Ändra sökning
Avgränsa sökresultatet
12 1 - 50 av 55
RefereraExporteraLänk till träfflistan
Permanent länk
Referera
Referensformat
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf
Träffar per sida
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sortering
  • Standard (Relevans)
  • Författare A-Ö
  • Författare Ö-A
  • Titel A-Ö
  • Titel Ö-A
  • Publikationstyp A-Ö
  • Publikationstyp Ö-A
  • Äldst först
  • Nyast först
  • Skapad (Äldst först)
  • Skapad (Nyast först)
  • Senast uppdaterad (Äldst först)
  • Senast uppdaterad (Nyast först)
  • Disputationsdatum (tidigaste först)
  • Disputationsdatum (senaste först)
  • Standard (Relevans)
  • Författare A-Ö
  • Författare Ö-A
  • Titel A-Ö
  • Titel Ö-A
  • Publikationstyp A-Ö
  • Publikationstyp Ö-A
  • Äldst först
  • Nyast först
  • Skapad (Äldst först)
  • Skapad (Nyast först)
  • Senast uppdaterad (Äldst först)
  • Senast uppdaterad (Nyast först)
  • Disputationsdatum (tidigaste först)
  • Disputationsdatum (senaste först)
Markera
Maxantalet träffar du kan exportera från sökgränssnittet är 250. Vid större uttag använd dig av utsökningar.
  • 1.
    Atai, Farrokh
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Integral representation of solution to the non-stationary Lamé equationManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We consider methods for constructing explicit solutions of the non-stationary Lame equation,which is a generalization of the classical Lame equation, that has appeared in works on integrablemodels, conformal eld theory, high energy physics and representation theory. We also present ageneral method for constructing integral representations of solutions to the non-stationary Lameequation by a recursive scheme. Explicit integral representations, for special values of the modelparameters, are also presented. Our approach is based on kernel function methods which can benaturally generalized to the non-stationary Heun equation.

  • 2.
    Atai, Farrokh
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Hallnäs, Martin
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models2014Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 104, nr 7, s. 811-835Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the relativistic generalization of the quantum A (N-1) Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh-Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.

  • 3.
    Atai, Farrokh
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Deformed Calogero-Sutherland model and fractional Quantum Hall effectManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The deformed Calogero-Sutherland (CS) model is a quantum integrable systemwith arbitrary numbers of two types of particles and reducing to the standard CSmodel in special cases. We show that a known collective field description of theCS model, which is based on conformal field theory (CFT), is actually a collectivefield description of the deformed CS model. This provides a natural application ofthe deformed CS model in Wen’s effective field theory of the fractional quantumHall effect (FQHE), with the two kinds of particles corresponding to electrons andquasi-hole excitations. In particular, we use known mathematical results aboutsuper Jack polynomials to obtain simple explicit formulas for the orthonormal CFTbasis proposed by van Elburg and Schoutens in the context of the FQHE.

  • 4.
    Atai, Farrokh
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Series solutions of the non-stationary Heun equationManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We consider the non-stationary Heun equation, also known as quantum PainlevéVI, which has appeared in dierent works on quantum integrable models and conformaleld theory. We use a generalized kernel function identity to transform the problemto solve this equation into a dierential-dierence equation which, as we show, canbe solved by ecient recursive algorithms. We thus obtain series representations ofsolutions which provide elliptic generalizations of the Jacobi polynomials. These seriesreproduces, in a limiting case, a perturbative solution of the Heun equation due toTakemura, but our method is dierent in that we expand in non-conventional basisfunctions that allow us to obtain explicit formulas to all orders;

  • 5.
    Blennow, Mattias
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Damping signatures in future neutrino oscillation experiments2006Ingår i: Nuclear physics B, Proceedings supplements, ISSN 0920-5632, E-ISSN 1873-3832, Vol. 155, s. 195-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In precision measurements of neutrino oscillation parameters and in the search for theta(13), it becomes important to study small effects which may alter the precision measurements or mimic a theta(13) signal. Here, I give an introduction to a common framework in which many of the possible small effects - such as neutrino wave packet decoherence, neutrino quantum decoherence, and neutrino decay - can be studied. This framework is based on the introduction of damping factors on the probability level of the neutrino oscillation formulas.

  • 6. Calogero, F.
    et al.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Goldfishing by gauge theory2006Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, nr 8Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A new solvable many-body problem of goldfish type is identified and used to revisit the connection between two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.

  • 7. Carey, Alan L.
    et al.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Wang, Bai-Ling
    Differential twisted K-theory and applications2009Ingår i: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 59, nr 5, s. 632-653Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

  • 8.
    de Woul, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Fermions in two dimensions and exactly solvable models2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena.

    Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors.

    A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied.

  • 9.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Hoppe, Jens
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lundholm, Douglas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Partial Hamiltonian reduction of relativistic extended objects in light-cone gauge2011Ingår i: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, nr 1, s. 031-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The elimination of the non-transversal field in the standard light-cone formulation of higher-dimensional extended objects is formulated as a Hamiltonian reduction.

  • 10.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Hoppe, Jens
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lundholm, Douglas
    Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark.
    Sundin, Martin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    A dynamical symmetry for supermembranes2011Ingår i: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 3, nr 134Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A dynamical symmetry for supersymmetric extended objects is given.

  • 11.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Exact Solution of a 2D Interacting Fermion Model2012Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 314, nr 1, s. 1-56Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study an exactly solvable quantum field theory (QFT) model describing interacting fermions in 2+1 dimensions. This model is motivated by physical arguments suggesting that it provides an effective description of spinless fermions on a square lattice with local hopping and density-density interactions if, close to half filling, the system develops a partial energy gap. The necessary regularization of the QFT model is based on this proposed relation to lattice fermions. We use bosonization methods to diagonalize the Hamiltonian and to compute all correlation functions. We also discuss how, after appropriate multiplicative renormalizations, all short- and long distance cutoffs can be removed. In particular, we prove that the renormalized two-point functions have algebraic decay with non-trivial exponents depending on the interaction strengths, which is a hallmark of Luttinger-liquid behavior.

  • 12.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Gauge invariance, correlated fermions, and Meissner effect in 2+1 dimensionsArtikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    We present a 2+1 dimensional quantum gauge theory model with correlated fermions that is exactly solvable by bosonization. This model gives an effective description of partially gapped fermions on a square lattice that have density-density interactions and are coupled to photons. We show that the photons in this model are massive due to gauge-invariant normal-ordering, similarly as in the Schwinger model. Moreover, the exact excitation spectrum of the model has two gapped and one gapless mode. We also compute the magnetic field induced by an external current and show that there is a Meissner effect. We find that the transverse photons have significant effects on the low-energy properties of the model even if the fermion-photon coupling is small.

  • 13.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Gauge Invariance, Correlated Fermions, and Photon Mass in 2+1 Dimensions2014Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 154, nr 3, s. 877-894Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a 2+1 dimensional quantum gauge theory with correlated fermions that is exactly solvable by bosonization. This model describes a system of Luttinger liquids propagating on two sets of equidistant lines forming a grid embedded in two dimensional continuum space; this system has two dimensional character due to density-density interactions and due to a coupling to dynamical photons propagating in the continuous embedding space. We argue that this model gives an effective description of partially gapped fermions on a square lattice that have density-density interactions and are coupled to photons. Our results include the following: after non-trivial renormalizations of the coupling parameters, the model remains well-defined in the quantum field theory limit as the grid of lines becomes a continuum; the photons in this model are massive due to gauge-invariant normal-ordering, similarly as in the Schwinger model; the exact excitation spectrum of the model has two gapped and one gapless mode.

  • 14.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Partial continuum limit of the 2D Hubbard modelArtikel i tidskrift (Övrigt vetenskapligt)
  • 15.
    de Woul, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Partially Gapped Fermions in 2D2010Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 139, nr 6, s. 1033-1065Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We compute mean field phase diagrams of two closely related interacting fermion models in two spatial dimensions (2D). The first is the so-called 2D t-t'-V model describing spinless fermions on a square lattice with local hopping and density-density interactions. The second is the so-called 2D Luttinger model that provides an effective description of the 2D t-t'-V model and in which parts of the fermion degrees of freedom are treated exactly by bosonization. In mean field theory, both models have a charge-density-wave (CDW) instability making them gapped at half-filling. The 2D t-t'-V model has a significant parameter regime away from half-filling where neither the CDW nor the normal state are thermodynamically stable. We show that the 2D Luttinger model allows to obtain more detailed information about this mixed region. In particular, we find in the 2D Luttinger model a partially gapped phase that, as we argue, can be described by an exactly solvable model.

  • 16. Forger, M
    et al.
    Paufler, Cornelius
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Romer, H
    Hamiltonian multivector fields and Poisson forms in multisymplectic field theory2005Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, nr 11, s. 112903-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing explicit expressions for the Poisson bracket between two Poisson forms.

  • 17. Hallnaes, Martin
    et al.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    A product formula for the eigenfunctions of a quartic oscillator2015Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 426, nr 2, s. 1012-1025Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the Schrodinger operator on the real line with an even quartic potential. Our main result is a product formula of the type psi(k)(x)psi(k)(y) = integral(R) psi(k)(z)K(x,y, z)dz for its eigenfunctions psi(k). The kernel function K is given explicitly in terms of the Airy function Ai(x), and it is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions psi(k).

  • 18.
    Hallnäs, Martin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Exactly solved quantum many-body systems in one dimension2005Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.

  • 19. Hallnäs, Martin
    et al.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero-Sutherland Type2010Ingår i: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 31, nr 3, s. 309-342Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.

  • 20.
    Hallnäs, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Exact solutions of two complementary one-dimensional quantum many-body systems on the half-line2005Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, nr 5Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider two particular one-dimensional quantum many-body systems with local interactions related to the root system C-N. Both models describe identical particles moving on the half-line with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as delta-prime interaction. We show that the Bethe ansatz solution of the delta-interaction model is consistent even for the general case where the particles are distinguishable, whereas for the delta-prime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta- and the fermionic delta-prime model, and we elaborate on the physical interpretations of these models.

  • 21.
    Hallnäs, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Explicit formulae for the eigenfunctions of the N-body Calogero model2006Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 39, nr 14, s. 3511-3533Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the quantum Calogero model, which describes N non-distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two-body interactions proportional to the inverse square of the inter-particle distance. We elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions, arbitrary coupling parameter and particle number. We also show that our method applies, with minor changes, to all Calogero models associated with classical root systems.

  • 22.
    Hallnäs, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Quantum Calogero-Sutherland type models and generalised classical polynomials2007Manuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In this paper we consider a large class of many-variable polynomials which contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero-Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.

  • 23.
    Hallnäs, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Paufler, Cornelius
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles2005Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 38, nr 22, s. 4957-4974Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.

  • 24.
    Hekmati, Pedram
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Abelian Extensions, Fractional Loop Group and Quantum Fields2010Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis deals with the theory of Lie group extensions, Lie conformal algebras and twisted K-theory, in the context of quantum physics. These structures allow for a mathematically precise description of certain aspects of interacting quantum field theories. We review three concrete examples, namely symmetry breaking (or anomalies) in gauge theory, classification of D-brane charges in string theory and the formulation of integrable hierarchies in the language of Poisson vertex algebras. The main results are presented in three appended scientific papers.

    In the first paper we establish, by construction, a criterion for when an infinite dimensional abelian Lie algebra extension corresponds to a Lie group extension.

    In the second paper we introduce the fractional loop group LqG, that is the group of maps from a circle to a compact Lie group G, with only a small degree of differentiability q ε R+ in the Sobolev sense. We construct abelian extensions and highest weight modules for the Lie algebra Lqg, and discuss an application to equivariant twisted K-theory on G.

    In the third paper, we construct a structure of calculus algebra on the Lie conformal algebra complex and provide a more detailed description in the special case of the complex of variational calculus.

  • 25.
    Hekmati, Pedram
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Group Extensions, Gerbes and Twisted K-theory2008Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis reviews the theory of group extensions, gerbes and twisted K-theory. Application to anomalies in gauge theory is briefly discussed. The main results are presented in two appended scientific papers. In the first paper we establish, by construction, a criterion for when an infinite dimensional abelian Lie algebra extension corresponds to a Lie group extension. In the second paper we introduce the fractional loop group L_qG, construct highest weight modules for the Lie algebra and discuss an application to twisted K-theory on G.

  • 26.
    Hekmati, Pedram
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Integrability Criterion for Abelian Extensions of Lie Groups2010Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 138, nr 11, s. 4137-4148Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras (g) over cap = g circle plus(omega) a integrates to a corresponding Lie group extension A (sic) (G) over cap (sic) G, where G is connected, simply connected and A congruent to a/Gamma for some discrete subgroup Gamma subset of a. When pi(1) (G) not equal 0, the kernel A is replaced by a central extension (A) over cap of pi(1) (G) by A.

  • 27.
    Hekmati, Pedram
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Fractional Loop Group and Twisted K-theory2010Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 299, nr 3, s. 741-763Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the structure of abelian extensions of the group L (q) G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed.

  • 28.
    Hällgren, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Ohlsson, Tommy
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Teoretisk partikelfysik.
    Seidl, Gerhart
    Department of Physics, Oklahoma State University.
    Neutrino oscillations in deconstructed dimensions2005Ingår i: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 2005, nr 02, s. 049-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a model for neutrino oscillations in the presence of a deconstructed non-gravitational large extra dimension compactified on the boundary of a two-dimensional disk. In the deconstructed phase, sub-mm lattice spacings are generated from the hierarchy of energy scales between similar to 1 TeV and the usual B - L breaking scale similar to 10(15) GeV. Here, short-distance cutoffs down to similar to 1eV are motivated by the strong coupling behavior of gravity in local discrete extra dimensions. This could make it possible to probe the discretization of extra dimensions and non-trivial field configurations in theory spaces which have only a few sites, i.e., for coarse latticizations. Thus, the model has relevance to present and future precision neutrino oscillation experiments.

  • 29. Lahtinen, V.
    et al.
    Månsson, Teresia
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Ardonne, E.
    Hierarchy of exactly solvable spin- 1 2 chains with s o (N) 1 critical points2014Ingår i: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 89, nr 1, s. 014409-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)1 critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples N transverse field Ising chains such that the resulting theory is critical and described by the so(N)1 conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary N into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd N our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even N the models can be viewed as longer ranger generalizations of the XY chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted conformal field theory (CFT) describing the criticality.

  • 30.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    A 2D Luttinger Model2010Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 141, nr 1, s. 17-52Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A detailed derivation of a two dimensional (2D) low energy effective model for spinless fermions on a square lattice with local interactions is given. This derivation utilizes a particular continuum limit that is justified by physical arguments. It is shown that the effective model thus obtained can be treated by exact bosonization methods. It is also discussed how this effective model can be used to obtain physical information about the corresponding lattice fermion system.

  • 31.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    A Two-Dimensional Analogue of the Luttinger Model2010Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 92, nr 2, s. 109-124Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a fermion model that is, as we suggest, a natural 2D analogue of the Luttinger model. We derive this model as a partial continuum limit of a 2D spinless lattice fermion system with local interactions and away from half filling. In this derivation, we use certain approximations that we motivate by physical arguments. We also present mathematical results that allow an exact treatment of parts of the degrees of freedom of this model by bosonization, and we propose to treat the remaining degrees of freedom by mean field theory.

  • 32.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    An explicit solution of the (quantum) elliptic Calogero-Sutherland model2005Ingår i: SPT 2004: SYMMETRY AND PERTURBATION THEORY / [ed] Gaera, G; Prinari, B; RauchWojciechowshi, S, 2005, s. 159-174Konferensbidrag (Refereegranskat)
    Abstract [en]

    We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials.

  • 33.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Explicit Solution of the (Quantum) Elliptic Calogero-Sutherland Model2014Ingår i: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 15, nr 4, s. 755-791Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The elliptic Calogero-Sutherland model is a quantum many body system of identical particles moving on a circle and interacting via two body potentials proportional to the Weierstrass -function. It also provides a natural many-variable generalization of the Lam, equation. Explicit formulas for the eigenfunctions and eigenvalues of this model as infinite series are obtained, to all orders and for arbitrary particle numbers and coupling parameters. These eigenfunctions are an elliptic deformation of the Jack polynomials. The absolute convergence of these series is proved in special cases, including the two-particle (=Lam,) case for non-integer coupling parameters and sufficiently small elliptic deformation.

  • 34.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Gauge theory approach towards an explicit solution of the (classical) elliptic Calogero-Moser system2005Ingår i: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 12, s. 423-439Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We discuss the relation of the trigonometric Calogero-Moser (CM) system to Yang-Mills gauge theories and its generalization to the elliptic case. This yields a linearization of the time evolution of the elliptic CM system and suggests two promising strategies for finding a fully explicit solution of this model. We also present a large class of integrable spin-particle systems generalizing the elliptic CM system.

  • 35.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model2006Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, nr 2Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter lambda, we also obtain dual relations involving mixed correlation functions of anyons with two different statistics parameters lambda and 1/lambda. We also give alternative, elementary proofs of these identities by direct computations.

  • 36.
    Langmann, Edwin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Source Identity and Kernel Functions for Elliptic Calogero-Sutherland Type Systems2010Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 94, nr 1, s. 63-75Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Kernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh-Feigin-Veselov-Sergeev-type deformations of the elliptic Calogero-Sutherland model for special parameter values.

  • 37.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Laptev, Ari
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Paufler, Cornelius
    Singular factorizations, self-adjoint extensions and applications to quantum many-body physics2006Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 39, nr 5, s. 1057-1071Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.

  • 38.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Lebowitz, Joel L.
    Departments of Mathematics and Physics, Rutgers University.
    Mastropietro, Vieri
    Dipartimento di Matematica, Università degli Studi di Milano.
    Moosavi, Per
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik.
    Steady states and universal conductance in a quenched Luttinger model2016Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, s. 1-32Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian (Formula presented.) with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian (Formula presented.) which differs from (Formula presented.) by the strength of the interaction. Asymptotically in time, as (Formula presented.), after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference (Formula presented.) between right- (+) and left- (−) moving fermions obtained from the two-point correlation function. Both I and (Formula presented.) depend on (Formula presented.) and (Formula presented.). Only for the case (Formula presented.) does (Formula presented.) equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, (Formula presented.), has a universal value equal to the conductance quantum (Formula presented.) for the spinless case.

  • 39.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Moosavi, Per
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik.
    Construction by bosonization of a fermion-phonon model2015Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 56, nr 9, artikel-id 091902Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We discuss an extension of the (massless) Thirring model describing interacting fermions in one dimension which are coupled to phonons and where all interactions are local. This fermion-phonon model can be solved exactly by bosonization.We present a construction and solution of this model which is mathematically rigorous by treating it as a continuum limit of a Luttinger-phonon model. A self-contained account of the mathematical results underlying bosonization is included, together with complete proofs.

  • 40.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Sundin, Martin
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Extrinsic curvature effects in brane-world scenariosArtikel i tidskrift (Övrigt vetenskapligt)
  • 41.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Takemura, Kouichi
    Source identity and kernel functions for Inozemtsev-type systems2012Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, nr 8, s. 082105-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Inozemtsev Hamiltonian is an elliptic generalization of the differential operator defining the BCN trigonometric quantum Calogero-Sutherland model, and its eigenvalue equation is a natural many-variable generalization of the Heun differential equation. We present kernel functions for Inozemtsev Hamiltonians and Chalykh-Feigin-Veselov-Sergeev-type deformations thereof. Our main result is a solution of a heat-type equation for a generalized Inozemtsev Hamiltonian which is the source of all these kernel functions. Applications are given, including a derivation of simple exact eigenfunctions and eigenvalues of the Inozemtsev Hamiltonian.

  • 42.
    Langmann, Edwin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Wallin, Mats
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Statistisk fysik.
    Mean field magnetic phase diagrams for the two dimensional t-t '-U Hubbard model2007Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 127, nr 4, s. 825-840Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the ground state phase diagram of the two dimensional t - t' - U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted Hartree-Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter space. Standard mean field theory, i.e., Hartree-Fock theory restricted to homogeneous states, fails to produce such inhomogeneous phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results obtained with more sophisticated methods.

  • 43.
    Loikkanen, Juha
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Paufler, Cornelius
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Yang-Mills action from minimally coupled bosons on R-4 and on the four-dimensional Moyal plane2005Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, nr 3, s. 032301-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider bosons on (Euclidean) R-4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cutoff regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane R-theta(4) We also include a detailed comparison of our cutoff regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R-4 still holds if one replaces all pointwise products by the noncommutative Moyal product.

  • 44.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    FAMILIES OF DIRAC OPERATORS AND QUANTUM AFFINE GROUPS2011Ingår i: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 90, nr 2, s. 213-220Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this paper we show how to deform the Fredholm family in the sense of quantum groups. The family of Dirac-type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transforms covariantly under a central extension of the algebra.

  • 45.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    From gauge anomalies to gerbes and gerbal actions2010Ingår i: Motives, quantum field theory, and pseudodifferential operators, Boston: Clay mathematical institute , 2010, s. 211-220Konferensbidrag (Refereegranskat)
  • 46.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    From Gauge Anomalies to Gerbes and Gerbal Representations: Group Cocycles in Quantum Theory2010Ingår i: Acta Polytechnica: journal of advanced engineering, ISSN 1210-2709, Vol. 50, nr 3, s. 42-47Artikel i tidskrift (Refereegranskat)
  • 47.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Star products and central extensions2006Ingår i: Analysis, geometry and topology of elliptic operators / [ed] Bernhelm Booss-Bavnbek, Hackensack, NJ: World Scientific Publ. , 2006, s. 401-410Konferensbidrag (Refereegranskat)
  • 48.
    Mickelsson, Jouko
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Twisted K theory invariants2005Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 71, nr 2, s. 109-121Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    An invariant for twisted K theory classes on a 3-manifold is introduced. The invariant is then applied to the twisted equivariant classes arising from the supersymmetric Wess-Zumino-Witten model based on the group SU(2). It is shown that the classes defined by different highest weight representations of the loop group LSU(2) are inequivalent. The results are compatible with Freed-Hopkins-Teleman identification of twisted equivariant K theory as the Verlinde algebra.

  • 49.
    Mickelsson, Jouko
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Paycha, Sylvie
    THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN2011Ingår i: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 90, nr 1, s. 53-80Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

  • 50.
    Mickelsson, Jouko
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Teoretisk fysik, Matematisk fysik.
    Pellonpää, Juha-Pekka
    Families index theorem in supersymmetric WZW model and twisted K-theory: The SU(2) case2007Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 271, nr 3, s. 775-789Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The construction of twisted K-theory classes on a compact Lie group is reviewed using the supersymmetric Wess-Zumino-Wittenmodel on a cylinder. The Quillen superconnection is introduced for a family of supercharges parametrized by a compact Lie group and the Chern character is explicitly computed in the case of SU( 2). For large euclidean time, the character form is localized on a D-brane.

12 1 - 50 av 55
RefereraExporteraLänk till träfflistan
Permanent länk
Referera
Referensformat
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf