Analytic continuation of group representations II

The paper deals with the general background connecting the ideas of analytic continuation and contraction of Lie algebras and their representations. A connection is also established between the Kodaira-Spencer deformation theory and the theory of cohomology of Lie algebras.Supported by the Office of Naval Research, NONR 3656(09).     

Analytic continuation of group representations. VI

The Gell-Mann formula for analytically continuing group representations is worked out explicitly for more cases than in previous work, and extended to certain pseudo-Riemannian symmetric spaces. The method of finding the asymptotic behavior of matrix elements of group representations introduced in Part V is developed in more detail and it is shown how it leads to new mathematical problems in the theory of dynamical systems and Hilbert space theory.Work supported by the U.S. Atomic Energy Commission.     

Analytic continuation of group representations. III

The connection between the ideas of contraction and analytic continuation of Lie algebras and their representations is discussed, with particular emphasis on the contraction of the Poincaré to the Galilean group.This research was supported in part by the Office of Air Force Scientific Research AF 49 (638)-1440.     

Analytic continuation of group representations. IV

The main problem, deforming a subalgebra of a Lie algebra, is treated algebraically, requiring an extensive development of methods of defining multiplications on Lie algebra cohomology cochains. Some applications to differential geometry are also presented.Work supported by the U.S. Atomic Energy Commission.     

Analytic continuation of group representations-V

The connection between analytic continuation of group representations and analytic continuation of their matrix elements is discussed, together with some related problems concerning the group-theoretic nature of theS-matrix, and the asymptotic behavior of the special functions of mathematical physics.Work supported by U. S. Atomic Energy Commission     

Analytic continuation of harmonic sums

We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a harmonic sum to be calculated as an expansion valid for large values of its argument. A program for implementing this method is also provided.     

Analytic continuation at first-order phase transitions

We study the analytic structure of thermodynamic functions at first-order phase transitions in systems with short-range interactions and in particular in the two-dimensional Ising model. We analyze the nature of the approximation of the d=2 system by anN × strip. Investigation of the structure of the eigenvalues of the transfer matrix in the vicinity of H=0 in the complexH plane allows us to define a new function which provides rapidly convergent approximations to the stable free energyf and its derivatives for allH 0. This new function is used for numerical calculation of the coefficients C_n in the power series expansions of the magnetizationm in the form m(H)=1 + C_n(H-H ₀ )ⁿ for various H₀ 0. The resulting series are studied by conventional methods. We confirm recent series analysis results on the existence of the droplet model type essential singularity at H=0. Evidence is found for a spinodal at H=H_sp(Ti < 0.     

Analytic continuation of Moyal and operator algebras

We consider the analytic continuation of the commutator[q ^m, pⁿ for values of m,n positive integers. This is carried out both in the Moyal algebra and the operator algebra.     

Analytic continuation of sturmian expansions for two-photon ionization

We show that the sturmian expansions of transition amplitudes for two-photon ionization may be handled as ordinary (single variable) power series, and we describe a simple method for performing their analytic continuation above the photoelectric threshold. Numerical results are presented for the hydrogenic states 1s, 2s, and 2p.     

Analytic continuation of the Regge pole residue function

We show that the Regge residue functiony(s) is particularly well suited for performing unitarity bootstrap calculations. The reason is that firstlyy(s) has only one, viz. the right hand, cut along which its value can be evaluated from direct channel unitarity using a parameterfree representation for the partial wave amplitudeS(l, s). Secondly its values for negative real argument follow directly from large-energy scattering with the exchange of one Regge pole in the crossed channel. These values can be evaluated from the sameS(l, s) representation by partial wave sums. Then all one needs for a bootstrap system is an analytic connection of these 2 different pieces of information. We show that this can be achieved by logarithmic dispersion relations. This bootstrap system is supposed to compete favorably with the old unitaryN/D calculations. We finally also propose a new parameter free representation ofS(l, s) which applies equally well as that of Cheng. One main result is that Im α(s) has to decrease exponentially for larges.     

Analytic vectors,anomalies and star representations

It is hinted that anomalies are not really anomalous since (at least in characteristic examples) they can be related to a lack of common analytic vectors for the Hamiltonian and the observables. We reanalyze the notions of analytic vectors and of local representations of Lie algebras in this light, and show how the notion of preferred observables introduced in the deformation (star product) approach to quantization may help give an anomaly-free formulation to physical problems. Finally, some remarks are made concerning the applicability of these considerations to field theory, especially in two dimensions.     

Analytic Bethe ansatz for fundamental representations of Yangians

We study the analytic Bethe, ansatz in solvable vertex models associated with the YangianY(X _r ) or its quantum affine analogueU _q (X _r ⁽¹⁾ ) forX _r =B _r ,C _r andD _r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations ofY(X _r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying theT-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.     

Yang-Baxterization of braid group representations

For a given braid group representation (BGR), a process of the Yang-Baxterization is formulated to generate solutions of the Yang-Baxter equation (YBE). When a BGR admits the Birman-Wenzl (BW) algebraic structure, this process can be explicitly passed through and two types of trigonometric solutions of YBE are generated from such a BGR. These two solutions have, the essential difference to each other and both of them, preserve the crossing symmetry property if the given BGR has. By taking certain, reduction on the BW algebra, the rational solution is also generated. A practical condition to judge whether a BGR satisfies the BW algebra is given, from which one finds that not only the familiar BGRs of [5,7,9], but also some new, ones obtained recently in [12] have the BW structure. Thus they can be explicitly Yang-Baxterized to solutions of the YBE.     

Analytic continuation and reciprocity relation for collinear splitting in QCD

It is well-known that direct analytic continuation of the DGLAP evolution kernel(splitting functions)from space-like to time-like kinematics breaks down at three loops.We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta.However,splitting functions can be constructed from the squares of(generalized)splitting amplitudes.We establish the rules of analytic continuation for splitting amplitudes,and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions.In this way we derive the time-like splitting functions at three loops without ambiguity.We also propose a reciprocity relation for singlet splitting functions,and provide non-trivial evidence that it holds in QCD at least through three loops.     

Notes on quaternionic group representations

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clebsch-Gordan series for reducible representations. We also derive all irreducible inequivalentQ-representations of a groupG, classifying them according to a suitable generalization of the Wigner-Frobenius-Schur classification.