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Projection Operator Formalism for Quantum Constraints

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PROJECTION OPERATOR FORMALISM FOR QUANTUM CONSTRAINTS By JEFFREY SCOTT LITTLE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 1 °c 2007 Je®rey Scott Little 2 To my wonderful wife and soul mate, Megan. 3 ACKNOWLEDGMENTS Obtaining a Ph.D. in any ¯eld is never a complete individual e®ort, I owe many thanks to many people that helped me pursue this life-long goal. First of all, I would like to thank my advisor John Klauder, for giving me the chance to study quantization of constraints systems. I am indebted to him for all the patience, time, and encouragement that he has a®orded me over the years that I spent under his tutelage at the University of Florida. His passion for the course of study has helped me obtain a broader view of physics, as well as a more developed palate for various topics in physics. In fact I am grateful to the entire physics department for allowing me the opportunity to study theoretical physics. I gratefully acknowledge the Alumni Fellowship Association, which allowed me to attend the University without an overwhelming teaching responsibility. I would also like to thank my grandparents, Ruby, Granville, Hazel, and Veral, who instilled in me from an extremely early age that I could accomplish anything if I set my mind to it. Thanks go to my dad, Je®, who gave me a sense of scienti¯c curiosity and to my mom, Linda, who was my wonderful math instructor from fractions to calculus, not to mention all of their love and support, and to my sisters, Lisa and Sierra, whose constant encouragement aided me through my early college and graduate career. I am assuredly indebted to my dear Aunt Brenda, who carefully edited several chapters of this manuscript, even though she is not a physics person. I would also like to thank my wife's family for all of their support these past 2 years. Thanks go to my friends at the University of Florida Larry, Ethan, Wayne, Ian, Lester, Jen, and Garret, whose compassion and conversations about a wide variety of topics are unmeasurable. Saving the best for last, I thank the love of my life, my darling wife, Megan. Without whom I would have never completed this dissertation. I thank her for all of the love, support, and encouragement that she has given me; she is the source of my inspiration to achieve, more than I ever dreamed could be achieved. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 8 CHAPTER 1 INTRODUCTION .................................. 10 1.1 Philosophy .................................... 11 1.2 Outline of the Remaining Chapters ...................... 12 2 CONSTRAINTS AND THE DIRAC PROCEDURE ............... 14 2.1 Classical Picture ................................ 14 2.1.1 Geometric Playground ......................... 16 2.1.2 Constraints Appear ........................... 18 2.1.3 Another Geometric Interlude ...................... 20 2.1.4 Observables ............................... 21 2.2 Quantization .................................. 22 2.2.1 Canonical Quantization Program ................... 22 2.2.2 What About Constraints? The Dirac Method ............ 24 3 OTHER METHODS ................................. 27 3.1 Faddeev-Popov Method ............................ 27 3.1.1 Yet Another Geometric Interlude from the Constraint Sub-Manifold 27 3.1.2 Basic Description ............................ 28 3.1.3 Comments and Criticisms ....................... 29 3.2 Re¯ned Algebraic Quantization ........................ 31 3.2.1 Basic Outline of Procedure ....................... 32 3.2.2 Comments and Criticisms ....................... 33 3.3 Master Constraint Program .......................... 34 3.3.1 Classical Description .......................... 35 3.3.2 Quantization ............................... 36 3.3.3 MCP Constraint Example ....................... 39 3.3.4 Comments and Criticisms ....................... 41 3.4 Conclusions ................................... 42 4 PROJECTION OPERATOR FORMALISM .................... 43 4.1 Method and Motivation ............................ 43 4.1.1 Squaring the Constraints ........................ 45 4.1.2 Classical Consideration ......................... 45 4.1.2.1 Quantum Consideration ................... 46 4.1.2.2 Projection Operator Justi¯cation .............. 47 5 4.2 Tools of the Projection Operator Formalism ................. 48 4.2.1 Coherent States ............................. 48 4.2.2 Reproducing Kernel Hilbert Spaces .................. 49 4.3 Constraint Examples .............................. 51 4.3.1 Constraint with a Zero in the Continuous Spectrum ......... 51 4.3.2 Closed, First-Class Constrant ..................... 52 4.3.3 Open, First Class constraint ...................... 54 4.4 Conclusions ................................... 58 5 HIGHLY IRREGULAR CONSTRAINTS ...................... 59 5.1 Classi¯cation .................................. 59 5.2 Toy Model .................................... 64 5.3 Observables ................................... 72 5.4 Observation and Conclusions .......................... 75 6 ASHTEKAR-HOROWITZ-BOULWARE MODEL ................. 76 6.1 Introduction ................................... 76 6.2 Classical Theory ................................ 77 6.3 Quantum Dynamics ............................... 79 6.4 The Physical Hilbert Space via the Reproducing Kernel ........... 79 6.4.1 The Torus T2 .............................. 79 6.5 Super-selection Sectors? ............................ 84 6.6 Classical Limit ................................. 85 6.7 Re¯ned Algebraic Quantization Approach .................. 88 6.8 Commentary and Discussion .......................... 90 7 PROBLEM WITH TIME .............................. 92 8 TIME DEPENDENT CONSTRAINTS ....................... 94 8.1 Classical Consideration ............................. 94 8.1.1 Basic Model ............................... 94 8.1.2 Commentary and Discussion ...................... 97 8.2 Quantum Considerations ............................ 99 8.2.1 Gitman and Tyutin Prescription for Time-Dependent Second-Class Constraints ............................... 99 8.2.2 Canonical Quantization ......................... 101 8.2.3 Dirac ................................... 102 8.2.4 Projection Operator Formalism .................... 102 8.2.5 Time-Dependent Quantum Constraints ................ 103 8.2.6 Observations and Comparisons ..................... 105 9 TIME-DEPENDENT MODELS ........................... 106 9.1 First-Class Constraint ............................. 106 9.2 Second Class Constraint ............................ 110 6 9.3 Conclusions ................................... 111 10 CONCLUSIONS AND OUTLOOK ......................... 113 10.1 Summary .................................... 113 10.2 Ending on a Personal Note ........................... 115 APPENDIX A REPARAMETERIZATION INVARIANT THEORIES .............. 116 REFERENCES ....................................... 118 BIOGRAPHICAL SKETCH ................................ 122 7 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy PROJECTION OPERATOR FORMALISM FOR QUANTUM CONSTRAINTS By Je®rey Scott Little December 2007 Chair: John Klauder Major: Physics Motivated by several theoretical issues surrounding quantum gravity, a course of study has been implemented to gain insight into the quantization of constrained systems utilizing the Projection Operator Formalism. Throughout this dissertation we will address several models and techniques used in an attempt to illuminate the subject. We also attempt to illustrate the utility of the Projection Operator Formalism in dealing with any type of quantum constraint. Quantum gravity is made more di±cult in part by its constraint structure. The constraints are classically ¯rst-class; however, upon quantization they become partially second-class. To study such behavior, we will focus on a simple problem with ¯nitely many degrees of freedom and will demonstrate how the Projection Operator Formalism is well suited to deal with this type of constraint. Typically, when one discusses constraints, one imposes regularity conditions on these constraints. We introduce the \new" classi¯cation of constraints called \highly irregular" constraints, due to the fact these constraints contain both regular and irregular solutions. Quantization of irregular constraints is normally not considered; however, using the Projection Operator Formalism we provide a satisfactory quantization. It is noteworthy that irregular constraints change the observable aspects of a theory as compared to strictly regular constraints. More speci¯cally, we will attempt to use the tools of the Projection Operator Formalism to study another gravitationally inspired model, namely 8 the Ashtekar-Horowitz-Boulware model. We will also o®er a comparison of the results obtained from the Projection Operator Formalism with that of the Re¯ned Algebraic Quantization scheme. Finally, we will use the Projection Operator Method to discuss time-dependent quantum constraints. In doing so, we will develop the formalism and study a few key time-dependent models to help us obtain a larger picture on how to deal with reparameterization invariant theories such as General Relativity. 9 CHAPTER 1 INTRODUCTION \It is very important that we do not all follow the same fashion... Its necessary
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