Index of content:
Volume 28, Issue 3, March 1987

Group theoretical basis of some identities for the generalized hypergeometric series
View Description Hide DescriptionIt is shown that Thomae’s identity between two _{3} F _{2} hypergeometric series of unit argument together with the trivial invariance under separate permutations of numerator and denominator parameters implies that the symmetric group S _{5} is an invariance group of this series. A similar result is proved for the terminating Saalschützian _{4} F _{3} series, where S _{6} is shown to be the invariance group of this series (or S _{5} if one parameter is eliminated by using the Saalschütz condition). Here Bailey’s identity is realized as a permutation of appropriately defined parameters. Finally, the set of three‐term relations between _{3} F _{2} series of unit argument discovered by Thomae [J. Thomae, J. Reine Angew. Math. 8 7, 26 (1879)] and systematized by Whipple [F. J. Whipple, Proc. London Math. Soc. 2 3, 104 (1925)] is shown to be transformed into itself under the action of the group S _{6}×Λ, where Λ is a two‐element group. The 12 left cosets of S _{6}×Λ with respect to the invariance group S _{5} are the structural elements underlying the three‐term relations. The symbol manipulator macsyma was used to obtain preliminary results.

Continuous Hahn polynomials and the Heisenberg algebra
View Description Hide DescriptionContinuous Hahn polynomials have surfaced in a number of somewhat obscure physical applications. For example, they have emerged in the description of two‐photon processes in hydrogen, hard‐hexagon statistical mechanical models, and Clebsch–Gordan expansions for unitary representations of the Lorentz group SO(3,1). In this paper it is shown that there is a simple and elegant way to construct these polynomials using the Heisenberg algebra.

The Gel’fand realization and the generating function of the Clebsch–Gordan coefficients of SL(2,R) in the hyperbolic basis
View Description Hide DescriptionIt is shown that the canonical realization of the representations of SL(2,R) proposed by Gel’fand and co‐workers yields a generating function of the Clebsch–Gordan coefficients of the group in the hyperbolic basis. This function is the coupled state and appears as the solution of an ordinary differential equation reducible to the hypergeometric equation. The desired expansion of the generating function that yields the Clebsch–Gordan coefficients is essentially a generalization of Barnes’ theory of analytic continuation of the hypergeometric function. In this paper the normalized Clebsch–Gordan coefficients for the coupling of two representations of the positive discrete class are calculated. The final result is an analytic continuation of the corresponding expression in the SO(2) basis. The possible application of the generating function to the reduction of the Kronecker product of three irreducible representations is discussed.

Nonlinear equations with superposition formulas and the exceptional group G_{2}. II. Classification of the equations
View Description Hide DescriptionNonlinear equations with superposition formulas are obtained, corresponding to the action of the complex and real forms of the exceptional Lie group G_{2} on the homogeneous spaces G_{2}/H. The isotropy group of the origin H is taken as one of the maximal parabolic subgroups of G_{2}, or as one of the maximal reductive subgroups, leaving some vector space V∈C^{7} (or V∈R^{7}) invariant. The parabolic subgroups, as well as the simple subgroups SL(3,C), SU(3), SL(3,R) or SU(2,1) lead to equations with quadratic or quartic nonlinearities. The semisimple subgroups SL(2,C)⊗SL(2,C), SU(2)⊗SU(2), and SU(1,1)⊗SU(1,1) lead to equations with quadratic nonlinearities and additional nonlinear constraints on the independent variables.

Superposition formulas for rectangular matrix Riccati equations
View Description Hide DescriptionA system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G⊇G _{0}. We consider the case when G=SL(n+k,C) and G _{0}=P(k) is a maximal parabolic subgroup of G, leaving a k‐dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix W(t)∈C^{ n×k }. The special case n=r k (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.

On a property of a classical solution of the nonlinear mass transport equation u _{ t }=u _{ x x }/1+u _{ x } ^{2}. II
View Description Hide DescriptionA mechanism of smoothing due to evaporation condensation of the roughly perturbed surface of solid is formulated by Mullins [W. W. Mullins, J. Appl. Phys. 2 8, 333 (1957); 3 0, 77 (1959)] as a certain Cauchy problem for a nonlinear parabolic equation which describes the evolution of the profile of the surface. In the preceding paper [A. Kitada, J. Math. Phys. 2 7, 1391 (1986)], through the careful investigations of the Cauchy problem, it was demonstrated that each peak in the initial surface did n o t i n c r e a s e in height with time. In the present paper, by slightly limiting the set of functions to which the classical solutions of the Cauchy problem belong, it is demonstrated that each peak height d e c r e a s e s with time in the strict sense.

The evolution partial differential equation u _{ t }=u _{ x x x }+3(u _{ x x } u ^{2} +3u ^{2} _{ x } u)+3u _{ x } u ^{4}
View Description Hide DescriptionThe evolution equationu _{ t }=u _{ x x x }+3(u _{ x x } u ^{2} +3u ^{2} _{ x } u)+3u _{ x } u ^{4}, u=u(x,t), is integrable; it can be (exactly) linearized by an appropriate change of (dependent) variable. Hence several explicit solutions of the partial differential equation(PDE) can be exhibited; some of them display a remarkable solitronic phenomenology.

On the WKBJ approximation
View Description Hide DescriptionA simple approach employing properties of solutions of differential equations is adopted to derive an appropriate extension of the WKBJ method. Some of the earlier techniques that are commonly in use are unified, whereby the general approximate solution to a second‐order homogeneous linear differential equation is presented in a standard form (SF) that is valid for all orders. In comparison to other methods, the present one is shown to be leading in the order of iteration, and thus possibly has the ability of accelerating the convergence of the solution.

Algebraic structures of degenerate systems and the indefinite metric
View Description Hide DescriptionIt is shown that the indefinite metric structures of degenerate systems as given by Strocchi and Wightman [F. Strocchi and A. S. Wightman, J. Math. Phys. 1 5, 2198 (1974); 1 7, 1930 (1976)] arise in a natural fashion from the algebraic structure of such systems, where the latter has been developed in a C*‐context by Grundling and Hurst [H. B. G. S Grundling and C. A. Hurst, Commun. Math. Phys. 9 8, 369 (1985)]. Auxiliary concepts like gauge equivalence are examined, and the preceding general theory is specialized to the situation of linear boson fields with linear Hermitian constraints. Two examples of this situation are given—a one‐dimensional scalar boson in a periodic universe and Landau gauge electromagnetism.

Generalized Bergman kernels and geometric quantization
View Description Hide DescriptionIn geometric quantization it is well known that, if f is an observable and F a polarization on a symplectic manifold (M,ω), then the condition ‘‘X _{ f } leaves F invariant’’ (where X _{ f } denotes the Hamiltonian vector field associated to f ) is sufficient to guarantee that one does not have to compute the BKS kernel explicitly in order to know the corresponding quantum operator. It is shown in this paper that this condition on f can be weakened to ‘‘X _{ f } leaves F+F ^{°} invariant’’and the corresponding quantum operator is then given implicitly by formula (4.8); in particular when F is a (positive) Kähler polarization, all observables can be quantized ‘‘directly’’ and moreover, an ‘‘explicit’’ formula for the corresponding quantum operator is derived (Theorem 5.8). Applying this to the phase space R^{2n } one obtains a quantization prescription which ressembles the normal ordering of operators in quantum field theory. When we translate this prescription to the usual position representation of quantum mechanics, the result is (a.o) that the operator associated to a classical potential is multiplication by a function which is essentially the convolution of the potential function with a Gaussian function of width ℏ, instead of multiplication by the potential itself.

Loop gauge theory and group cohomology
View Description Hide DescriptionA generalized fiber bundle model in which the fibers are Hilbert spaces is studied. Unitary transformations are used to define a unitary isomorphism (‘‘parallelism’’) among them. The Weyl group is first used to ‘‘connect’’ the projective Hilbert spaces (ray spaces) and to introduce a one‐form connection that defines which coset at a point y is parallel to a given coset at another point x. Then, the central extension of the Weyl group by U(1)’ is studied in order to introduce the most general mapping between the elements of these cosets. This leads to a two‐form connection and makes the model a good candidate for a fiber bundle approach to string theories.

Fractal and nonfractal behavior in Levy flights
View Description Hide DescriptionThe d‐dimentional space‐continuous time‐discrete Markovianrandom walk with a distribution of step lengths, which behaves like x ^{−(α+d)} with α>0 for large x, is studied. By studying the density–density correlation function of these walks, it is determined under what conditions the walks are fractal and when they are nonfractal. An ensemble average of walks is considered and the lower entropy dimension D of the set of stopovers of the walks in this ensemble is calculated, and D=min{2,α,d} is found. It is also found that the fractal nature of the walks is related to a finite value of the mean first passage time. The crossover of the correlation function from the fractal to nonfractal regimes is studied in detail. Finally, it is conjectured that these results for the lower entropy dimension apply to a wide class of symmetric Markov processes.

The hydrogen atom in phase space
View Description Hide DescriptionThe Hamiltonian of the three‐dimensional hydrogen atom is reduced, in parabolic coordinates, to the Hamiltonians of two bidimensional harmonic oscillators, by doing several space‐time transformations,separating the movement along the three parabolic directions (ξ,η,φ), and introducing two auxiliary angular variables ψ and ψ’, 0≤ψ, ψ’≤2π. The Green’s function is developed into partial Green’s functions, and expressed in terms of two Green’s functions that describe the movements along both the ξ and η axes. Introducing auxiliary Hamiltonians allows one to calculate the Green’s function in the configurational space, via the phase‐space evolution function of the two‐dimensional harmonic oscillator. The auxiliary variables ψ and ψ’ are eliminated by projection. The thus‐obtained Green’s function, save for a multiplicating factor, coincides with that calculated following the path‐integral formalism.

The generalized Morse oscillator in the SO(4,2) dynamical group scheme
View Description Hide DescriptionA family of the Morse oscillators with certain quantized coupling constants are described as composite objects in the framework of the SO(4,2) dynamical group scheme. Although a single Morse oscillator can be solved by the subgroup SO(2,1) of SO(4,2) this SO(2,1) is not the spectrum generating group], the set of all energy levels is given by the representation of another particular one‐parameter subgroup of SO(4,2), which is the dynamical group of a single Morse oscillator. The continuous spectra of this oscillator and other variations of the Morse potential are also discussed by making an analytic continuation from the Morse potential well to the Morse barrier.

On a class of 6j coefficients with one multiplicity index for groups SP(2N), SO(2N), and SO(2N+1)
View Description Hide DescriptionAn important class of 6j symbols for the groups SP(2N), SO(2N), and SO(2N+1) with one nontrivial multiplicity index is investigated. An appropriate choice of a basis in the multiplicity space is made and the so‐called canonical form for 6j symbols is obtained. Their expressions depending on the roots of an Nth‐order equation and explicit expressions for some simple class of representations are obtained.

SU(2) and SU(1,1) time‐ordering theorems and Bloch‐type equations
View Description Hide DescriptionAlgebraic time‐ordering techniques for SU (2) and SU (1,1) coherence preserving Hamiltonians are reviewed. The link with Bloch‐type equations is pointed out and the extension of the method to higher groups is briefly discussed.

Mathematical structures for long‐range dynamics and symmetry breaking
View Description Hide DescriptionThe algebraic dynamics of systems with long‐range (instantaneous) interactions requires an enlargement of the (quasi) local algebra which, in most relevant cases, includes variables at infinity that enter in an essential way in the time evolution of local variables. The mathematical structures emerging for the treatment of long‐range dynamics are investigated also in connection with spontaneous symmetry breaking.

Transmission coefficients in anharmonic symmetrical potentials
View Description Hide DescriptionBarrier transmission in potentials of the type V(x)=A x ^{2}+B x ^{4} is studied using the phase integral method, the same as the JWKB approximation in lower orders. Elliptic functions are used for the classical solutions. The transmission coefficient is calculated for all signs and values of A and B that give a potential barrier.

Analytical continuation of the Faddeev equation for local potentials
View Description Hide DescriptionIt is shown how to analytically continue the Faddeev equation in the second sheet of the complex energy plane when one has a local two‐body interaction.

Geometric space‐time perturbation. I. Multiparameter perturbations
View Description Hide DescriptionThe standard definition of space‐time perturbation is reexamined. It is seen that the noninvariance of the metric under identification gauge transformations is a consequence of the adopted zero signature in the fifth dimension of the space of space‐times. An n‐parameter extension of that definition is proposed, with a (4+n)‐dimensional flat space of space‐times with a nonsingular metric. It is shown that in the vicinity of a point in the background space‐time there is a geometrically defined family of perturbations, which are solutions of the Einstein–Yang–Mills equations.