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Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium by Y

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Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium by Y Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium by Yeuan-Ming Sheu B. Sc., National Taiwan University, 1992 M. Sc., National Taiwan University, 1994 Sc. M., Brown University, 1996 Submitted in partial ful¯llment of the requirements for the Degree of Doctor of Philosophy in the Department of Physics at Brown University Providence, Rhode Island May 2008 °c Copyright 2008 by Yeuan-Ming Sheu This dissertation by Yeuan-Ming Sheu is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Herbert M. Fried, Director Date Antal Jevicki, Advisor Recommended to the Graduate Council Date Gerald Guralnik, Reader Brown University, Department of Physics Date Chung-I Tan, Reader Brown University, Department of Physics Approved by the Graduate Council Date Sheila Bonde Dean of the Graduate School iii Vita Yeuan-Ming Sheu was born in a remote mountain village in Tunglo, Miaoli county, Taiwan, Re- public of China, on March 10, 1970. He is the ¯rst child of an army-o±cer-turned-civil-servant father and a fulltime mom. He grew up in the country with his two brothers and two sisters, and was adored (perhaps spoiled a little bit) by his grandparents until leaving home for Hsin-Chu Senior High. In his high school years, he competed in annual national scienti¯c exhibitions in both physics and biology, and earned a recommendation to the science task camp for collage admissions. He got admitted to the Department of Physics at the National Taiwan University with a full scholarship from the Education Ministry. Right after his freshman year, he worked on rebuilding instruments and projects in the semiconductor physics lab, and soon fell in love with Physics. After getting his Bachelor of Science in June, 1992, he continued his graduate study and received his Master of Science in June 1994 under the guidance of Prof. Yuan-Huei Chang to study the impurity properties of semiconductor quantum wells under high magnetic ¯elds. Mr. Sheu attended Brown University with a fellowship in September 1994, and continued to study condensed matter physics. After a few unproductive years, he took a leave of absence and joined Advanced Power Technologies, Inc. (later merged into BAE Systems) in Washington, DC, as a research physicist in the summer of 2001. After getting his company's support, he resumed his graduate study under Prof. Antal Jevicki and Prof. Herbert M. Fried, and has been working on problems in Quantum Field Theory since Spring 2003. Over the years, he has published a number of articles in peer-reviewed journals, and has applied for several patents on inventions of semiconductor and optical devices. When not pondering the mysteries of nature, he enjoys spending time with his lovely wife, boating, and day dreaming. iv Acknowledgements In the long journey of my graduate study, there were ups and downs; it has become an enjoyable experience the past few years. Besides my desk and blackboards, the research leading to this thesis was carried out on airplane tray tables, hotel desks, the Washington, DC metro, and on breakfast tables in CÄoted'Azur. An undertaking such as this could not have been possible without the assistance of countless people. I would ¯rst like to thank the faculty in the department of physics at both Brown University and National Taiwan University who have guided me to the wonderful world of Physics, in particular, thanks to Prof. Herbert M. Fried and Prof. Antal Jevicki for their guidance, inspiration, and friendship. Though he is in his 70s, Prof. Fried still works hard to conduct research with his notebooks, blackboards and napkins in various parts of the world. My lively discussion with Prof. Fried inspired me to do Physics more intuitively, and not just via formulation. I would also like to thank several individuals here at Brown and around the world, especially, Dr. Thierry Grandou, at Institut Non-Lin¶eaire de Nice of CNRS. Without them, I would not be able to complete this work. In addition to the support from BAE Systems, colleagues at Advanced Technologies deserve my special thanks, especially Mr. Oved Zucker, Dr. Ramy Shanny, Dr. Michael Grove, Dr. Robert D'Amico, and the many others who have encouraged me to resume my graduate study. I would also like to thank my parents for unconditional support of my academic pursuits, and my brothers and sisters who have taken care for my aging parents while I am on the opposite side of the globe. Furthermore, I wish to thank my wife, Yu-Jie, who has accompanied me through those tough years with all her love. Finally, I would like to dedicate this thesis to my grandparents and my father who have watched over me in heaven. v Contents List of Tables x List of Figures xi 1 Introduction 1 1.1 Overview . 1 1.2 Prior Attempts . 2 1.3 Current Work . 3 1.4 Thesis Organization . 4 2 Basics 5 2.1 The Functional Method in Quantum Field Theory . 5 2.1.1 The Functional approach in quantum ¯eld theory . 5 2.1.2 The Fermion Green's Function and Closed-Loop Functional . 7 2.2 Finite-Temperature Quantum Field Theory . 7 2.2.1 Statistical Thermodynamics . 7 2.2.2 The Functional Approach to Finite-Temperature Field Theory . 9 2.2.3 Finite-Temperature Propagators . 12 2.2.4 Imaginary-Time (Matsubara) Formalism . 13 2.2.5 Real-Time Formalism . 15 2.3 Finite-Temperature Green's Functions . 18 2.3.1 QED Finite-Temperature Generating Functional . 18 2.3.2 Fully-dressed Finite-Temperature Green's function . 18 2.3.3 Linkage Operator . 19 2.3.4 Coupled Thermal Fermion Green's Function . 20 2.3.5 Closed-Fermion-Loop Functional and Thermal Normalization Constant . 23 2.4 Proper-Time Representations of Schwinger and Fradkin . 23 2.4.1 Schwinger's Proper-Time Representation . 24 2.4.2 Fradkin's Representation . 25 2.4.3 Coupled Green's Functions in Mixed Space Representation . 27 vi 2.5 Mixed Representation of Propagators . 28 2.5.1 Free Finite-Temperature Fermion Propagators . 28 2.5.2 Free Finite-Temperature Boson Propagators . 30 2.5.3 Free Finite-Temperature Photon/Gauge Field Propagators . 31 2.5.4 Interpretation of Thermal Parts of Propagators . 32 2.6 Relationship Between Formalisms . 33 2.7 Damping Rate . 34 2.8 Hot Thermal Loops and the Resummation Program . 36 3 Finite-Temperature Propagator in a Hot QED Medium 38 3.1 Overview . 38 3.2 Dressed Finite-Temperature Fermion Propagator . 38 3.3 New Variant of Fradkin Representation . 41 3.3.1 Thermal Green's Functions in a Mixed Representation . 41 3.3.2 Free-Field Limit . 46 3.3.3 Bloch-Nordsieck Approximation . 48 3.4 Dressed Propagator in Mixed Formalisms . 49 3.4.1 Approximation for Closed-Fermion-Loop Functional . 49 3.4.2 Dressed Finite-Temperature Fermion Propagator . 51 3.4.3 Linkage Operations . 52 3.4.4 Dropping Spin-related Contributions . 53 3.5 Quenched Dressed Finite-Temperature Fermion Propagator . 54 3.5.1 Quenched Dressed Finite-Temperature Fermion Propagator . 54 3.5.2 Linkage with Real-time Photon Propagators . 55 3.5.3 Thermal-Photon assisted Damping . 56 3.5.4 Bremsstrahlung Processes as a Damping Mechanism . 61 3.5.5 Damping E®ects under Quenched Approximation . 66 3.6 Non-Quenched Full Finite-Temperature Propagator . 71 3.6.1 Thermal Closed-Fermion-Loop and the Photon Polarization Tensor . 71 3.6.2 Pair-Productions as a Damping Mechanism . 73 4 Discussion and Perspectives 81 4.1 Model Approximation and Damping Mechanisms . 81 4.2 Damping E®ects . 83 4.3 Comparison to Perturbative Theory . 85 4.4 Longitudinal and Transverse Disturbance in the Medium . 87 4.5 Mass Shift . 91 4.6 Impact of Gauge . 96 4.7 Non-Zero Chemical Potential . 97 4.8 Hot Thermal Loop Approximation Revisited . 99 vii 4.9 Possible Extension to QCD . 101 5 Conclusions 104 A Units and Metric 107 A.1 Natural Units . 107 A.2 Metric . 107 A.3 Gordon decomposition . 109 B Matsubara Summation 110 B.1 Standard Contour Integral . 110 B.2 Saclay Method . 111 B.3 Mixed Representation of Finite-Temperature Propagator . 111 C Gauge 114 C.1 Gauge Conditions . 114 C.2 Photon Propagator and Gauge Parameter . 115 C.3 Current Conservation and Gauge Conditions in QED . 116 C.4 Gauge Structure of Green's Function(al) . 117 C.5 Gauge Structure of Closed-Fermion-Loop Functional . 119 D Reviews of Functional Methods 120 D.1 Functional Di®erentiation . 120 D.2 Functional Integration . 121 D.3 Functional Di®erentiation vs Functional Integral . 122 D.4 Two Useful Relations . 122 D.5 Functional Form of Unity . 123 D.6 Linkage Operation . 123 E Misc Relations 125 E.1 Useful Relations in Fradkin's Representation . 125 E.2 Representations of Delta- and Heaviside Step- Function . 125 E.3 Representation of h Function . 126 E.4 Operator Relations . 127 E.5 Legendre Function of the Second Kind . 127 E.6 Abel's Trick . 128 E.7 Bogoliubov Transformation . 128 F Calculations in Full Imaginary-Time Formalism 130 G Calculations in Feynman Gauge 133 viii Bibliography 134 ? Parts of this thesis are expected to be published in the May 2008 issue of Phys. Rev. D with H. M. Fried at Brown University and T. Grandou at Institut Non-Lin¶eairede Nice Sophia-Antipolis, UMR-CNRS 6618, and can also be found in arXiv e-print server at arXiv:0804.1591v1 [hep-th]. ix List of Tables A.1 Dimension of physical quantity in natural units.
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