Irreducible Representation Space Research Articles

An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schrödinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus Tn=Rn/Λ is defined as a quotient of the Euclidean space Rn by an arbitrary n-dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schrödinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.

Time asymmetric quantum theory – II. Relativistic resonances from S‐matrix poles

AbstractRelativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.

Time asymmetric quantum theory – III. Decaying states and the causal Poincaré semigroup

AbstractThese Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of the Lippmann‐Schwinger plane wave scattering states, form only a semigroup of Poincaré transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed.

Surface Group Representations and U(p, q)-Higgs Bundles

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a companion paper "Moduli spaces of holomorphic triples over compact Riemann surfaces" (math.AG/0211428) we prove that these moduli spaces of triples are non-empty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p,q)-representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor-Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is non-empty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

Symmetric Analogues of Rarita-Schwinger Equations

In this paper a generalization of the classicalRarita–Schwinger equations for spin 3/2 fields to the case of spin fieldswith values in irreducible representation spaces with weight k+1/2 isgiven. It corresponds to the study of serie of first orderconformal invariant operators, which are constructed from twisted Diracoperators. The representation character of polynomial solutions of the equations onflat space and their relations are described in details.

Rarita–Schwinger Type Operators in Clifford Analysis

In this paper we investigate a generalization of the classical Rarita–Schwinger equations for spin 3/2 fields to the case of functions taking values in irreducible representation spaces with weight k+1/2. These fields may be realised as functions taking values in spaces of spherical monogenics earlier considered in F. Sommen and N. Van Acker (1993, in “Clifford Algebras and Their Applications in Mathematical Physics,” Kluwer Academic, Dordrecht/Norwell, MA). In this paper we develop the main function theoretic results.

Spinor-valued and Clifford algebra-valued harmonic polynomials

We give decompositions of the spinor-valued and the Clifford algebra-valued harmonic polynomials on R n . In order to do so, we consider some differential complexes and show that these are exact. As a corollary, we have Poincaré lemma for harmonic polynomials. Besides, we prove that each component of the decompositions is an irreducible representation space with respect to Spin( n).

On Verma bases for representations of sl(n,C)

Conjectural bases for irreducible representation spaces of simple Lie algebras, due to Verma, have been described by Li, Moody, Nicolescu, and Patera. In this paper the basis conjecture for the algebras of type An is established.

NONLOCALITY AND GRAVITY-INDUCED CP VIOLATION

We consider a space–time dependent relative phase between the right- and left-handed spinors and show that it results in a violation of locality in the presence of gravity once the demand of parity covariance is dropped. This violation of locality is such that it readily interprets itself as a gravity-induced CP violation, and at the same time confirms an earlier remark by Wigner that a representation space carries more information than a wave equation. This happens, as Kirchbach has noted, because while the dimensionality of an irreducible representation space does not depend upon the concrete realization of the symmetry generators, Noether currents (Dirac, versus Majorana, versus the CP violating construct presented here) do. The gravity-induced CP violation provides a dynamical reason on how a neutron star carrying its baryon and lepton numbers can collapse into a black hole and loose information on the latter characteristics.

Coordinates of the representation space in the semisimple Lie group of rank one

In this paper we show that the space of irreducible representations from a finitely presented group into the group of isometries of a rank one symmetric space of non-compact type, embeds into ℝn for some n, where the coordinates are the translation lengths of isometries in the representation. The ingredients of the proof consist of the two facts that the representation is determined by its marked length spectrum and that the nested sequence of algebraic subvarieties is stabilised at a finite step by the Noetherian property of the polynomial ring. As a minor application, we use this fact to simplify McMullen's proof about the exponential algebraic convergence of Thurston's double limit to the geometrically infinite manifold in the space of discrete faithful representations of π1(S) in Iso+.

Representation properties, Racah sum rule, and Biedenharn–Elliott identity for Uq(osp(1|2))

It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra Uq(osp(1|2)) can be expressed by Clebsch–Gordan coefficients. The Racah sum rule satisfied by Uq(osp(1|2)) Racah coefficients and 6−j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn–Elliott identity holds for Uq(osp(1|2)) Racah coefficients and 6−j symbols. A recursion relation for Uq(osp(1|2)) 6−j symbols is derived from the Biedenharn–Elliott identity.

Symmetrized projector quantum Monte Carlo studies of the ground state of C60.

We develop a symmetrized version of the projector quantum Monte Carlo (SPQMC) method which preserves the symmetries of the system by simultaneously sampling all symmetry-related Ising configurations at each MC step and use it to study the effect of electron correlations on a single ${\mathrm{C}}_{60}$ molecule and its structural motifs, within the Hubbard model. This SPQMC method allows more accurate estimates of correlation functions as seen from calculations on small systems. The method applied to some molecular fragments of ${\mathrm{C}}_{60}$, viz., pyracylene, fluoranthene, and corannulene, gives a good chemical description of these systems. Analysis of the ground-state bond orders allows us to visualize pyracylene as a naphthalene moiety with weakly bridged ethylenic units, fluoranthene as weakly bridged naphthalene and benzene units, and corannulene as radialene with ethylenic bridges. We study the ground-state properties of a single fullerene molecule, with and without bond alternation. The bond orders in the ground state for the two types of nearest-neighbor bonds are unequal, even for uniform ${\mathrm{C}}_{60}$. The spin and charge correlations give a consistent picture of the interacting ground state in all these systems. We construct projections of the bond orders and spin-spin correlation functions on the space of irreducible representations of the icosahedral point group. These projections, analogous to the structure factors of translationally invariant systems, give the amplitudes for distortions which transform as the irreducible representations of the point group. The amplitude for the ${\mathit{H}}_{\mathit{g}}$ distortion is the largest, while the spin structure has large weights in the ${\mathit{T}}_{2\mathit{g}}$ and ${\mathit{G}}_{\mathit{u}}$ representations. \textcopyright{} 1996 The American Physical Society.

On quasi-rational conformal field theories

A simple family of quasi-rational conformal field theories, namely the unitary c = 1 theories, is studied from a W-algebra point of view. The moduli space of irreducible representations and an appropriate generalization of the Verlinde formula are considered.

Homology and cohomology of Lie superalgebra $\mathfrak{sl}(2,1)$ with coefficients in the spaces of finite-dimensional irreducible representations

Homology and cohomology of Lie superalgebra $\mathfrak{sl}(2,1)$ with coefficients in the spaces of finite-dimensional irreducible representations

BRST COHOMOLOGY IN QUANTUM AFFINE ALGEBRA $U_q \left( {\widehat{sl_2 }} \right)$

Using free field representation of quantum affine algebra [Formula: see text], we investigate the structure of the Fock modules over [Formula: see text]. The analysis is based on a q-analog of the BRST formalism given by Bernard and Felder in the affine Kac-Moody algebra [Formula: see text]. We give an explicit construction of the singular vectors using the BRST charge. By the same cohomology analysis as the classical case (q=1), we obtain the irreducible highest weight representation space as a non-trivial cohomology group. This enables us to calculate a trace of the q-vertex operators over this space.

Angular momentum on the quantum Euclidean spaceR q 3

We construct the «angular momentum» on the Euclidean quantum spaceRq3,i.e. theq-deformed analog of the angular momentum on ordinary 3-dimensional spaceR3, as a q-deformed differential operator onRq3. It generates anUq(3) ≈Uq(SU(2)) Hopf algebra, as expected. The base spaces of irreducible (vector) representations are the spaces of q-symmetrized homogeneous polynomials in the coordinates ofRq3.

Physical states in G/G models and two-dimensional gravity

An analysis of the BRST cohomology of the G/G topological models is performed for the case of A 1 (1). Invoking a special free field parametrization of the various currents, the cohomology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irreducible representations. Using the physical states we calculate the characters and partition function, and verify the index interpretation. We twist the energy-momentum tensor to establish an intriguing correspondence between the SL(2)/SL(2) model with level k = p/ q −2 and ( p, q) models coupled to gravity.

Quantum Clebsch-Gordan coefficients for nongeneric q values

In the decomposition of a coproduct in the direct product of two irreducible representation spaces of the quantum enveloping algebra Uqsl(2), the mixed representation, reducible but indecomposable, appears when q is a root of unity. In this paper, the necessary and sufficient condition for mixture of two irreducible representations are presented, the quantum Clebsch-Gordan coefficients which are neither all vanishing nor divergent for the nongeneric q values are defined, and the method for computing the new states are discussed in some detail.

A remark on the representation theory of the algebra Uq(sl(n)) when q is a root of unity

An associative algebra is defined on the set of certain irreducible representation spaces of the algebra Uq(sl(n)) when q is a root of unity. This associative algebra is shown to be connected with some algebras defined by the decomposition rule of tensor products of the irreducible representations of certain finite groups.

The missing link: operators for labelling multiplicity in the Clebsch-Gordan series

It is indicated that there are invariant operators on the direct product space of irreducible representations of Lie Groups which distinguish between repeating representations. They complete the labelling of the Clebsch-Gordan series.