DOCSLIB.ORG

The Topological G/G WZW Model in the Generalized Momentum

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Topological G/G WZW Model in the Generalized Momentum The Erwin Schrodinger International Pasteurgasse ESI Institute for Mathematical Physics A Wien Austria The Top ological GG WZW Mo del in the Generalized Momentum Representation Anton Yu Alekseev Peter Schaller Thomas Strobl Vienna Preprint ESI May Supp orted by Federal Ministry of Science and Research Austria Available via httpwwwesiacat ETHTH TUW PITHA ESI hepth The Top ologi cal GG WZW Mo del in the Generalized Momentum Representation Anton Yu Alekseev Institut fur Theoretische Physik ETHHonggerb erg CH Zuric h Switzerland Peter Schaller Institut fur Theoretische Physik TU Wien Wiedner Hauptstr A Vienna Austria y Thomas Strobl Institut fur Theoretische Physik RWTH Aachen Sommerfeldstr D Aachen Germany May Abstract We consider the top ological gauged WZW mo del in the general ized momentum representation The chiral eld g is interpreted as a counterpart of the electric eld E of conventional gauge theories The gauge dep endence of wave functionals g is governed by a new gauge co cycle We evaluate this co cycle explicitly using the GW Z W machinery of Poisson mo dels In this approach the GWZW mo del is reformulated as a Schwarz typ e top ological theory so that the ac tion do es not dep end on the worldsheet metric The equivalence of this new formulation to the original one is proved for genus one and conjectured for an arbitrary genus Riemann surface As a bypro duct we discover a new way to explain the app earence of Quantum Groups in the WZW mo del On leave of absence from Steklov Mathematical Institute Fontanka StPetersburg Russia email alekseevitpphysethzch y email tstroblplutophysikrwthaachende Intro duction In this pap er we investigate the quantization of the gauged GG WZW mo del in the generalized momentum representation The consideration is inspired by the study of twodimensional YangMills and BFtheories in the momentum representation The problem of quantization of gauge theories in the momentum repre 1 sentation has b een attracting attention for a long time While in the connection representation the idea of gauge invariance may b e implemented in a simple way g A A we get a nontrivial b ehaviour of the quantum wave functions under gauge transformations in the momentum representation Indeed one can apply the following simple argument The wave functional in the momentum represen tation may b e thought of as a functional Fourier transformation of the wave functional in the connection representation Z Z A E D A exp i tr E A i i Taking into account the b ehaviour of A and E under gauge transformations g 1 1 g A g g g A i i i g 1 E g E g i i we derive R 1 g i tr E g g i i E E e We conclude that the wave functional in the momentum representation is not invariant with resp ect to gauge transformations Instead it gains a simple phase factor E g which is of the form Z 1 E g tr E g g i i The innitesimal version of the same phase factor Z E tr E i i 1 The authors are grateful to Prof RJackiw for drawing their attention to this pap er and for making them know ab out the scientic content of his letter to Prof D Amati corresp onds to the action of the gauge algebra It is easy to verify that satises the following equation g E g h E g E h This prop erty assures that the comp osition of two gauge transformations with gauge parameters g and h is the same as a gauge transformation with a parameter g h Equation is usually referred to as a co cycle condition It establishes the fact that is a oneco cycle of the innite dimensional gauge group A oneco cycle is said to b e trivial if g E g E E for some In this case the gauge invariance of the wave function may b e restored by the redenition ~ i(E ) E e E An innitesimal co cycle E is trivial if it can b e represented as a function of the commutator E E E It is easy to see that the co cycle is nontrivial Indeed let us cho ose b oth a a E and having only one nonzero comp onent E and in the Lie algebra Then the commutator in is always equal to zero whereas the expression is still nontrivial As a consequence also the gauge group co cycle is nontrivial On the other hand on some restricted space of values of the eld E the co cycle may b ecome trivial generically if we admit nonlo cal expressions for This is imp ortant to mention as one may rewrite the integrated Gauss law as a triviality condition on the co cycle Let us parametrize as E exp iE which is p ossible whenever and insert this ex R 1 2 pression into The result is precisely with tr E g g In fact i i eg in two dimensions the wave functions of the momentum representation 2 More accurately one obtains only mo d But anyway this mo dication of is quite natural in view of the origin of the co cycle within Alternatively one might regard also a multiplicative co cycle exp i right from the outset cf App endix A 1 are supp orted on some conjugacy classes E x g x E g x with sp ecic 0 values of E But away from these sp ecic conjugacy classes and in partic 0 ular in the original unrestricted space of values for E the general argument of the co cycle b eing nontrivial applies More details on this issue may b e found in App endix A It is worth mentioning that in the ChernSimons theory a co cycle app ears in the connection representation as well g i(Ag ) A e A The co cycle A g is ususally called WessZumino action It is intimately related to the theory of anomalies Recently a co cycle of typ e has b een observed in twodimensional BF and YMtheories In this pap er we consider the somewhat more complicated example of the gauged WZW GWZW mo del for a semisimple Lie group Like the BF theory it is a twodimensional top ological eld theory for a detailed account see It has a connection oneform gauge eld as one of its dynamical variables and p ossesses the usual gauge symmetry However there is a complication which makes the analysis dierent from the pattern In the GWZW mo del the variable which is conjugate to the gauge eld and which shall b e denoted by g in the following takes values in a Lie group G instead of a linear space So we get a sort of curved momentum space We calculate the co cyle which governs the gauge dep endence of wave GW Z W functions in a g representation and nd that it diers from the standard expression We argue that while the co cycle corresp onds to a Lie group G our co cycle is related to its quantum deformation G In the course q of the analysis we nd that the GWZW mo del b elongs to the class of Poisson mo dels recently discovered in This theory provides a technical to ol for the evaluation of the co cycle GW Z W Let us briey characterize the content of each section In Section we develop the Hamiltonian formulation of the GWZW mo del nd canonically conjugate variables and write down the gauge invariance equation for the wave functional in the g representation Section is devoted to the description of Poisson mo dels A two dimensional top ological mo del of this class is dened by xing a Poisson bracket on the target space Using the Hamiltonian formulation the top ology of the spacetime b eing a torus or cylinder we prove that the GWZW mo del is equivalent to a certain Poisson mo del coupled to a top ological term S that has supp ort of measure zero on the target space of the eld theory The target space of the coupled Poisson mo del is the Lie group G We start from the GWZW action evaluate the Poisson structure on G and discover its relation to the theory of Quantum Groups The origin of the term S is considered in details in App endix B In Section we solve the gauge invariance equation and nd the gauge dep endence of the GWZW wave functional in the g representation This provides a new co cycle Calculations are p erformed for the Poisson GW Z W mo del without the singular term In App endix C we reconsider the problem in the presence of the top ological term It is shown that in the case of G SU at most one quantum state is aected We compare the results with other approaches In some nal remarks we conjecture that the Poisson mo del coupled to S gives an alternative formulation of the GWZW mo del valid for a Riemann surface of arbitrary genus We comment on the new relation b etween the WZW mo del and Quantum Groups which emerges as a bypro duct of our consideration Hamiltonian Formulation of GWZW Mo del The WZW theory is dened by the action Z Z k k 1 1 2 1 1 3 tr g g g g d x tr d dg g WZW g where the elds g take values in some semisimple Lie group G and indices are raised with the standard Minkowski metric The case of a Euclidean metric may b e treated in the same fashion Some remarks concerning the second term in may b e found in App enidx B The simplest way to gauge the global symmetry transformations g 1 l g l is to intro duce a gauge eld h taking its values in the gauge group the action 1 GW Z W h g WZW hg h 1 1 is then invariant under the lo cal transformations g l g l h hl With 1 the celebrated PolyakovWigmann formula and a h h where GWZW can b e brought into the standard form o 1 GW Z W g a a WZW g + R k 1 1 1 2 tr a g g a g g a g a g a a d x + + + + 4 + In the course of our construction of GWZW a a dx a dx has b een + 2 sub ject to the zero curvature condition da a This condition results also from the equations of motion arising from So further on we treat a as unconstrained elds taking their values in the Lie algebra of the chosen gauge group In order to nd a Hamiltonian formulation of the GWZW
Load more

Recommended publications
  • Quantum Groups and Algebraic Geometry in Conformal Field Theory
    QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY DlU'KKERU EI.INKWIJK BV - UTRECHT QUANTUM GROUPS AND ALGEBRAIC GEOMETRY IN CONFORMAL FIELD THEORY QUANTUMGROEPEN EN ALGEBRAISCHE MEETKUNDE IN CONFORME VELDENTHEORIE (mrt em samcnrattint] in hit Stdirlands) PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE UTRECHT. OP GEZAG VAN DE RECTOR MAGNIFICUS. TROF. DR. J.A. VAN GINKEI., INGEVOLGE HET BESLUIT VAN HET COLLEGE VAN DE- CANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 19 SEPTEMBER 1989 DES NAMIDDAGS TE 2.30 UUR DOOR Theodericus Johannes Henrichs Smit GEBOREN OP 8 APRIL 1962 TE DEN HAAG PROMOTORES: PROF. DR. B. DE WIT PROF. DR. M. HAZEWINKEL "-*1 Dit proefschrift kwam tot stand met "•••; financiele hulp van de stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) Aan mijn ouders Aan Saskia Contents Introduction and summary 3 1.1 Conformal invariance and the conformal bootstrap 11 1.1.1 Conformal symmetry and correlation functions 11 1.1.2 The conformal bootstrap program 23 1.2 Axiomatic conformal field theory 31 1.3 The emergence of a Hopf algebra 4G The modular geometry of string theory 56 2.1 The partition function on moduli space 06 2.2 Determinant line bundles 63 2.2.1 Complex line bundles and divisors on a Riemann surface . (i3 2.2.2 Cauchy-Riemann operators (iT 2.2.3 Metrical properties of determinants of Cauchy-Ricmann oper- ators 6!) 2.3 The Mumford form on moduli space 77 2.3.1 The Quillen metric on determinant line bundles 77 2.3.2 The Grothendieck-Riemann-Roch theorem and the Mumford
    [Show full text]
  • Quantum Group and Quantum Symmetry
    QUANTUM GROUP AND QUANTUM SYMMETRY Zhe Chang International Centre for Theoretical Physics, Trieste, Italy INFN, Sezione di Trieste, Trieste, Italy and Institute of High Energy Physics, Academia Sinica, Beijing, China Contents 1 Introduction 3 2 Yang-Baxter equation 5 2.1 Integrable quantum field theory ...................... 5 2.2 Lattice statistical physics .......................... 8 2.3 Yang-Baxter equation ............................ 11 2.4 Classical Yang-Baxter equation ...................... 13 3 Quantum group theory 16 3.1 Hopf algebra ................................. 16 3.2 Quantization of Lie bi-algebra ....................... 19 3.3 Quantum double .............................. 22 3.3.1 SUq(2) as the quantum double ................... 23 3.3.2 Uq (g) as the quantum double ................... 27 3.3.3 Universal R-matrix ......................... 30 4Representationtheory 33 4.1 Representations for generic q........................ 34 4.1.1 Representations of SUq(2) ..................... 34 4.1.2 Representations of Uq(g), the general case ............ 41 4.2 Representations for q being a root of unity ................ 43 4.2.1 Representations of SUq(2) ..................... 44 4.2.2 Representations of Uq(g), the general case ............ 55 1 5 Hamiltonian system 58 5.1 Classical symmetric top system ...................... 60 5.2 Quantum symmetric top system ...................... 66 6 Integrable lattice model 70 6.1 Vertex model ................................ 71 6.2 S.O.S model ................................. 79 6.3 Configuration
    [Show full text]
  • Non-Commutative Phase and the Unitarization of GL {P, Q}(2)
    Non–commutative phase and the unitarization of GLp,q (2) M. Arık and B.T. Kaynak Department of Physics, Bo˘gazi¸ci University, 80815 Bebek, Istanbul, Turkey Abstract In this paper, imposing hermitian conjugate relations on the two– parameter deformed quantum group GLp,q (2) is studied. This results in a non-commutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group Up,q (2) with pq real via a non–commutative phase, the representation of the algebra is built by means of the action of the operators constituting the Up,q (2) matrix on states. 1 Introduction The mathematical construction of a quantum group Gq pertaining to a given Lie group G is simply a deformation of a commutative Poisson-Hopf algebra defined over G. The structure of a deformation is not only a Hopf algebra arXiv:hep-th/0208089v2 18 Oct 2002 but characteristically a non-commutative algebra as well. The notion of quantum groups in physics is widely known to be the generalization of the symmetry properties of both classical Lie groups and Lie algebras, where two different mathematical blocks, namely deformation and co–multiplication, are simultaneously imposed either on the related Lie group or on the related Lie algebra. A quantum group is defined algebraically as a quasi–triangular Hopf alge- bra. It can be either non-commutative or commutative. It is fundamentally a bi–algebra with an antipode so as to consist of either the q–deformed uni- versal enveloping algebra of the classical Lie algebra or its dual, called the matrix quantum group, which can be understood as the q–analog of a classi- cal matrix group [1].
    [Show full text]
  • Non-Commutative Dual Representation for Quantum Systems on Lie Groups
    Loops 11: Non-Perturbative / Background Independent Quantum Gravity IOP Publishing Journal of Physics: Conference Series 360 (2012) 012052 doi:10.1088/1742-6596/360/1/012052 Non-commutative dual representation for quantum systems on Lie groups Matti Raasakka Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam, Germany, EU E-mail: [email protected] Abstract. We provide a short overview of some recent results on the application of group Fourier transform to quantum mechanics and quantum gravity. We close by pointing out some future research directions. 1. Introduction The group Fourier transform is an integral transform from functions on a Lie group to functions on a non-commutative dual space. It was first formulated for functions on SO(3) [1, 2], later generalized to SU(2) [2–4] and other Lie groups [5, 6], and is based on the quantum group structure of Drinfel’d double of the group [7, 8]. The transform provides a unitarily equivalent representation of quantum systems with a Lie group configuration space in terms of non- commutative algebra of functions on the classical dual space, the dual of the Lie algebra. As such, it has proven useful in several different ways. Most importantly, it provides a clear connection between the quantum system and the corresponding classical one, allowing for a better physical insight into the system. In the case of quantum gravity models this has turned out to be particularly helpful in unraveling the geometrical content of the models, because the dual variables have an intuitive interpretation as classical geometrical quantities.
    [Show full text]
  • 1.25. the Quantum Group Uq (Sl2). Let Us Consider the Lie Algebra Sl2
    55 1.25. The Quantum Group Uq (sl2). Let us consider the Lie algebra sl2. Recall that there is a basis h; e; f 2 sl2 such that [h; e] = 2e; [h; f] = −2f; [e; f] = h. This motivates the following definition. Definition 1.25.1. Let q 2 k; q =6 ±1. The quantum group Uq(sl2) is generated by elements E; F and an invertible element K with defining relations K − K−1 KEK−1 = q2E; KFK−1 = q−2F; [E; F] = : −1 q − q Theorem 1.25.2. There exists a unique Hopf algebra structure on Uq (sl2), given by • Δ(K) = K ⊗ K (thus K is a grouplike element); • Δ(E) = E ⊗ K + 1 ⊗ E; • Δ(F) = F ⊗ 1 + K−1 ⊗ F (thus E; F are skew-primitive ele­ ments). Exercise 1.25.3. Prove Theorem 1.25.2. Remark 1.25.4. Heuristically, K = qh, and thus K − K−1 lim = h: −1 q!1 q − q So in the limit q ! 1, the relations of Uq (sl2) degenerate into the relations of U(sl2), and thus Uq (sl2) should be viewed as a Hopf algebra deformation of the enveloping algebra U(sl2). In fact, one can make this heuristic idea into a precise statement, see e.g. [K]. If q is a root of unity, one can also define a finite dimensional version of Uq (sl2). Namely, assume that the order of q is an odd number `. Let uq (sl2) be the quotient of Uq (sl2) by the additional relations ` ` ` E = F = K − 1 = 0: Then it is easy to show that uq (sl2) is a Hopf algebra (with the co­ product inherited from Uq (sl2)).
    [Show full text]
  • An Introduction to the Theory of Quantum Groups Ryan W
    Eastern Washington University EWU Digital Commons EWU Masters Thesis Collection Student Research and Creative Works 2012 An introduction to the theory of quantum groups Ryan W. Downie Eastern Washington University Follow this and additional works at: http://dc.ewu.edu/theses Part of the Physical Sciences and Mathematics Commons Recommended Citation Downie, Ryan W., "An introduction to the theory of quantum groups" (2012). EWU Masters Thesis Collection. 36. http://dc.ewu.edu/theses/36 This Thesis is brought to you for free and open access by the Student Research and Creative Works at EWU Digital Commons. It has been accepted for inclusion in EWU Masters Thesis Collection by an authorized administrator of EWU Digital Commons. For more information, please contact [email protected]. EASTERN WASHINGTON UNIVERSITY An Introduction to the Theory of Quantum Groups by Ryan W. Downie A thesis submitted in partial fulfillment for the degree of Master of Science in Mathematics in the Department of Mathematics June 2012 THESIS OF RYAN W. DOWNIE APPROVED BY DATE: RON GENTLE, GRADUATE STUDY COMMITTEE DATE: DALE GARRAWAY, GRADUATE STUDY COMMITTEE EASTERN WASHINGTON UNIVERSITY Abstract Department of Mathematics Master of Science in Mathematics by Ryan W. Downie This thesis is meant to be an introduction to the theory of quantum groups, a new and exciting field having deep relevance to both pure and applied mathematics. Throughout the thesis, basic theory of requisite background material is developed within an overar- ching categorical framework. This background material includes vector spaces, algebras and coalgebras, bialgebras, Hopf algebras, and Lie algebras. The understanding gained from these subjects is then used to explore some of the more basic, albeit important, quantum groups.
    [Show full text]
  • Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz
    FUNDAMENTALS OF PHYSICS – Vol. I - Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz QUANTUM MECHANICAL LAWS Bogdan Mielnik and Oscar Rosas-Ortiz Departamento de Física, Centro de Investigación y de Estudios Avanzados, México Keywords: Indeterminism, Quantum Observables, Probabilistic Interpretation, Schrödinger’s cat, Quantum Control, Entangled States, Einstein-Podolski-Rosen Paradox, Canonical Quantization, Bell Inequalities, Path integral, Quantum Cryptography, Quantum Teleportation, Quantum Computing. Contents: 1. Introduction 2. Black body radiation: the lateral problem becomes fundamental. 3. The discovery of photons 4. Compton’s effect: collisions confirm the existence of photons 5. Atoms: the contradictions of the planetary model 6. The mystery of the allowed energy levels 7. Luis de Broglie: particles or waves? 8. Schrödinger’s wave mechanics: wave vibrations explain the energy levels 9. The statistical interpretation 10. The Schrödinger’s picture of quantum theory 11. The uncertainty principle: instrumental and mathematical aspects. 12. Typical states and spectra 13. Unitary evolution 14. Canonical quantization: scientific or magic algorithm? 15. The mixed states 16. Quantum control: how to manipulate the particle? 17. Measurement theory and its conceptual consequences 18. Interpretational polemics and paradoxes 19. Entangled states 20. Dirac’s theory of the electron as the square root of the Klein-Gordon law 21. Feynman: the interference of virtual histories 22. Locality problems 23. The idea UNESCOof quantum computing and future – perspectives EOLSS 24. Open questions Glossary Bibliography Biographical SketchesSAMPLE CHAPTERS Summary The present day quantum laws unify simple empirical facts and fundamental principles describing the behavior of micro-systems. A parallel but different component is the symbolic language adequate to express the ‘logic’ of quantum phenomena.
    [Show full text]
  • Quantum Supergroups and Canonical Bases
    Quantum supergroups and canonical bases Sean Clark University of Virginia Dissertation Defense April 4, 2014 Let g be a semisimple Lie algebra (e.g. sl(n); so(2n + 1)). Π = fαi : i 2 Ig the simple roots. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. ±1 Uq(g) is the Q(q) algebra with generators Ei, Fi, Ki for i 2 I, Various relations; for example, hi −1 hhi,αji I Ki ≈ q , e.g. KiEjKi = q Ej 2 2 I quantum Serre, e.g. Fi Fj − [2]FiFjFi + FjFi = 0 (here [2] = q + q−1 is a quantum integer) Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g). WHAT IS A QUANTUM GROUP? A quantum group is a deformed universal enveloping algebra. Let g be a semisimple Lie algebra (e.g. sl(n); so(2n + 1)). Π = fαi : i 2 Ig the simple roots. Some important features are: −1 −1 I an involution q = q , Ki = Ki , Ei = Ei, Fi = Fi; −1 I a bar invariant integral Z[q; q ]-form of Uq(g).
    [Show full text]
  • A Brief History of Hidden Quantum Symmetries in Conformal Field
    A brief history of hidden quantum symmetries in Conformal Field Theories∗ C´esar G´omez and Germ´an Sierra Instituto de Matem´aticas y F´ısica Fundamental, CSIC Serrano 123. Madrid, SPAIN Abstract We review briefly a stream of ideas concerning the role of quantum groups as hidden symmetries in Conformal Field Theories, paying particular attention to the field theo- retical representation of quantum groups based on Coulomb gas methods. An extensive bibliography is also included. arXiv:hep-th/9211068v1 16 Nov 1992 ∗Lecture presented in the XXI DGMTP-Conference (Tianjin ,1992) and in the NATO-Seminar (Salamanca,1992). 0 Integrability, Yang-Baxter equation and Quantum Groups In the last few years it has become clear the close relationship between conformal field theories (CFT’s), specially the rational theories , and the theory of quantum groups.From a more general point of view this is just one aspect of the connection between integrable systems and quantum groups. In fact if one recalls the history, quantum groups originated from the study of integrable models and more precisely from the quantum inverse scattering approach to integrability. The basic relation: R(u − v) T1(u) T2(v)= T2(v) T1(u) R(u − v) (1) which is at the core of this approach may also be taken as a starting point in the definition of quantum groups [FRT]. In statistical mechanics the operator T (u) is nothing but the monodromy matrix which depends on the spectral parameter u and whose trace gives the transfer matrix of a vertex model. Meanwhile the matrix R(u) stands for the Boltzmann weights of this vertex model.
    [Show full text]
  • Canonical Bases for the Quantum Group of Type Ar and Piecewise
    Canonical bases for the quantum group of type Ar and piecewise-linear combinatorics Arkady Berenstein and Andrei Zelevinsky 1 Department of Mathematics, Northeastern University, Boston, MA 02115 0. Introduction This work was motivated by the following two problems from the classical representa- tion theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A precise formulation of this problem was given in [3]; we shall explain it in more detail a bit later. 2. Construct a basis in every polynomial representation of GLr+1, such that the maximal element w0 of the Weyl group Sr+1 (considered as an element of GLr+1)acts on this basis by a permutation (up to a sign), and explicitly compute this permutation. This problem is motivated by recent work by John Stembridge [10] and was brought to our attention by his talk at the Jerusalem Combinatorics Conference, May 1993. We show that the solution to both problems is given by the same basis, obtained by the specialization q = 1 of Lusztig’s canonical basis for the modules over the Drinfeld-Jimbo q-deformation Ur of U(slr+1). More precisely, we work with the basis dual to Lusztig’s, and our main technical tool is the machinery of strings developed in [4]. The solution to Problem 1 appears below as Corollary 6.2, and the solution to Problem 2 is given by Proposition 8.8 and Corollary 8.9.
    [Show full text]
  • The Algebraic Versus Geometric Approach to Quantum Field Theory *
    v ; ; LAPP.TH-292/90 r\ j June ^ The Algebraic Versus Geometric Approach To Quantum Field Theory * B. SCHROER" Laboratoire d'Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux Cedex FRANCE Abstract Some recent developments in algebraic QFT are reviewed and confronted with results obtained by geometric methods. In particular I try to give a critical evaluation of the present status of the quantum symmetry discussion and comment on the possible relation of the (Gepner-Witten) modularity in conformai QFTj and the Tomita modularity (existence of quantum reflections) of the algebraic approach. * presented at the 4<k Annecy Meeting on Theoretical Pliyiics, 5-9 March 1990; proceeding! edited by P. Binetruy et al. to appear as Nud.Phyi.B. (Proc. Suppl.) ** Permanent address : Insu fur Théorie der Elementarteilchen, Fachbereich 20-WE 4, Freie Univ. Berlin, AmimaUee 14,1000 Berlin 33, RFA. .A.?. F. il" III- I "J'l-ll -.NMlN I I -\ I! 1 \ ( I Dl \ O II I I I1MOM SI!.:' 12.4* • ® I 11 I- \ 'SS ISO I- © Ml. MOl1I! 50 27 Introduction This conference has a special occasion and before I present some recent results obtained by the algebraic method in QFT, I take the permission to say some words about Raymond Stora. I met Raymond the first time more than a quarter of a century ago at some conference in Boulder/Colorado which, according to the best of my memory, was dedicated to a survey of Wigner's work on the representation theory of the Poincaré group and its application to QFT.
    [Show full text]
  • The Spin-Statistics Relation and Noncommutative Quantum Field Theory
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Helsingin yliopiston digitaalinen arkisto Pro gradu -tutkielma Teoreettisen fysiikan suuntautumisvaihtoehto The Spin-Statistics Relation and Noncommutative Quantum Field Theory Tapio Salminen Ohjaaja: Dos. Anca Tureanu Tarkastajat: Dos. Anca Tureanu ja Prof. Peter Preˇsnajder HELSINGIN YLIOPISTO FYSIKAALISTEN TIETEIDEN LAITOS PL 64 (Gustaf H¨allstr¨omin katu 2) 00014 Helsingin yliopisto Acknowledgements I have been very fortunate to have a collection of inspiring teachers and mentors throughout my education. I thank you all for your support. When it comes to this thesis I am greatly indebted to Anca Tureanu whose help and support have been invaluable. Thank you. My family and my friends I thank for all the gloomier times as well as all the fun and beautiful moments that have got me where I am now. On your continuing support I will be counting on also in the future, don’t forget. A special thank you is preserved for my very best friend for the past years. Mariia, you are my greatest source of awe and inspiration. You are the only reason I am. You are all my reasons. Contents 1 Historical introduction 1 1.1 Experimentalevidence . 2 1.2 Relativistictheory ........................ 5 2 The Poincar´egroup and spin 9 2.1 Representations of the Poincar´egroup . 10 3 The spin-statistics theorem 12 3.1 Rotational invariance and the spin-statistics relation...... 13 3.2 Commuting and anticommuting fields . 17 4 Noncommutative quantum field theory and twists 22 4.1 Noncommutativity of space-time . 23 4.2 Weyl-Moyal correspondence and the ⋆-product .
    [Show full text]