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The Quartic Interaction in the Large-N Limit Of THE QUARTIC INTERACTION IN THE LARGE-N LIMIT OF QUANTUM FIELD THEORY ON A NONCOMMUTATIVE SPACE By Philip Albert DeBoer B. Sc. University of Prince Edward Island, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 2001 © Philip Albert DeBoer, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V6T 1Z1 Date: Abstract With a view to understanding generic properties of quantum field theories defined on spaces with noncommuting spatial coordinates (termed noncommutative quantum field theories), two simple models are considered. The first model is a theory of bosonic vector fields having an 0(iV)-symmetric quartic interaction. The second model is the fermionic counterpart of the bosonic theory, and is known as the Gross-Neveu model. In both cases the study is conducted in the simplifying large-TV limit. Unlike in the commutative case, the noncommutative theory gives rise to two inequiv- alent quartic interactions of the form (</>2)2 and (<j)%<f)i)2'. The latter interaction is difficult to work with, but significant progress is made for the theories containing only the former interaction. ii Table of Contents Abstract ii Table of Contents iii Acknowledgments v 1 Introduction 1 1.1 Motivation 1 1.2 Quantum Field Theory 4 1.2.1 Saddle-Point Approximation 5 1.2.2 Effective Action 7 1.2.3 Large-iV 8 1.2.4 Renormalization 10 1.3 Moyal Product 10 1.3.1 Construction 11 1.3.2 Properties 14 1.4 Noncommutative Quantum Field Theories 14 2 Noncommutative O(N) 04 Theory 16 2.1 Introduction 16 2.2 Local Auxiliary Field 18 2.3 Two-point Propagator 19 2.4 Self-Energy 20 iii 2.5 Symmetric Vertex 22 2.5.1 Four Dimensions 26 2.6 Beyond the Classical Case 27 3 Noncommutative Gross-Neveu Theory 32 3.1 Introduction 32 3.2 Local Auxiliary Field 33 3.3 Perturbation About the Classical Solution 33 3.4 Two Dimensions 35 3.4.1 The Commutative Theory 35 3.4.2 The Noncommutative Theory 36 3.4.3 A Double-Scaling Limit 38 3.5 Three Dimensions 38 3.5.1 The Commutative Theory 39 3.5.2 Noncommutative Theory 40 4 Conclusions 44 Bibliography 46 iv Acknowledgments Thanks are due to my supervisor Dr. G. Semenoff for his guidance and insightful com• ments during the course of this work. I would also like to thank Emil for his willingness to answer all my little questions. Thanks also to Mark and Ben for their encouragement. Finally, I would like to thank Drs. J. McKenna and especially K. Schleich for their support during the final stages of this work. Great is our Lord and mighty in power; his understanding has no limit. Psalm 147:5 NIV Chapter 1 Introduction 1.1 Motivation Recently there has been much interest in noncommutative quantum field theories. The low-energy limit of string theories with background antisymmetric tensor fields [1, 2, 3, 4, 5, 6] gives rise to certain noncommutative quantum field theories. They retain some of the interesting features of string theory such as nonlocality, which can be studied in the more familiar context of quantum field theory. Since the string theories are consistent quantum mechanical theories, the noncommutative field theories which are their consistent zero slope limits should also be internally consistent, that is, unitary and renormalizable. In fact, for some theories, unitarity has been demonstrated explicitly at one-loop order [7]. Because the theory generated by the string theory limit is difficult to study, simpler quantum field theories are being studied in the noncommutative regime. The field theories which arise from string theory are the Yang-Mills theories; these quantum fields have a nonabelian gauge symmetry. These theories are technically diffi• cult to study; one of the problems is a lack of local gauge-invariant observables in the noncommutative case. As a result, in order to better understand generic features of these theories, simpler examples have been chosen for study. In particular, the theories studied here are theories of N fields with an O(N) symmetry and having quartic interactions. Both bosonic and fermionic models will be examined. These models have the advantage of being solvable in the large-N limit, but since these theories do not arise as string Chapter 1. Introduction 2 theory limits there is no reason a priori to assume internal consistency. The concept of field theory on noncommutative geometry is not entirely new, however. Because quantum field theories are plagued by divergences arising from short-distance behaviour, noncommutativity has been suggested as a way to tame these divergences. The argument is that the uncertainty relation can be used to establish a short-distance cutoff so that distances shorter than the cutoff can be interpreted as long distances. It was shown by Filk [15] that this need not be the case. For the theories considered here it will be seen that although the noncommutativity partially reduces the short-distance ultraviolet divergences, new divergences appear in the long-distance infrared regime. The remainder of this chapter introduces the important techniques required for study• ing noncommutative quantum field theories. In Chapter 2 the noncommutative O(N) c/>4 theory is studied in the large-N limit. In the critical dimension the coupling is dimensionless and the theory is marginally renormalizable by power counting. Here this occurs in four dimensions, and this is where the theory is studied in this paper. Although two possible interactions exist in the noncommutative case, only the sym• metric vertex will be examined in detail. This allows the sum of diagrams contributing to the self-energy of the field to be written as a geometric series, greatly simplifying the analysis. A momentum-independent solution to the self-energy is assumed. In the commutative case this solution is stable but nonrenormalizable. It is found that the noncommutative theory behaves similarly, but the nonrenormalizability is less severe. In Chapter 3 the fermionic Gross-Neveu theory is explored in the noncommutative regime. This theory is the fermionic analogue of the 04 vector model in that it has a quartic interaction. Again there are two possible interactions but only the symmetric one is studied. The critical dimension is two. Chapter 1. Introduction 3 In two dimensions the Gross-Neveu model is marginally renormalizable by power counting. The commutative version, however, is perturbatively renormalizable in the coupling constant; this is also true in the noncommutative case. Here, though, the model is studied in a 1/N expansion which is nonperturbative in the coupling since it sums contributions to all orders in the coupling. Unlike in the commutative case, the theory is found to be nonrenormalizable in the sense that the dependence on the coupling cannot be removed. A running coupling is introduced, and this effective coupling runs to zero as the cutoff is taken to infinity, leaving behind the trivial free field theory. This is a very interesting result, since the noncommutativity has destroyed the renormalizability despite early expectations that noncommutativity could actually improve it! Since the study is conducted in two dimensions, space and time are noncommutative. In this case the action contains infinite numbers of time derivatives, which ruins the ordinary Hamiltonian interpretation. But these are physical constraints and do not affect the mathematical analysis, so the study is justified on the basis that generic properties of the theories may still be exhibited. In Chapter 3 the Gross-Neveu model is also studied in three dimensions, where the coupling constant has the dimension of inverse mass. The commutative model is non• renormalizable in the coupling constant expansion, but is renormalizable in the non• perturbative large-iV expansion at the second-order phase transition. As in the two- dimensional case, the noncommutativity destroys the renormalizability of the large-./V expansion even near the critical point. Finally, in Chapter 4, the conclusions are summarized and discussed in the context of string theory. Chapter 1. Introduction 4 1.2 Quantum Field Theory Quantum field theory is described by an action S expressed as a functional of a Lagrangian C through (1.1) where D is the dimension of the space-time manifold. In this thesis the manifold will always be RD with Euclidean metric. Here d> represents the field configurations and d^d) their space-time derivatives. The field configurations <j> which minimize the action correspond to the observed fields. Using the calculus of variations the task of finding the fields which minimize the action of a particular Lagrangian can be reduced to finding fields which satisfy their Euler-Lagrange equations of motion (1.2) Analogous to the partition function of statistical mechanics, quantum field theory has a generating functional (1.3) This integral is over all possible field configurations. A source term J has been included explicitly. This allows correlation functions to be obtained from Z by taking functional derivatives with respect to J.
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