Universal Covering Group of U(n) and Projective Representations

Using fiber bundle theory, we construct the universal covering group of U(n),U(n), and show that U(n) is isomorphic to the semidirect product SU(n) ∝.We give a bijection between the set of projective representations of U(n) and theset of equivalence classes of certain unitary representations of SU(n) ∝.Applying Bargmann's theorem, we give explicit expressions for the liftings ofprojective representations of U(n) to unitary representations of SU(n) ∝. Forcompleteness, we discuss the topological and group theoretic relations betweenU(n), SU(n), U(t), and Z _n .     

Relativity groups in the presence of matter

We show that the action of the boosts on an infinite system can be described continuously by bundle maps of Hilbert bundles based on the manifoldsG/G ₀, whereG is the full relativity group andG ₀ its closed subgroup which can be unitarily implemented on the fibre, which is a Hilbert space. We then develop a general theory of group representations on product bundlesM × ?, whereM is a manifold and ? a Hilbert space. We construct certain bundle representations of the Galilei and the Poincaré group and find that they correspond to various classes of elementary excitations. In particular, we define nonrelativistic zero-mass systems and obtain an analogue of the Faraday effect for the passage of hot electrons through matter. Our construction remains valid for the case whenG ₀ is the product of a lattice translation group and the time translations. We conclude that many qualitative features of the physics of condensed matter can be directly understood in terms of relativity group action on a bundle space as state space, which also suggests some avenues for further work.     

Machinery for the analysis of field theories with zero-mass particles

We establish a machinery for the analysis of field theory models with zero-mass particles with great generality and in a model independent way. To this end a special class of functions called the class A_n-functions, introduced earlier by the author, is quite suitable for such a machinery. We show that Feynman integrands and absolutely convergent Feynman integrals, involving zero-mass particles, belong to such a class as functions of external (and internal) momenta and the non-zero masses of the theory. We show, in particular, how one may find real numbers b_r, d_r′, which set 《high-energy》 and 《low-energy》 scales, respectively, with b_r>1 and 0<d_r′<1, such that for certain parameters η_r′,λ_r′>0, with η_r?b_r and λ_r′?d_r′, the absolute value of a Feynman integrand may be bounded by introducing suitable high- and low-energy asymptotic coefficients. The analysis provides, in particular, explicit forms for the latter coefficients for the Feynman integrals themselves and hence leads, to powerful results for the study of their asymptotic behaviour.     

On representations of the groups Sp(n, 1) and Sp(n)

New classes of unitary irreducible representations of Sp(n, 1) which can be useful for applications in physics are obtained. The infinitesimal operators of these representations of Sp(n, 1) and of irreducible representations of Sp(n+1) with highest weights (m, m, m₃,…,m_n+1) and (m₁, m₂, 0,…,0) are expressed in terms of the simple Clebsch–Gordancoefficients for Sp(n). For Sp(3) and Sp(2, 1) they are found in an explicit form.     

On the representation of finite transformations of O2 in the canonical basis of the group On

The matrix elements of the orthogonal transformations of the two-coordinate subspace of the n-dimensional space in the canonical basis of the orthogonal (O_n) and the rotation (SO_n) groups are considered. The matrix elements of the projection operator of the representations of SO₂ as the non-canonical subgroup of SO_n in the canonical basis of SO_n have been used. The relations between basis states of O_n and SO_n representations are described, and it is shown how to use the substitution group and the Regge symmetries and other types of symmetries.     

Classification of representations of the algebra U′q(so 3) through examples II

This paper completes series of articles devoted to classification of the representations of the nonstandard deformation U′_q(so ₃) providing examples of such representations in low dimensions. The classification differs substantially when the deformation parameter q is/is not root of unity (q ⁿ=1). When it is a root of unity, the situation differs for odd and even n. The examples presented here cover the first nontrivial case when n is even (namely, n=4), from which the general case follows easily.     

Representations and inequalities for Ising model Ursell functions

We describe and investigate representations for the Ursell functionu _n of a family ofn random variables {σ _i}. The representations involve independent but identically distributed copies of the family. We apply one of these representations in the case that the random variables are spins of a finite ferromagnetic Ising model with quadratic Hamiltonian to show that (?1) ^n/2+1 u _n(σ ₁, ...,σ _n) ≧ 0 forn=2, 4, and 6 by proving the stronger statement \(( - 1 )^{\frac{n}{2} + 1} \frac{{\partial ^m }}{{\partial J_{i1j1} \cdots \partial J_{imjm} }}Z^{\frac{n}{2}} u_n \left| {_{J = 0} } \right. \geqq {}^\backprime 0\) forn=2, 4, and 6, theJ _ij being coupling constants in the Hamiltonian andZ the partition function. For generaln we combine this result with various reductions to show that sufficiently simple derivatives of (?1) ^n/2+1 Z ^n/2u_n, evaluated at zero coupling, are nonnegative. In particular, we conclude that (?1) ^n/2+1 u _n ≧ 0 if all couplings are nonzero and the inverse temperature β is sufficiently small or sufficiently large, though this result is not uniform in the ordern or the system size. In an appendix we give a simple proof of recent inequalities which boundn-spin expectations by sums of products of simpler expectations.     

On the skew representations of the quantum affine algebra

An explicit realization of the skew representations of the quantum affine algebra U _q (gl _n ) is given. It is used to identify these representations in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (orq-character).     

Massless Fields as Unitary Representations of the Poincaré Group

Relativistic zero-mass fields are described as manifestly covariant unitary irreducible representations of the Poincaré group. The wave-equations, which are a necessary condition for unitarity, are constructed for spinor fields of arbitrary spin and for tensor fields of integer spin. Poincaré covariance together with causality and positive energy are used to determine the commutators of quantized fields up to a positive multiple and to prove the spin-statistics theorem. The use of potentials for boson fields is discussed and it is shown that, at the expense of manifest covariance, potentials may be introduced as zero-mass limits of the massive Wigner representations.     

The matrix elements of irreducible representations of the group SO(n,1) in integral form

Using the method of induced representations, the matrix elements of unitary irreducible representations of the group SO(n,1) are found in integral form.     

Algebraic solution of the Hubbard model on the infinite interval

We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2) ) and that the bound states are Yangian singlet.     

Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-l B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ affine Lie algebras, are baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D_n+1⁽²⁾, A_2n⁽²⁾ and B_n⁽¹⁾R-matrices of Bazhanov and Jimbo. For the D_n+1⁽²⁾ and B_n⁽¹⁾ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.     

Dynamical supersymmetry breaking in a finite field theoretical model

We present a supersymmetric field theory in two or three space-time dimensions with an internal symmetry of the O(N) type. In the large-N limit the model is finite and supersymmetry is spontaneously broken. The fields representing the order parameters of the broken supersymmetry phase acquire dynamics through quantum corrections. In particular the Goldstone fermion is a zero-mass fermionic bound state.     

Geometry of coset spaces and massless modes of the squashed seven-sphere in supergravity

We derive some general results for Killing vectors on arbitrary coset manifolds and explicitly exhibit the squashed seven-sphere as the coset space Sp₄×Sp₂/Sp₂×Sp₂. Using these results, we then analyze the zero-mass sector of supergravity of the squashed S⁷ and argue that it is not interpretable as a spontaneously broken version of N=8 supergravity. We also point out the existence of a new solution which combines squashing and torsion.     

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of n-dimensional (n ≥ 3) riemannian manifolds of bounded geometry, fixed Euler characteristic, and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.     

Polar angle dependence of the longitudinal polarization of quarks produced in e+e−-annihilation

We calculate one-loop radiative QCD corrections to the three polarized and unpolarized structure functions that determine the beam-quark polar angle dependence of the longitudinal polarization of light and heavy quarks produced in e⁺e^?-annihilations. We present analytical and numerical results for the longitudinal polarization and its polar angle dependence. We discuss in some detail the zero-mass limit of our results and the role of the anomalous spin-flip contributions to the polarization observables in the zero-mass limit. Our discussion includes transverse and longitudinal beam polarization effects.     

Positive energy unitary irreducible representations of the superalgebras <Emphasis Type="Italic">osp</Emphasis>(1|,2<Emphasis Type="Italic">n</Emphasis>, ℝ)

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n, ?).     

Invariant differential operators for non-compact Lie groups: The main su(n, n) cases

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n ²-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.     

On the formulation of theories with zero-mass propagators

A new renormalization scheme is proposed for theories with zero-mass propagators. For each Feynman diagram the method yields an ultraviolet and infrared convergent contribution to the Green functions. The method is first developed for the massless A⁴ model and then applied to the Goldstone and pre-Higgs models.