and Y _{ L 3 M 3 }(r̂_{3}), where the coefficients of the expansion were expressed in terms of 9j symbols. Here we present a simpler form of those coefficients expressed in terms of 6j symbols.

Eigenvalues and degeneracies for n‐dimensional tensor spherical harmonics
View Description Hide DescriptionSymmetric transverse traceless tensor harmonics of arbitrary rank are constructed on spheres S ^{ n } of dimensionality n≥3, and the associated eigenvalues of the Laplacian are computed. It is shown that these tensor harmonics span the space of symmetric transverse traceless tensors on S ^{ n } and are eigenfunctions of the quadratic Casimir operator of the group O(n+1). The dimensionalities of the eigenspaces of the Laplacian are computed for harmonics of rank 1 and rank 2.

Differential equations for the cuspoid canonical integrals
View Description Hide DescriptionDifferential equations satisfied by the cuspoid canonical integrals I _{ n }(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(a _{1},a _{2},...,a _{ n−1}). A set of linear coupled ordinary differential equations is derived for each step in the sequence I _{ n }(0,0,...,0,0) →I _{ n }(0,0,...,0,a _{ n−1}) →I _{ n }(0,0,...,a _{ n−2},a _{ n−1}) →...→I _{ n }(0,a _{2},...,a _{ n−2},a _{ n−1}) →I _{ n }(a _{1},a _{2},...,a _{ n−2},a _{ n−1}). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives.

Discriminant, transmission coefficient, and stability bands of Hill’s equation
View Description Hide DescriptionThe discriminant Δ(k ^{2}) of Hill’s equation is shown to be related to the transmission coefficientT(k)e ^{ iθ(k)} of one period of the potential by Δ(k ^{2})=[2/T(k)]cos[kπ+θ(k)]. This result is used to find the boundaries of the stability bands.

Lax‐pairs, spectral problems, and recursion operators
View Description Hide DescriptionWe present various examples for the connection between Lax‐pairs and recursion operators. From this connection a new method for constructing recursion operators is derived. As an application we find recursion operators to some integrable equations newly found by Wadati e t a l. [J. Phys. Soc. Jpn. Lett. 4 6, 1965 (1979); J. Phys. Soc. Jpn. 4 7, 1698 (1979)].

On the Schrödinger operator with periodic point interactions in the three‐dimensional case
View Description Hide DescriptionWe prove that it is possible to define the self‐adjoint operator which gives sense to the merely formal expression −Δ−∑_{ y∈L }λδ(⋅−y) (where L is a certain lattice of R^{3}) as the limit when ε→0_{+} in the resolvent sense of the net H _{ε} =−Δ+∑_{ y∈L } λ(ε)ε^{−} ^{2} V(⋅−y/ε) λ(ε) being a real‐valued, C ^{∞} [0,1] function with λ(0)=1 and V∈L ^{∞} is such that supp V is contained in the Wigner–Seitz cell. By using the direct integral decomposition, we reduce the problem to the convergence of the reduced Hamiltonian H _{ε} (θ)=−Δ_{θ} +λ(ε)ε^{−} ^{2} V(⋅/ε). In order to find the limit when ε→0_{+} of [H _{ε}(θ)−E]^{−} ^{1}, we also study the properties of its integral kernel.

Global reduction of a dynamical system on a foliated manifold
View Description Hide DescriptionConditions for the global reductions of a dynamical system defined on foliated manifoldM are given. They are expressed by a local condition on the topology of one single leaf and a global condition on the transverse bundle to the foliation. The link of this condition is shown with the existence of a normalizer of the Lie algebra of the vector fields tangent to the foliation in the Lie algebra of all the vector fields of M. This normalizer contains all the derivations of the functions constant on the leaves.

The Haar integral for Lie supergroups
View Description Hide DescriptionEach supermanifold can be considered as equivalent to a certain family of real manifolds. The Haar integral of a Lie supergroup is defined using this equivalence. A useful general formula is derived together with explicit construction methods of the simple Lie supergroups of types SPL(n‖m;E _{ L }), OSP(n‖2r;E _{ L }), and B(n;E _{ L }), and the extended Poincaré group.

A novel class of Bessel function integrals
View Description Hide DescriptionA number of definite integrals over the unit interval involving Bessel functions with argument [α^{2} x ^{−} ^{1}+β^{2}(1−x)^{−} ^{1}]^{1} ^{/} ^{2} are evaluated in closed form.

Post‐Newtonian extensions of the Runge–Lenz vector
View Description Hide DescriptionWe obtain the most general post‐Newtonian extension of the Runge–Lenz vector corresponding to a very large class of two‐body relativistic systems whose equations of the relative motion can be derived from a post‐Newtonian Hamiltonian depending on four parameters which include the electromagnetic and gravitational cases. Assuming a couple of conditions, we fix the arbitrariness and obtain a unique post‐Newtonian Runge–Lenz vector, whose associated symmetries generate the same algebra as in the Keplerian case.

Geometrical models for quantum logics with conditioning
View Description Hide DescriptionAfter obtaining a representation of a quantum logic by means of projection operators on the state space, geometrical conditions are imposed on a cone in an abstract Banach space which allow us to show that certain projections leaving this cone invariant will form a quantum logic with conditioning. Several examples are also presented.