Volume 28, Issue 10, October 1987
Index of content:

SL(2,C), SU(2), and Chebyshev polynomials
View Description Hide DescriptionWhen expressed in terms of the trace, the characters of SU(2) are known to be related with the Chebyshev polynomials of second kind. It is shown that those of the first kind also play a fundamental role. If A∈SU(2) and t=Tr A, then f _{ n }(t)=Tr(A _{ n }), f _{ n } (2 cos θ)=sin nθ/sin θ, l _{ n }(t) =Tr(A ^{ n }), l _{ n } (2 cos θ)=2 cos nθ, where A _{ n } denotes the representative of A in the irrep of dimension n. Other polynomials related with them are of interest. They are (i) the ‘‘primordial’’ polynomialsP _{ n } (every f _{ n } or l _{ n } can be expressed in a unique way in terms of P _{ d }, where d is a divisor of n), (ii) the ‘‘factorial’’ polynomialsf _{ n }!=f _{1} f _{2}⋅⋅⋅f _{ n } which occur in a natural way in the representations, (iii) the g _{ n }polynomials appearing in the generating functions of powers of f _{ n }.

Pedestrian approach to two‐cocycles for unitary ray representations of Lie groups
View Description Hide DescriptionNecessary and sufficient conditions for unitary ray representations of connected Lie groups are reexamined. Thus a systematic constructive method is obtained for calculating the admissible exponent factors (two‐cocycles). The gauge freedom of the unitary ray representation formalism is also considered. This introduces the distinction between trivial and genuine ray representations. A special gauge is then adopted, within which the two‐cocycle is almost unique. The only prerequisite of the exponent factor calculus is the knowledge of the binary combination rules for the essential parameters of the group. The attained method affords a simple, general, and explicit (i.e., coordinate‐dependent) two‐cocycle calculus. The aim of this paper is merely instrumental.

Finite‐dimensional representations of the special linear Lie superalgebra sl(1,n). I. Typical representations
View Description Hide DescriptionIn a series of two papers all finite‐dimensional irreducible representations of the special linear Lie superalgebra sl(1,n) are written down in a matrix form. This paper develops a background for constructing the representations. Expressions for the transformation of the basis under the action of the generators are given for all induced and, hence, for all typical sl(1,n) modules.

Parastatistics and the Clifford algebra unitary group approach to the many‐electron correlation problem
View Description Hide DescriptionIt is shown that the Clifford algebra unitary group approach, which is based on the subgroup chain U(2^{ n })⊇SO(2n+1)⊇SO(2n)⊇U(n), may be described in terms of the para‐Fermi algebra. Applications to the development of efficient algorithms for the evaluation of matrix elements of U(n) generators and of their products are also briefly discussed.

On the initial value problem for a class of nonlinear integral evolution equations including the sine–Hilbert equation
View Description Hide DescriptionA method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented. The underlying scattering problem is a matrix Riemann–Hilbert problem. Scattering analysis shows that the spectrum is purely discrete. An application is to the so‐called sine–Hilbert equationHθ_{ t } =−c sin θ, where c is a constant and H denotes the Hilbert transform.

Exact solutions for the nonlinear Klein–Gordon and Liouville equations in four‐dimensional Euclidean space
View Description Hide DescriptionA systematic method for constructing particular solutions of the nonlinear Klein–Gordon and Liouville equations in four‐spatial dimensions is developed. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations (ODE’s) by introducing symmetry variables and then seeking exact solutions for more tractable ODE’s. Various exact solutions are presented, in which new solutions with nonspherical symmetries are included. Furthermore, the exact method is applied to the above equations in general n‐spatial dimensions. Among them, a conformally invariant nonlinear Klein–Gordon equation is particularly interesting from the viewpoint of field theories. The exact solutions for these equations are generalizations of those for the corresponding equations in four‐spatial dimensions.

Tensors with icosahedral symmetry that are invariant under a certain wreath product
View Description Hide DescriptionFaithful icosahedral symmetry exists only for tensors of rank higher than 5. The most relevant tensor of this type is the one for third‐order elastic constants C ^{(3)} _{ i j k l m n } defined by the series expansion T _{ i j }=C ^{(2)} _{ i j k l }ε_{ k l }+ 1/2 C ^{(3)} _{ i j k l m n }ε_{ k l }ε_{ m n } +⋅⋅⋅ of the stress tensorT in terms of the deformationtensor ε. A basis for those tensors in (R^{3})^{⊗6} that are invariant under a certain action of both the icosahedral group S _{2}×A _{5} and the wreath product S _{2}∼S _{3} of the symmetric groups S _{2} and S _{3} are evaluated.

Local Green’s function. I
View Description Hide DescriptionThe local Green’s function is used in many physical problems. In this paper, the properties of the local Green’s function are studied, and it is proved that the N×N local Green’s function can represent the results of the full N _{1}×N _{1}Green’s function, where N is small (or at least finite) and N _{1} is large (or infinite). The accuracy of cutting the general Green’s function into the local Green’s function is also discussed.

Local Green’s function. II
View Description Hide DescriptionAs shown in the preceding paper [J. Math. Phys. 2 8, XXXX (1987)], the local Green’s function can represent the results of the original general Green’s function. However, it is difficult to find the exact local Green’s function in the general case. In this paper, a special case—‘‘the chain matrix’’—is studied, which is a generalization of the tridiagonal matrice.

Local Green’s function. III
View Description Hide DescriptionIn this paper a way of solving for the local Green’s function by means of a projection operator method is presented. Both a procedure for attacking the problem and a general formula for calculation are obtained. It is also proved that the calculation can be done with a finite number of basis functions. A short discussion of the accuracy is also presented.

The ‘‘spectral Wronskian’’ tool and the ∂̄ investigation of the KdV hierarchy
View Description Hide DescriptionThe recently introduced ‘‘spatial transform’’ (ST) method for providing solutions to nonlinear evolution equations is developed when the basic ∂̄ equation is the ‘‘spectral Schrödinger’’ equation (S). A fundamental tool is the ‘‘spectral Wronskian,’’ which allows one to take advantage of the structure of a two‐dimensional module for some set of solutions of (S). This leads easily to the KdV hierarchy. Contrary to the usual spectral transform (or inverse scattering transform) method there is no a p r i o r i assumption on the long distance behavior of the solutions. A recursion operator is exhibited. Local conservation laws and Bäcklund relations are also derived.

Yang–Mills–Higgs theory on a compact Riemann surface
View Description Hide DescriptionJaffe and Taubes [V o r t i c e s a n d M o n o p o l e s (Birkhauser, Boston, 1980)] have shown the existence and uniqueness of n‐vortex solutions on the complex plane. In this paper, their results are generalized to an arbitrary U(1) bundle over a compact Riemann surface with a Hermitian metric. Berger’s ‘‘nonlinear analysis’’ [N o n l i n e a r i t y a n d F u n c t i o n a l A n a l y s i s (Academic, New York, 1977)] has provided an effective method to prove the existence part of the main theorem of this paper.

Differential geometric aspects of the Cartan form: Symmetry theory
View Description Hide DescriptionThis paper demonstrates that the classical Cartan form θ^{1} _{ L } is not adequate for the determination of all the natural symmetries and conservation laws for a Lagrangian L. It is shown that the various extensions θ^{2} _{ L },..., θ^{ r } _{ L } of the classical Cartan form, introduced in recent papers, give larger symmetry groups: G _{1}⊆G _{2}⊆⋅⋅⋅⊆G _{ r }. This paper also introduces the notion of contact equivalent Lagrangians, which serves to clarify the idea that different Lagrangians can give rise to the same variational and symmetry theories.

On the dimensional reduction of invariant fields and differential operators
View Description Hide DescriptionIn the present paper the most general type of group action introducing some relevant exact sequences for the dimensional reduction of invariant fields and differential operators is studied.

Cohomology of supermanifolds
View Description Hide DescriptionThe cohomological properties of supermanifolds (intended in the sense of De Witt [S u p e r m a n i f o l d s (Cambridge U. P., London, 1984)] and Rogers [J. Math. Phys. 2 1, 1352 (1980)]) are investigated, paying particular attention to the de Rham cohomology of supersmooth differential forms (SDR cohomology). The SDR cohomology of De Witt supermanifolds is shown to be equivalent to the de Rham cohomology of their body. The SDR cohomology is explicitly computed for some topologically nontrivial supermanifolds and some general conclusions concerning the geometric structure of supermanifolds and the properties of the SDR cohomology are drawn. In particular, it is shown that the SDR cohomology is neither a topological nor a real differentiable invariant, but rather a ‘‘superdifferentiable’’ invariant.

Canonoid transformations and constants of motion
View Description Hide DescriptionThe necessary and sufficient conditions for a canonoid transformation with respect to a given Hamiltonian are obtained in terms of the Lagrange brackets of the transformation. The relation of these conditions with the constants of motion is discussed.

Decomposition of Lorentz transformations
View Description Hide DescriptionIt is shown that every Lorentz transformation can be decomposed into a helicity‐preserving transformation that changes the momentum of a free particle and a helicity‐changing transformation that leaves the momentum invariant. Since momentum‐preserving transformations constitute a subgroup of the Lorentz group, helicity‐preserving transformations form a coset space. It is shown further that, for massive particles, every Lorentz transformation can be decomposed into the Wigner rotation and helicity‐preserving transformations. For massless particles, every Lorentz transformation can be decomposed into the gauge transformation and helicity‐preserving transformation. The gauge transformation in this case is a Lorentz‐boosted Wigner rotation.

Generalized Lorentz transformation for an accelerated, rotating frame of reference
View Description Hide DescriptionAn exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time‐dependent, nongravitational acceleration and an arbitrary time‐dependent angular velocity is given. This transformation is a generalization of the Lorentztransformation and is obtained in two steps. First, the Minkowski metric is transformed under an intermediate coordinate transformation to obtain a new set of noninertial metric coefficients in which one can easily identify the Thomas precession, as well as the expression for the acceleration of the moving frame with respect to the instantaneous rest frame in terms of the acceleration as seen from a stationary inertial frame. Second, a rotation of axes is performed to absorb the Thomas precession and to add an ordinary spatial rotation. The coordinate transformation obtained by combining these effects is nonlinear, since certain terms involve time integrals, and leads to the appropriate space‐time metric for an accelerated, rotating frame of reference. It is shown that the usual forms of the Lorentztransformation are contained as special cases of this result.

Jauch–Piron states in W*‐algebraic quantum mechanics
View Description Hide DescriptionA state φ on a W*‐algebra M is said to fulfill the J a u c h–P i r o n c o n d i t i o n if φ(p)=φ(q)=1 for projections p,q∈M implies φ(p∧q)=1. Here p∧q denotes the infimum of p and q in the projection lattice of M. The Jauch–Piron condition is a compatibility condition between the algebraic and the lattice‐theoretic approach for the description of physical systems. Normal (i.e., σ‐weakly continuous) states always fulfill the Jauch–Piron condition. It is argued that states not fulfilling this condition should be regarded as unphysical. It is shown that a state φ on a σ‐finite f a c t o rM is singular if and only if projections e, f∈ M exist such that φ(e)=φ( f )=1 and e∧f=0. In particular, any p u r e state φ on M fulfilling the Jauch–Piron condition is normal, which implies that the underlying factor M is of type I. Furthermore, the following result is proved: Let φ be a p u r e Jauch–Piron state on W*‐algebra M with separable predual and without any commutative summand. Then φ is normal and a central projection z _{0}∈ M exists such that φ(z _{0})=1 and z _{0} Mz _{0} is a factor of type I. Thus, c u m g r a n o s a l i s, p u r e Jauch–Piron states exist only on commutative W*‐algebras and type I factors. The former case corresponds to classical theories, the latter to Hilbert‐space quantum mechanics. The implications of these results on the interpretation of quantum mechanics are discussed.

The Wigner transformation is of finite order
View Description Hide DescriptionThe Wigner integral transformation, which intertwines the twisted product and the composition of kernels, is of order 24. Indeed, it commutes with, and its sixth power equals, the Fourier cotransformation.