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Quantum Field Theory Via Vertex Algebras

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Quantum Field Theory Via Vertex Algebras C a r d if f UNIVERSITY PRI FYSGOL C a eRDY|§> S c h o o l o f M a t h e m a t ic s C a r d i f f U n i v e r s i t y A THESIS SUBMITTED FOR THE DEGREE OF Doctor of Philosophy Quantum field theory via vertex algebras Author: Heiner Olbermann September 17, 2010 UMI Number: U585B83 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U585B83 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Declaration This work has not previously been accepted in substance for any degree and is not concur­ rently submitted in candidature for any degree. .......... This thesis is being submitted in partial fulfillment of the requirements for the degree of PhD. Date Statemel This thesis is the result of my own independent work/investigation, except where otherwise stated. Other sources are acknowledged by explicit references. t f Statement 3 I hereby give consent for my thesis, if accepted, to be available for photocopying and for interlibrary loan, for the title and summary to be made available to outside organisations. Date .as../..1'/ it> Statement 4: Previously approved bar on access I hereby give consent for my thesis, if accepted, to be available for photocopying and for interlibrary loans after expiry of a bar on access previously approved by the Graduate De­ velopment Committee. lO J to Date / his LcbXL Abstract We investigate an alternative formulation of quantum field theory that elevates the Wilson- Zimmermann operator product expansion (OPE) to an axiom of the theory. We observe that the information contained in the OPE coefficients may be straightforwardly repackaged into “vertex operators”. This way of formulating quantum field theory has quite obvious similarities to the theory of vertex algebras. As examples of this framework, we discuss the free massless boson in D dimensions and the massless Thirring model. We set up perturbation theory for vertex algebras. We discuss a general theory of pertur­ bations of vertex algebras, which is similar to the Hochschild cohomology describing the deformation theory of ordinary algebras. We pass on to a more explicit discussion by look­ ing at perturbations of the free massless boson in D dimensions. The perturbations we consider correspond to some interaction Lagrangian A P(<p) = A Cp tpP. We construct the perturbations by exploiting the associativity of the vertex operators and the field equation in perturbative form. We develop a set of graphical rules that display the vertex operators as certain multiple series reminiscent of the hypergeometric series. Contents 1 Introduction 3 2 Vertex algebras in QFT 14 2.1 Axiomatic framework ............................................................................................... 14 2.2 Comparison to other notions of vertex algebras .................................................. 17 2.2.1 Chiral algebras ............................................................................................. 17 2.2.2 Full field algebras ......................................................................................... 25 3 Examples: The free boson and the massless Thirring model 30 3.1 The free boson ........................................................................................................ 30 3.1.1 Proof of associativity for the free boson ................................................... 37 3.2 The massless Thirring model ................................................................................... 42 3.2.1 Classical m o d e l ............................................................................................. 45 3.2.2 Klaiber’s solution ......................................................................................... 47 3.3 Construction of the vertex algebra for the massless Thirring model .................... 53 3.3.1 Proof of associativity for the massless Thirring mo d el ........................... 59 3.3.2 Current, field equation, primary fields ....................................................... 70 4 Perturbation theory 73 1 4.1 Perturbation theory via Hochschild cohomology ............................................... 74 4.1.1 The cohomology of vertex algebra perturbations ..................................... 75 4.1.2 Gauge th e o rie s ............................................................................................. 78 4.2 Graphical rules for computing vertex operators .................................................. 84 4.2.1 Right inverse for the Laplacian ................................................................ 88 4.2.2 The case D = 2, P = </?3: ............................................................................. 90 4.2.3 The general case .......................................................................................... 106 4.2.4 Alternative expressions for tree-like terms ............................................. 113 4.3 Renormalization ........................................................................................................ 122 4.3.1 E x am p les ...................................................................................................... 144 5 Conclusions and outlook 148 A Spherical harmonics and Gegenbauer functions 151 B Identities for Gegenbauer functions 156 C The free field vertex operators 168 2 Chapter 1 Introduction Quantum field theory (QFT) is the theoretical formalism that describes collisions of elemen­ tary particles and critical phenomena. There are many ways to define a QFT all of which are equivalent to some extent, at least at a formal level. Among the most common are the Lagrangian approach via path inte­ grals [22,31,46,72], the definition of a set of n-point functions satisfying some set of ax­ ioms [63,68], the so-called bootstrap program [3,49], the operator algebraic approach [33], to name only a few. Common to all of these is that one deals with a set of “fields” @a, a € X, where T is some index set. In the axiomatic approach, a QFT is defined by the set of all correlation functions n \ / corr. ’ n € N, ai,...,a„ G I 2 = 1 which are distributions in the variables x * 6 MD. The path integral can be understood a tool to define these correlation functions. This does not always work; one instance where it works particularly well is perturbation theory. Here, one tries to give rigorous meaning to 3 limits ( n ^^^con- := ii^o / n eXP [“ 5 'a(<P)] <W<*>) , (1.0.1) : t where is a Gaussian measure (associated to the free part of a classical action) and S\ is the full action including all counterterms depending on a cutoff scale A. This may be done by imposing some boundary conditions (renormalization conditions) on the correlation functions for finite A and deriving the limits (1.0.1) by the so-called Polchinski flow equa­ tions [50,51,65]. Another option is to make a diagrammatic expansion of the right hand side of eq. (1.0.1). The terms in this expansion can be identified with “Feynman diagrams”. Each of them has a finite value for A < oo, but typically diverges for A —> oo. In fact, the A-dependent terms in S\ have to be chosen precisely so that any divergence in a Feynman diagram is canceled by another divergence with the opposite sign in some other Feynman diagram (see e.g. [72]). Formula (1.0.1) is supposed to define the correlation functions of a Euclidean QFT, i.e. a QFT defined on RD equipped with the Euclidean metric. The correlation functions of the corresponding relativistic QFT can be found by a so-called “Wick rotation”: D-dimensional Euclidean space can be viewed as the D-dimensional Riemannian subspace of a complex D- dimensional space that as a further subspace also possesses D-dimensional Minkowski space. The correlation functions on the Euclidean space can be continued analytically to the whole complex D-dimensional space and the restrictions of these continuations to Minkowski space define the relativistic QFT. However no such construction exists between Riemannian and Lorentzian QFT when one considers curved spaces, as a generic Lorentzian spacetime cannot be expressed as a section of a complex manifold that also possesses a Riemannian section. 4 So formula (1.0.1) is not appropriate under these more general conditions, and one has to look at alternative ways to define QFT. Such an alternative framework is provided by Algebraic QFT (AQFT) [24,33]. Here, the main focus is on the algebraic relations between quantum fields. Technically speaking, the basic idea is to associate a *-algebra A(0) to any region O of the considered Lorentzian spacetime. This may be thought of as the algebra of observables (smeared quantum fields) of the subsystem associated to the region O. For the case of perturbative quantum field theory in curved spacetime, the construction of these local algebras has been achieved in [12,13,40,41]. The success of AQFT in this context suggests that at the heart of QFT are the algebraic relations between quantum fields. One way to express these algebraic relations on the level of correlation functions (which also works for
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