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UNIVERSITY of CALIFORNIA SAN DIEGO Aspects of Scale Invariant UNIVERSITY OF CALIFORNIA SAN DIEGO Aspects of Scale Invariant Quantum Field Theory A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Sridip Pal Committee in charge: Professor Kenneth Intriligator, Chair Professor Benjamin Grinstein, Co-Chair Professor Tarun Grover Professor John McGreevy Professor James Mckernan 2019 Copyright Sridip Pal, 2019 All rights reserved. The dissertation of Sridip Pal is approved, and it is accept- able in quality and form for publication on microfilm and electronically: Co-Chair Chair University of California San Diego 2019 iii DEDICATION To my parents and my friends Ahanjit, Brato, Shouvik for being there Always . iv EPIGRAPH The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. —Sydney Coleman v TABLE OF CONTENTS Signature Page....................................... iii Dedication.......................................... iv Epigraph...........................................v Table of Contents...................................... vi List of Figures........................................x List of Tables........................................ xii Acknowledgements..................................... xiii Vita............................................. xviii Abstract of the Dissertation................................. xix Chapter 1 Introduction.................................1 Chapter 2 Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory. 13 2.1 Introduction.............................. 13 2.2 Generalities.............................. 17 2.2.1 Dynamical exponent..................... 25 2.3 d = 2, z = 2 Non Relativistic theory................. 28 2.3.1 Listing out terms....................... 28 2.3.2 Using counter-terms..................... 31 2.3.3 Consistency conditions and vanishing anomalies...... 34 2.3.4 Applications......................... 35 2.4 Generalisation to arbitrary z value.................. 38 2.5 A candidate for a C-theorem in d + 1D............... 39 2.6 Summary and Discussion....................... 46 Chapter 3 On the Heat Kernel and Weyl Anomaly of Schrodinger¨ invariant theory. 49 3.1 Introduction.............................. 50 3.2 Newton-Cartan Structure & Weyl Anomaly............. 53 3.3 Discrete Light Cone Quantization (DLCQ) & its cousin Lightcone Reduction (LCR)........................... 57 3.4 Heat Kernel for a Galilean CFT with z = 2.............. 61 3.4.1 Preliminaries: Heat Kernel, Zeta Regularisation....... 61 3.4.2 Heat Kernel in Flat Space-time................ 66 3.4.3 Heat Kernel in Curved spacetime.............. 70 vi 3.5 Perturbative proof of Vanishing anomaly............... 73 3.6 Modified LCR and Generalisation.................. 77 3.6.1 Proving the heat kernel prescription............. 77 3.6.2 Generalisation........................ 78 3.7 Summary, Discussion and Future directions............. 83 Chapter 4 Unitarity and Universality in non relativistic Conformal Field theory... 87 4.1 Introduction.............................. 87 4.2 Unitarity bounds & Null conditions................. 90 4.3 Non Unitarity in non integer dimensions.............. 96 4.4 Universality.............................. 98 4.5 Discussion & Outlook........................ 101 4.6 Acknowledgments.......................... 104 Chapter 5 Existence and Construction of Galilean invariant z 6= 2 Theories...... 106 5.1 Introduction.............................. 107 5.2 Generalized Schrodinger¨ algebra and its Representation....... 110 5.3 Null reduction and Embedding into Conformal group........ 116 5.4 Explicit d + 1 dimensional examples................. 117 5.4.1 z = 0............................. 117 2` 5.4.2 z = `+1 ;` 2 Z, ` ≥ 1..................... 118 5.5 Conclusion.............................. 119 Chapter 6 Hilbert Series and Operator Basis for NRQED and NRQCD/HQET.... 121 6.1 Introduction.............................. 121 6.2 Effective Theory for a Nonrelativistic Fermion............ 123 6.3 Hilbert series for a nonrelativistic theory............... 125 6.4 NRQED................................ 128 6.5 NRQCD/HQET............................ 131 6.6 Discussion and Conclusions..................... 133 Chapter 7 Conformal Structure of the Heavy Particle EFT Operator Basis...... 137 7.1 Introduction.............................. 137 7.2 Operator Basis for Heavy Particle EFTs............... 139 7.3 Operator Basis and the Schrodinger¨ Algebra............. 142 7.4 Discussion and Conclusions..................... 149 Chapter 8 Nonrelativistic Conformal Field Theories in the Large Charge Sector... 152 8.1 Introduction and Summary...................... 152 8.2 Lightning Review of Schrodinger¨ Algebra.............. 157 8.3 Lightning Review of Coset Construction............... 159 8.4 Schrodinger¨ Superfluid from Coset Construction.......... 162 8.5 Superfluid Hydrodynamics...................... 166 8.5.1 Superfluid in a Harmonic Trap................ 167 vii 8.6 Operator Dimensions......................... 169 8.6.1 Ground State Energy & Scaling of Operator Dimension... 169 8.6.2 Excited State Spectrum.................... 171 8.7 Correlation Functions......................... 173 8.7.1 Two Point Function...................... 173 8.7.2 Three Point Function..................... 173 8.8 Conclusions and Future Directions.................. 175 Chapter 9 The Spinful Large Charge Sector of Non-Relativistic CFTs: From Phonons to Vortex Crystals.............................. 178 9.1 Introduction and Summary...................... 178 9.2 Superfluid Hydrodynamics and Large Charge NRCFT........ 182 9.3 Phonons................................ 185 9.4 Single Vortex in the Trap....................... 188 9.4.1 Single vortex in d = 2.................... 190 9.4.2 Single vortex in d = 3.................... 191 9.5 Multi-Vortex Profile.......................... 193 9.6 Conclusions and Future Directions.................. 198 Chapter 10 Modular Invariance and Cardy Formula.................. 200 Chapter 11 Bounds on density of states and spectral gap in CFT2 ........... 212 11.1 The premise and the results...................... 212 11.2 Derivation of the improvement.................... 216 11.3 Bound on bounds........................... 221 11.4 Bound on spectral gap: towards optimality.............. 225 11.5 Brief discussion............................ 231 Appendix A Weyl Consistency Conditions in Non Relativistic Quantum Field Theory. 233 A.1 Consistency Conditions for 2 + 1d NRCFT............. 233 A.2 Anomaly ambiguities......................... 236 2 A.2.1 ¶t Sector........................... 236 2 A.2.2 ¶t∇ Sector.......................... 236 A.2.3 ∇4 Sector........................... 238 A.3 S-theorem: 0 + 1D conformal quantum mechanics.......... 240 Appendix B On the Heat Kernel and Weyl Anomaly of Schrodinger¨ invariant theory. 244 B.1 Technical Aspects of Heat Kernel for one time derivative theory.. 244 B.2 Riemann normal co-ordinate and coincident limit.......... 245 B.3 Explicit Perturbative Calculation of h-regularized Heat Kernel... 249 viii Appendix C Unitarity and Universality in non relativistic Conformal Field theory... 255 C.1 SL(2;R) invariant field theory.................... 255 C.2 Schrodinger¨ algebra, Primaries and Quasi-primaries......... 258 C.3 Time reversal and Parity....................... 260 C.4 Free Schrodinger¨ Field theory & Its Euclidean avatar........ 263 Appendix D Existence and Construction of Galilean invariant z 6= 2 Theories..... 268 D.1 Diagonalizable and finite dimensional dilatation generator..... 268 Appendix E Hilbert Series and Operator Basis for NRQED and NRQCD/HQET.... 273 E.1 Characters and Haar Measures.................... 273 Appendix F Conformal Structure of the Heavy Particle EFT Operator Basis...... 275 F.1 Normalsize Symmetries of Non-Relativistic Systems........ 275 F.2 Symmetries, Operators, and States.................. 279 F.3 Constraints from Algebra and Unitarity bound............ 283 Appendix G Nonrelativistic Conformal Field Theories in the Large Charge Sector... 287 G.1 Appendix A: Phonons in the Trap.................. 287 G.2 Appendix B: Correlation Functions in Oscillator Frame....... 289 G.2.1 Two point function...................... 289 G.2.2 Three point function..................... 290 Appendix H The Spinful Large Charge Sector of Non-Relativistic CFTs: From Phonons to Vortex Crystals.............................. 291 H.1 Particle-Vortex Duality........................ 291 H.2 A Contour integral.......................... 294 Bibliography........................................ 296 ix LIST OF FIGURES Figure 1.1: The rotation of a sphere with a dot, without the dot, the two scenarios are exactly the same................................2 Figure 1.2: The image of a fractal............................3 Figure 2.1: Self energy correction to propagator at one loop. The ingoing momenta is denoted by (E;k) while the internal loop momenta is given by the variable p. 26 Figure 4.1: Unitarity bound on (a;D) plane: the projection of the region bounded by two curves onto the Y axis excludes D < 0. The blue thick curve is D+ = 1 q 1 1 q 1 −a − 2 + a + 4 while the orange dashed curve is D− = −a − 2 − a + 4 . 92 Figure 4.2: Unitarity bound on (b;D) plane: the projection of the region bounded by the d ellipse onto the Y axis excludes D 2 −1; 2 . Here d = 10. The blue thick db−1 1 p 2 curve is D+ = 2 + 2 1 + 2db − 2db while the orange dashed curve is db−1 1 p 2 D− = 2 − 2 1 + 2db − 2db ......................
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