Volume 31, Issue 6, June 1990
Index of content:

Beyond Gaussian integrals in Grassman algebra
View Description Hide DescriptionUsing the properties of determinants, the exact expressions for integrals of functions of the type exp[ηAη̄+(ηBη̄)^{l}] and their moments, where η and η̄ are Grassman generators with N components are derived.

Minimization of Landau potentials invariant under O(3).II
View Description Hide DescriptionIn a previous paper, the absolute minima for Landau potentials were computed for the irreducible representations of SO(3) or O(3) of spin up to four. Here, this analysis is extended to the case of spin five and six. Some novel properties of the extrema are pointed out.

Similarity transformations of irreducible corepresentations in Wigner canonical form
View Description Hide DescriptionThe general form of the matrices that transform an irreducible corepresentation (coirrep) into an equivalent one, where both representations are assumed to be in the form proposed by Wigner, is analyzed for the three types of coirreps. In addition, the relation of these transformations to inner automorphisms of the corresponding group algebras is clarified.

Shifted tableaux, Schur’s Q functions, and Kronecker products of S _{ n } spin irreps
View Description Hide DescriptionProperties of shifted tableaux have been explored in order to improve the algorithm for the calculation of Q function outer products. A simple technique has been established for finding out the highest and lowest partitions in the expansion of Q function outer products. Using these techniques and Young’s raising operators, we have completed the Kronecker product for S _{ n } spin irreps.

Finite geometries and Clifford algebras. II
View Description Hide DescriptionThe Clifford algebra in dimension d=2^{ m+1}−1, m≥2, is treated using the finite m‐dimensional projective geometry PG(m, 2) over the field of order 2. Full details are given for the case m=4, d=31, generalizing previous work for m=2 and 3. Details are given of some configurations that arise in PG(4, 2).

Explicit orthonormal Clebsch–Gordan coefficients of SU(3)
View Description Hide DescriptionThe construction of the explicit algebraic‐polynomial expressions for the nonmultiplicity‐free orthonormal Clebsch–Gordan (Wigner) coefficients of SU(3)⊇U(2) is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the Hecht or Gram–Schmidt process. The direct and inverse orthogonalization coefficients (the first of them being equivalent to the boundary orthonormal isofactors) are expressed, up to explicitly given multiplicative factors, in terms of the numerator and denominator polynomials related with the auxiliary A _{λ} function of Louck, Biedenharn, and Lohe that appears as a fragment of the denominator G‐functions of canonical SU(3) tensor operators.

Deformations of some infinite‐dimensional Lie algebras
View Description Hide DescriptionThe concept of a versal deformation of a Lie algebra is investigated and obstructions to extending an infinitesimal deformation to a higher‐order one are described. The rigidity of the Witt algebra and the Virasoro algebra is deduced from cohomology computations for certain Lie algebras of vector fields on the real line. The Lie algebra of vector fields on the line that vanish at the origin also turns out to be rigid. All the affine Lie algebras are rigid; this is derived from the cohomology of their maximal nilpotent subalgebra. On the other hand, the maximal nilpotent subalgebras in both the Virasoro and affine cases are not rigid and have interesting nontrivial deformations (in fact, most vector fieldLie algebras are not rigid).

Solvable Lie algebras of dimension six
View Description Hide DescriptionSix‐dimensional solvable Lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes. This completes Mubarakzyanov’s classification of the real six‐dimensional solvable Lie algebras.

Nonexistence and existence of various order integrals for two‐ and three‐dimensional polynomial potentials
View Description Hide DescriptionThe nonexistence of integrals of the motion, which are sixth and fourth degree polynomials in the velocities, was established for a range of polynomial potentials. The first known systematic search for sixth‐order integrals was performed for cubic and quartic polynomial potentials. It revealed that there exist no nondegenerate cases with such integrals for either potential. Similarly, there exists no nondegenerate fifth or sixth degree polynomial potentials possessing quartic invariants. A complete list of three‐dimensional cubic potentials with quadratic integrals is given. All these integrable three‐dimensional potentials can be interpreted as orthogonal superpositions of known integrable two‐dimensional potentials possessing quadratic integrals. They correspond to 3 of 11 possible coordinate systems in which three‐dimensional potentials separate.

Dispersion relations for causal Green’s functions: Derivations using the Poincaré–Bertrand theorem and its generalizations
View Description Hide DescriptionA famous theorem by Poincaré and Bertrand formally describes how to interchange the order of integration in a double integral involving two principal‐value factors. This theorem has important applications in many‐body physics, particularly in the evaluation of response functions (or ‘‘loop integrals’’) at either zero or finite temperatures. Of special interest is the loop containing an integration with respect to the energy of two causal propagators. It is shown that such a response function with two boson or two fermion lines behaves statistically like a boson, while the response function containing a boson and a fermion behaves like a fermion. Examples are given of typical loop integrals occurring in the solution of Dyson’s equations for nuclear matter in the presence of delta, nucleon, and pion interactions. A ‘‘form factor’’ that is essential for the convergence of the nucleon–pion loop integral is chosen to have little effect on the analogous nucleon–delta loop integral. The Poincaré–Bertrand (PB) theorem is then generalized to multiple integrals and higher‐order poles. From the generalization of the theorem to triple integrals, it is shown that causality is rigorously maintained, at zero temperature, for convolutions with respect to the time of products of Green’s functions and thus for Dyson’s equations. Also, for finite temperature, the three‐propagator loop integral satisfies the statistics appropriate for the loop as a whole, in direct analogy with the result for the two‐propagator loop. The intimate connection between the PB theorem and analyticity (or causality) is clearly demonstrated. Although this work considers explicitly only nuclear physics examples, the results are relevant to other fields where many‐body theory is used.

Classical Liouville completely integrable systems associated with the solutions of Boussinesq–Burgers’ hierarchy
View Description Hide DescriptionTwo new finite‐dimensional completely integrable systems in the Liouville sense are obtained. The solutions of Boussinesq–Burgers’ hierarchy are generated by using involutive solutions of the commutable flows in the completely integrable systems.

Auxiliary linear equations for a class of nonlinear partial differential equations via jet‐bundle formulation
View Description Hide DescriptionIt is shown that a class of two‐variable nonlinear partial differential equations, such as the Liouville equation, the sine–Gordon equation, the Ernst equation, and the Ernst–Maxwell equations, can be Ricatti‐type quasilinear systems through maps from J ^{1} (R ^{2}, R ^{ n }) to J ^{0} (R ^{2}, R ^{ l }). The auxiliary linear equations (Laxpair) for them are formulated, respectively, by using the W–E prolongation procedure. The Lax pairs for the Liouville equation and the sine–Gordon equation contain an arbitrary parameter besides the usual spectral parameter.

The quantization condition in the presence of a magnetic field and quasiclassical eigenvalues of the Kepler problem with a centrifugal potential and Dirac’s monopole field
View Description Hide DescriptionIn the presence of a magnetic field, the Maslov quantization condition is not available in the original form. An alternative quantization condition is proposed with the aid of a principal U(1) bundle over a phase space and a connection whose curvature form is the charged symplectic form. By means of this quantization condition, quasiclassical eigenvalues of the Kepler problem with a centrifugal potential and Dirac’s monopole field are calculated, which turn out to coincide with the eigenvalues of the quantized problem.

Proof of the noninteraction theorem for a system of N relativistic particles in direct interaction
View Description Hide DescriptionThis paper proves a noninteraction theorem for a system of N relativistic particles in direct interaction, and each particle has internal Grassmann degrees of freedom for describing spin.

On equations of type u _{ x t }=F(u,u _{ x }) which describe pseudospherical surfaces
View Description Hide DescriptionThe equations u _{ x t }=F(u,u _{ x }), which describe η‐pseudospherical surfaces, are characterized. In particular, when F does not depend on u _{ x }, the sine–Gordon, sinh–Gordon, and Liouville equations are essentially obtained. Moreover, it is shown that an equation u _{ x t }=F(u) has a self‐Bäclund transformation if and only if it describes an η‐pseudospherical surface.

Stochastic mechanics of a relativistic spinless particle
View Description Hide DescriptionAn extension of Nelson’s stochastic mechanics to the relativistic domain is proposed. To each pure state of a spinless relativistic quantum particle corresponds a Markov processt ^{]}ξ_{ t }, where the random variable ξ_{ t } represents, at every time t, the space position of the particle in the sense of Newton and Wigner. The process t ^{]}ξ_{ t } is not a diffusion but the usual Nelson’s theory is restored in the nonrelativistic limit.

Rotations, squeezing, and the unitary transformation operator from individual particles to Jacobi variables
View Description Hide DescriptionA unitary operator for the transformation from individual particles to Jacobi variables is constructed explicitly for particles of arbitrary masses. It is expressed as a product of rotation and squeezing operators using only canonical variables.

Sternberg construction and reduction
View Description Hide DescriptionA connection between the Sternberg construction, which allows one to introduce a symplectic structure on an associated fiber bundle with the base and the fiber being symplectic manifolds, and the reduction of symplectic manifolds is considered. It is shown that the Sternberg construction commutes with the reduction.

The problem of gravity‐gyroscopic waves, which are excited by the oscillations of a curve
View Description Hide DescriptionThe problem of the oscillations of an ideal stratified rotating fluid, which are excited by a curve in the case when the distribution of pressure on both sides of curve is prescribed, is considered. The solution to the problem, as well as results about symmetry properties of the potentials used for the solution of such problems, is obtained. The question of the uniqueness of the solution is also considered.