Weave States in Loop Quantum Gravity

Weaves are eigenstates of geometrical operators in nonperturbative quantum gravity, which approximate flat space (or other smooth geometries) at large scales. We describe two such states, which diagonalize the area as well as the volume operators. The existence of such states shows that some earlier worries about the difficulty of realizing kinematical states with non-vanishing volume can be overcome. We also show that the Q operator used in earlier work for extracting geometrical information from quantum states does not capture more information than the area and volume operators.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

Particle-field duality and form factors from vertex operators

Using a duality between the space of particles and the space of fields, we show how one can compute form factors directly in the space of fields. This introduces the notion of vertex operators, and form factors are vacuum expectation values of such vertex operators in the space of fields. The vertex operators can be constructed explicitly in radial quantization. Furthermore, these vertex operators can be exactly bosonized in momentum space. We develop these ideas by studying the free-fermion point of the sine-Gordon theory, and use this scheme to compute some form-factors of some non-free fields in the sine-Gordon theory. This work further clarifies earlier work of one of the authors, and extends it to include the periodic sector.     

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.     

Adiabatic theorem and spectral concentration

The spectral concentration of arbitrary order for the Stark effect is proved to exist for a large class of Hamiltonians appearing in nonrelativistic and relativistic quantum mechanics. The results are consequences of an abstract result about the spectral concentration for self-adjoint operators. A general form of the adiabatic theorem of quantum mechanics, generalizing an earlier result of the author as well as some results by Lenard, is also proved.     

Quantum statistical mechanics on stochastic phase space

Within the context of the theory of stochastic phase spaces, introduced in some earlier papers, a systematic mathematical procedure is developed for expressing quantum mechanical observables as generalized functions on a stochastic phase space. The states in such a theory are normalized, positive semidefinite, continuous functions of the phase space variables, satisfying marginality conditions appropriate to the stochastic nature of the underlying phase space. The action of a general quantum mechanical observable on the state space is then shown to lead in general to formal differential operators of finite or infinite order. Explicit computations of some typical operators are made to illustrate the theory. As a useful practical application, the theory is employed to derive a Bloch equation from which the Husimi transform of the canonical equilibrium state is then computed, after expressing it as an infinite series in powers of .Supported in part by a research grant from the National Research Council of Canada.     

Zur Quantentheorie der stimulierten RAMAN-Streuung in Molekülkristallen. I. Einige Bemerkungen zur Theorie nach K. GROB

Stimulated RAMAN scattering in molecular crystals is quantum theoretically described as stimulated polariton scattering. The quantum theoretical treatment of molecular crystals is given within the framework of a second quantization described in an earlier work. In the HEISENBERG picture the equations of motion for the operators of the polaritons are derived and specialized for RAMAN scattering. By supposing quasi-stationary behaviour the operators of the polaritons that are mixtures of phonons and photons are eliminated. The resulting equations are shown to have the same structure but a generalized physical meaning as those derived by GROB on other way.     

Intertwining operator realization of non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov–Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.     

Algebras of creation and destruction operators

A general analysis of bilinear algebras of creation and destruction operators is performed. Generalizing the earlier work on the single-parameterq-deformation of the Heisenberg algebra, we study two-parameter and four-parameter algebras. Two new forms of quantum statistics called orthofermi and orthobose statistics and aq-deformation interpolating between them have been found. In the Fock representation, quadratic relations among destruction operators, wherever they are allowed, are shown to follow from the bilinear algebra of creation and destruction operators. Postitivity of the Hilbert space for the four-parameter algebra has been studied in the two-particle sector, but for the two-parameter algebra, results are presented up to the four-particle sector.     

A remark on Schrödinger operators on aperiodic tilings

We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl ²( ^d ) to operators on thel ²-spaces of sets of vertices of strictly ergodic tilings.     

Construction of field algebras with quantum symmetry from local observables

It has been discussed earlier that (weak quasi-) quantum groups allow for a conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses the reconstruction of quantum symmetries and algebras of field operators. For every algebraA of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey a local braid relation.     

The bases of effective field theories

With reference to the equivalence theorem, we discuss the selection of basis operators for effective field theories in general. The equivalence relation can be used to partition operators into equivalence classes, from which inequivalent basis operators are selected. These classes can also be identified as containing Potential-Tree-Generated (PTG) operators, Loop-Generated (LG) operators, or both, independently of the specific dynamics of the underlying extended models, so long as it is perturbatively decoupling. For an equivalence class containing both, we argue that the basis operator should be chosen from among the PTG operators, because they may have the largest coefficients. We apply this classification scheme to dimension-six operators in an illustrative Yukawa model as well in the Standard Model (SM). We show that the basis chosen by Grzadkowski et al. [5] for the SM satisfies this criterion. In this light, we also revisit and verify our earlier result [6] that the dimension-six corrections to the triple-gauge-boson couplings only arise from LG operators, so the magnitude of the coefficients should only be a few parts per thousand of the SM gauge coupling if BSM dynamics respects decoupling. The same is true of the quartic-gauge-boson couplings.     

A folded-diagram perturbation expansion for effective operators

Linked-diagram expansions for the model-space effective operators are derived for both open-shell nuclei and closed-shell nuclei. These expansions are derived using time-dependent perturbation theory and are given as a diagrammatic series which contains folded diagrams. Our results are in general agreement with the results obtained earlier by Brandow, but the general structure of the present expansions is perhaps simpler and more convenient for calculations. The projections of the true nuclear eigenfunctions onto the model space are required for the derivation of effective operators. These projections are obtained from the solution of the model-space eigenvalue problem defined by the energy-independent effective Hamiltonian. Methods for calculating the above diagrammatic series for effective operators are discussed.     

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Received: 4 November 1999 / Accepted: 22 September 2000     

The Phase of the Scattering Operator from the Geometry of Certain Infinite-Dimensional Groups

We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson (J Math Phys 37(8):3933–3953, 1996) is that we can avoid the somewhat arbitrary choice in the regularization of the time evolution for intermediate times using a natural choice of the connection form on the space of appropriate unitary operators.     

Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

We give a Cwikel–Lieb–Rozenblum type bound on the number of bound states of Schrödinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.     

Limiting Absorption Principle for Some Long Range Perturbations of Dirac Systems at Threshold Energies

We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to studying a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non-self-adjoint operators, which we develop in the Appendix. We also discuss some applications to the dispersive Helmholtz model in the quantum regime.     

Baxter operators with deformed symmetry

Baxter operators are constructed for quantum spin chains with deformed _s?2${\mathrm{s?}}_2$ symmetry. The parallel treatment of Yang–Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier and relying on the factorization regarding parameter permutations is extended to the global chain operators following the scheme worked out recently in the undeformed case.