TI, 3 911 Topics in Modern Quantum Theory Andrew H. Wilkins Department of Physics and Mathematical Physics University of Adelaide Adelaide Australia October 13, 1998 u Contents Figures vlt Abstract tx Statement of originality and wishes regarding free distribution xl Acknowledgments xlll 0 General Introduction 1 1 Bosonic String Theory, Two Dimensions and the c: 1 Matrix Model 3 1.1 Canonical quantisation 4 1.1.1 Light-cone quantisation 5 L.1.2 Old-covariant quantisation 6 L.2 Vertex operators and physical-state conditions 8 1.3 Path-integral quantisation 8 7.4 Factorising the functional measures 10 I.4.t The critical string 11 L4.2 The non-critical string 72 1.5 Strings in more complicated backgrounds 13 1.6 Beta-functions, tachyons and weak fields 13 1.6.1 The B-function method t4 I.6.2 Tachyons and the weak-field expansion 16 1.7 A summary of the perturbative 2d string 18 1.8 The c: 1 matrix model 20 1.8.1 An example - pure gravity 2L Iil IV Contents I.8.2 The c : 1 model and its double scaling limit 1.8.3 The space-time interpretation 1.9 The proposal 1.9.1 The leg-pole transformation I.9.2 The tachyon and higher states 2 Massive Fields and the 2D String 2,1 Massless field equations 2.2 Higher-order corrections to the tachyon field equation 2.3 Higher-order corrections in general 2.4 The Massive Fields 2.5 Massive-field corrections to the tachyon field equation 2.6 The "first massive" discrete state in two dimensions 2.7 Tachyon scattering in a discrete-state background 2.8 Conclusions 3 Induced Chern-Simons Terms 3.1 Background Material 3.1.1 Chern-Simons terms 3.L2 The global SU(2) anomaly in four dimensions 3.1.3 Baryon-number violation 3.2 The claim and a counter-example 3.3 The two-dimensional model 3.4 Perturbative calculations 3.4.I The one-point function defined by Pauli-Villars regularisation 3.4.2 The "Adler" argument 3.4.3 Pauli-Villars and the two-point function Contents 3.5 Nonperturbative results 72 3.5.1 Determinant on a one-dimensional manifold 74 3.5.2 The cylindrical, or high-temperature, limit 78 3.6 Conclusions 78 4 Matrix Theory 81 4.I A lightning tour of the superstring 81 4.Ll The NSR action and worldsheet supersymmetry 81 4.7.2 Mode expansions for the fermions 82 4.7.3 Quantisation, the GSO projection and particle content 83 4.I.4 IIA supergravity 85 4.2 D0-branes 86 4.2.1 The superparticle solution of IIA supergravity 87 4.2.2 Charge quantisation 87 4.2.3 Preserved supersymmetry 89 4.2.4 Uplifting to eleven dimensions 90 4.2.5 Coupling to R-R fields 92 4.2.6 Dp-branes 93 4.3 Matrix theory 95 4.3.I M-theory 95 4.3.2 Light-like compactification 96 4.3.3 Matrix theory 98 5 D0-brane scattering 101 5.1 Notation, the light-cone and the infinite boost t02 5.2 The first-order solution 105 5.3 The form of higher-order solutions r07 v1 Contents 5.3.1 Acceleration r07 5.3.2 The scaling behaviour of the O(o|.J part of the metric 108 5,3.3 Scaling properties of the higher-order terms 108 5.4 The boundary term 109 5.5 The effective action to O(V6 lLr4) 109 5.5.1 The source action 110 5.5.2 The boundary actio 111 5.5.3 The gravity action in the bulk 712 5.5.4 The total effective action L74 5.6 The large-I/ Iimit LT4 5.7 Conclusions 115 A The phase-space density formalism and the leg-pole transformation LL7 B Derivation of the path integral L20 C The determinant of a Dirac operator L23 D Some useful formulae t26 E The three-particle integral 130 References 133 Figures 1.1 A string interaction I 1.2 Propagator, vertex and loop in the matrix model 2L 1.3 Double scaling limit of the c : 1 model 25 L.4 Semi-classical dynamics in the matrix model 27 3.1 A three-point diagram 67 3.2 A two-point diagram 70 3.3 Contours of integration 79 4.I Fermions on the torus 84 vll Vì II Figures Abstract This thesis contains three pieces of work. The first concerns the first massive level of closed bosonic string theory. Flee-field equations are derived by imposing Weyl invariance on the world-sheet. A two-parameter solution to the equation of motion and constraints is found in two dimensions with a flat linear-dilaton background. One-to-one tachyon scattering is studied in this background. The results support Dhar, Mandal and \Madia's proposal that 2d critical string theory corresponds to the c : I matrix model in which both sides of the Fermi sea are excited. In the second, a claim regarding the effective action of four-dimensional SU(2)¿ gauge theory is examined. Specificall¡ it has been proposed that at high and low temperature the effective action contains a three-dimensional Chern-Simons term whose coefficient is the chemical potential for baryon number. By performing exact calculations in a related two-dimensional theory it is demonstrated that the existence of the Chern-Simons term in four dimensions may be rather subtle. Finally, the effective action describing the scattering of three well-separated extremal brane solutions, in 11d supergravit¡ with zero p- transfer and small transverse velocities is calculated. It is proved that to obtain this action only the leading-order solution to Einstein's equations is needed. The result obtained agrees with Matrix theory. Using an interpretation of the conjecture of Maldacena the effective action can be viewed as the large-.ôf limit of the Matrix theory description of three supergraviton scattering at leading order. lx X xìl Acknowledgments It goes without saying that I wouldn't have been able to complete this Ph.D. without the countless hours my supervisor Jim McCarthy has spent discussing physics with me. It has truly been an honour to work with him. We have enjoyed a very rewarding relationship, not only because of his large knowledge of physics but also because he has been a good friend. More than this, however, it is through working with him that I have gained a great love of physics. I think this must be the mark of an excellent supervisor - that the student should go out into the world brimming with excitement about his field of study. The other person I thank is my wife Belinda. She has been a constant source of emotional and financial support. xlt1 xl\¡ General Introduction It is a testament to the rapidly changing face of modern physics that this thesis should contain not one, but rather three somewhat disjoint topics Í240,24I,3411. Not too long ago, much attention was being paid to the non-perturbative realisation of the two-dimensional (2d) string called the c : 1 matrix model. Although the 2d string was only a toy model, it was hoped that its description through the matrix model could shed light on questions such as the dynamics of black-holes. Such a progra,mme proved frustratingly problem- atic, however, because the string seemed to admit a multitude of non-perturbative extensions. Subsequently most of these have been found to be non-unitary and it was only relatively recently that a unitary extension was proposed [98]. Chapter 1 contains a reasonably self-contained introduction to perturbative bosonic string theory and the c: 1 matrix model. As the reader will no doubt appreciate, to make a detailed, pedagogical exposition of either of these fields would not only be an enormous undertaking, but would also be entirely unjustified in this thesis. Nevertheless, even though many interesting side-issues have been ignored, it is hoped that enough material is presented to make the original work in Chapter 2 understandable to a reader at graduate-student level. As just mentioned, currently there is only one version [98] of the c : 1 matrix model that is unitary and seems to contain the entire perturbative 2d string. In [98] it was explained how the space-time black-hole is realised in the matrix model by comparing the tachyon-graviton effective dynamics with scattering processes in the matrix model. A natural extension of this led to a proposal for the map between excitations in the matrix model and the higher string states, but it could not be checked because the scattering oftachyons in backgrounds other than the black-hole had not been worked out. This is the subject of Chapter 2. First, the constraints on the first massive level of the string which result from the imposition of Weyl invariance on the world-sheet are derived. An explanation is given as to why some previous attempts to obtain these equations have failed. Then the effective dynamics of tachyons in this background is calculated. The form of the discrete state which is the 2d realisation of the string's first massive level is found by solving its equation of motion and constraints. Studying one-to-one tachyon scattering in this discrete-state background allows an identification of the discrete state in the matrix model. The results confirm the conjecture of [98]. The second topic, contained in Chapter 3, is also a calculation in two dimensions. This time though the theory is not string theory but QED with a background axial charge.