Introduction to quantum field theory with applications to quantum gravity:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Oxford
Oxford University Press
2021
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| Edition: | First edition |
| Series: | Oxford graduate texts
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | x, 525 Seiten Diagramme |
| ISBN: | 9780198838319 9780198872344 |
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| 245 | 1 | 0 | |a Introduction to quantum field theory with applications to quantum gravity |c Iosif L. Buchbinder, Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia, Ilya L. Shapiro, Departamento de Física - Instituto Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330,MG, Brazil |
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| adam_text | Contents PART I INTRODUCTION TO QUANTUM FIELD THEORY 1 Introduction 1.1 What is quantum field theory, and some preliminary notes 1.2 The notion of a quantized field 1.3 Natural units, notations and conventions 2 Relativistic symmetry 2.1 Lorentz transformations 2.2 Basic notions of group theory 2.3 The Lorentz and Poincaré groups 2.4 Tensor representation 2.5 Spinor representation 2.6 Irreducible representations of the Poincaré group 8 8 11 15 17 20 26 3 Lagrange formalism in field theory 3.1 The principle of least action, and the equations of motion 3.2 Global symmetries 3.3 Noether’s theorem 3.4 The energy-momentum tensor 31 31 34 36 39 4 Field models 4.1 Basic assumptions about the structure of Lagrangians 4.2 Scalar field models 4.3 Spinor field models 4.4 Models of free vector fields 4.5 Scalar and spinor filelds interacting with an electromagnetic field 4.6 The Yang-Mills field 42 42 43 46 52 55 59 5 Canonical quantization of free fields 5.1 Principles of canonical quantization 5.2 Canonical quantization in field theory 5.3 Canonical quantization of a free real scalar field 5.4 Canonical quantization of a free complex scalar field 5.5 Quantization of a free spinor field 5.6 Quantization of a free electromagnetic field 70 70 74 77 82 85 90 6 The 6.1 6.2 6.3 scattering matrix and the Green functions Particle interactions and asymptotic states Reduction of the ճ-matrix to Green functions Generating functionals of Green functions and the S-matrix 3 3 3 5 99 99 104 109
viii Contents 6.4 The ճ-matrix and the Green functions for spinor fields 112 7 Funcţiona! integrals 7.1 Representation of the evolution operator by a functional integral 7.2 Functional representation of Green functions 7.3 Functional representation of generating functionals 7.4 Functional integrals for fermionic theories 7.5 Perturbative calculation of generating functionals 7.6 Properties of functional integrals 7.7 Techniques for calculating functional determinants 117 117 123 127 128 134 137 143 8 Perturbation theory 8.1 Perturbation theory in terms of Feynman diagrams 8.2 Feynman diagrams in momentum space 8.3 Feynman diagrams for the ճ-matrix 8.4 Connected Green functions 8.5 Effective action 8.6 Loop expansion 8.7 Feynman diagrams in theories with spinor fields 147 147 151 154 155 158 162 166 Θ Renormalization 9.1 The general idea of renormalization 9.2 Regularization of Feynman diagrams 9.3 The subtraction procedure 9.4 The superficial degree of divergences 9.5 Renormalizable and non-renormalizable theories 9.6. The arbitrariness of the subtraction procedure 9.7 Renormalization conditions 9.8 Renormalization with the dimensional regularization 9.9 Renormalization group equations 171 171 172 178 194 197 204 205 209 212 10 Quantum gauge theories 10.1 Basic notions of Yang-Mills gauge theory 10.2 Gauge invariance and observables 10.3 Functional integral for gauge theories 10.4 BRST symmetry 10.5 Ward identities 10.6 The gauge dependence of effective action 10.7 Background field method 10.8 Feynman diagrams in Yang-Mills theory 10.9 The background field method for Yang-
Mills theory lO.lORenormalization of Yang-Mills theory 223 223 226 228 235 236 247 249 252 254 257 PART II SEMICLASSICAL AND QUANTUM GRAVITY MODELS 11 A brief review of genera! relativity 11.1 Basic principles of general relativity 11.2 Covariant derivative and affine connection 271 271 272
Contents ix 11.3 11.4 11.5 11.6 11.7 11.8 The curvature tensor and its properties The covariant equation for a free particle: the classical limit Classical action for the gravity field Einstein equations and the Newton limit Some physically relevant solutions and singularities The applicability of GR and Planck units 274 277 280 283 284 289 12 Classical fields in curved spacetime 12.1 General considerations 12.2 Scalar fields 12.3 Spontaneous symmetry breaking in curved space and induced gravity 12.4 Spinor fields in curved space 12.5 Massless vector (gauge) fields 12.6 Interactions between scalar, fermion and gauge fields 300 304 312 313 13 Quantum fields in curved spacetime: renormalization 13.1 Effective action in curved spacetime 13.2 Divergences and renormalization in curved space 13.3 Covariant methods: local momentum representation 13.4 The heat-kernel technique, and one-loop divergences 314 314 321 328 342 14 One-loop divergences 14.1 One-loop divergences in the vacuum sector 14.2 Beta functions in the vacuum sector 14.3 One-loop divergences in interacting theories 363 363 372 374 15 The 15.1 15.2 15.3 388 388 391 395 renormalization group in curved space The renormalization group based on minimal subtractions The effective potential from a renormalization group The global conformal (scaling) anomaly 294 294 295 16 Non-local form factors in flat and curved spacetime 16.1 Non-local form factors: simple example 16.2 Non-local form factors in curved spacetime 16.3 The massless limit and leading logs vs. the infrared limit 397 397 407 412 17 The 17.1 17.2 17.3 414 414
416 420 conformal anomaly and anomaly-induced action Conformal transformations and invariants Derivation of the conformal anomaly Anomaly-induced action 18 General notions of perturbative quantum gravity 18.1 Symmetries of the classical gravitational models 18.2 Choice of the action and gauge fixing for quantum gravity 18.3 Bilinear forms and linear approximation 18.4 Propagators of quantum metric and Barnes-Rivers projectors 18.5 Gravitational waves, quantization and gravitons 425 425 427 434 438 443
x Contents 18.6 Gauge-invariant renormalization in quantum gravity 18.7 Power counting, and classification of quantum gravity models 446 447 19 Massive ghosts in higher-derivative models 19.1 How to meet a massive ghost 19.2 The dangers of having a ghost or a tachyon 19.3 Massive ghosts in polynomial models 19.4 Complex poles and the unitarity of the 5-matrix 19.5 Ghosts in nonlocal models 19.6 Effective approach to the problem of ghosts 19.7 Stable solutions in the presence of massive ghosts 455 455 457 460 462 463 465 467 20 One-loop renormalization in quantum gravity 20.1 Preliminary considerations 20.2 Gauge-fixing dependence in quantum GR 20.3 Gauge-fixing dependence in higher-derivative models 20.4 One-loop divergences in quantum general relativity 20.5 One-loop divergences in a fourth-derivative model 20.6 One-loop divergences in superrenormalizable models 474 474 475 476 478 481 486 21 The 21.1 21.2 21.3 renormalization group in perturbative quantum gravity On-shell renormalization group in quantum GR The renormalization group in fourth-derivative gravity The renormalization group in superrenormalizable models 488 489 490 492 22 The 22.1 22.2 22.3 induced gravity approach Gravity induced from the cut-off Gravity induced from phase transitions Once again on the cosmological constant problem 494 495 498 500 23 Final remarks on Part II 503 References 504 Index 522
Applications of quantum field theoretical methods to gravitational physics, in both the semidassical and the full quantum frameworks, require a careful formulation of the fundamental basis of quantum theory, with special attention to such important issues as renormalization, the quantum theory of gauge theories and, especially, effective action formalism. The first part of this graduate textbook provides both a conceptual and a technical introduction to the theory of quantum fields. The presentation is consistent, starting from elements of group theory and classical fields and then moving on to effective action formalism in general gauge theories. Compared to other books on these topics, this book describes the general formalism of renormalization in more detail, with spe cial attention paid to gauge theories. Part I of this book can serve as a textbook for a one-semester introductory course in quantum field theory, In Part II, we discuss the basic aspects of quantum field theory in curved space, and per turbative quantum gravity. More than half of this part is written with a full exposition of details and includes elaborated examples of the simplest calculations. All chapters include exercises ranging from very simple ones to those requiring small original investigations. The selection of material in the second part was done using the must-know principle: this means we included detailed expo sitions of relatively simple techniques and calculations, expecting that the interested reader will be able to learn more advanced issues independently after working through the
basic material and completing the exercises. It is a good moment for summarizing the various advances, and these two authors are among the best experts in the specific field of quantum effective actions in gravity. Michele Maggiore, University of Geneva The subject of the book is timely, especially since many results of quantum field theory are actively used in modern cosmology, and both authors have long-time experience of teaching courses on the subject at several universities. Valeri Frolov, University of Alberta OXFORD UNIVERSITY PRESS www.oup.com ISBN 978-0-19-883831-9 9780198838319
|
| adam_txt |
Contents PART I INTRODUCTION TO QUANTUM FIELD THEORY 1 Introduction 1.1 What is quantum field theory, and some preliminary notes 1.2 The notion of a quantized field 1.3 Natural units, notations and conventions 2 Relativistic symmetry 2.1 Lorentz transformations 2.2 Basic notions of group theory 2.3 The Lorentz and Poincaré groups 2.4 Tensor representation 2.5 Spinor representation 2.6 Irreducible representations of the Poincaré group 8 8 11 15 17 20 26 3 Lagrange formalism in field theory 3.1 The principle of least action, and the equations of motion 3.2 Global symmetries 3.3 Noether’s theorem 3.4 The energy-momentum tensor 31 31 34 36 39 4 Field models 4.1 Basic assumptions about the structure of Lagrangians 4.2 Scalar field models 4.3 Spinor field models 4.4 Models of free vector fields 4.5 Scalar and spinor filelds interacting with an electromagnetic field 4.6 The Yang-Mills field 42 42 43 46 52 55 59 5 Canonical quantization of free fields 5.1 Principles of canonical quantization 5.2 Canonical quantization in field theory 5.3 Canonical quantization of a free real scalar field 5.4 Canonical quantization of a free complex scalar field 5.5 Quantization of a free spinor field 5.6 Quantization of a free electromagnetic field 70 70 74 77 82 85 90 6 The 6.1 6.2 6.3 scattering matrix and the Green functions Particle interactions and asymptotic states Reduction of the ճ-matrix to Green functions Generating functionals of Green functions and the S-matrix 3 3 3 5 99 99 104 109
viii Contents 6.4 The ճ-matrix and the Green functions for spinor fields 112 7 Funcţiona! integrals 7.1 Representation of the evolution operator by a functional integral 7.2 Functional representation of Green functions 7.3 Functional representation of generating functionals 7.4 Functional integrals for fermionic theories 7.5 Perturbative calculation of generating functionals 7.6 Properties of functional integrals 7.7 Techniques for calculating functional determinants 117 117 123 127 128 134 137 143 8 Perturbation theory 8.1 Perturbation theory in terms of Feynman diagrams 8.2 Feynman diagrams in momentum space 8.3 Feynman diagrams for the ճ-matrix 8.4 Connected Green functions 8.5 Effective action 8.6 Loop expansion 8.7 Feynman diagrams in theories with spinor fields 147 147 151 154 155 158 162 166 Θ Renormalization 9.1 The general idea of renormalization 9.2 Regularization of Feynman diagrams 9.3 The subtraction procedure 9.4 The superficial degree of divergences 9.5 Renormalizable and non-renormalizable theories 9.6. The arbitrariness of the subtraction procedure 9.7 Renormalization conditions 9.8 Renormalization with the dimensional regularization 9.9 Renormalization group equations 171 171 172 178 194 197 204 205 209 212 10 Quantum gauge theories 10.1 Basic notions of Yang-Mills gauge theory 10.2 Gauge invariance and observables 10.3 Functional integral for gauge theories 10.4 BRST symmetry 10.5 Ward identities 10.6 The gauge dependence of effective action 10.7 Background field method 10.8 Feynman diagrams in Yang-Mills theory 10.9 The background field method for Yang-
Mills theory lO.lORenormalization of Yang-Mills theory 223 223 226 228 235 236 247 249 252 254 257 PART II SEMICLASSICAL AND QUANTUM GRAVITY MODELS 11 A brief review of genera! relativity 11.1 Basic principles of general relativity 11.2 Covariant derivative and affine connection 271 271 272
Contents ix 11.3 11.4 11.5 11.6 11.7 11.8 The curvature tensor and its properties The covariant equation for a free particle: the classical limit Classical action for the gravity field Einstein equations and the Newton limit Some physically relevant solutions and singularities The applicability of GR and Planck units 274 277 280 283 284 289 12 Classical fields in curved spacetime 12.1 General considerations 12.2 Scalar fields 12.3 Spontaneous symmetry breaking in curved space and induced gravity 12.4 Spinor fields in curved space 12.5 Massless vector (gauge) fields 12.6 Interactions between scalar, fermion and gauge fields 300 304 312 313 13 Quantum fields in curved spacetime: renormalization 13.1 Effective action in curved spacetime 13.2 Divergences and renormalization in curved space 13.3 Covariant methods: local momentum representation 13.4 The heat-kernel technique, and one-loop divergences 314 314 321 328 342 14 One-loop divergences 14.1 One-loop divergences in the vacuum sector 14.2 Beta functions in the vacuum sector 14.3 One-loop divergences in interacting theories 363 363 372 374 15 The 15.1 15.2 15.3 388 388 391 395 renormalization group in curved space The renormalization group based on minimal subtractions The effective potential from a renormalization group The global conformal (scaling) anomaly 294 294 295 16 Non-local form factors in flat and curved spacetime 16.1 Non-local form factors: simple example 16.2 Non-local form factors in curved spacetime 16.3 The massless limit and leading logs vs. the infrared limit 397 397 407 412 17 The 17.1 17.2 17.3 414 414
416 420 conformal anomaly and anomaly-induced action Conformal transformations and invariants Derivation of the conformal anomaly Anomaly-induced action 18 General notions of perturbative quantum gravity 18.1 Symmetries of the classical gravitational models 18.2 Choice of the action and gauge fixing for quantum gravity 18.3 Bilinear forms and linear approximation 18.4 Propagators of quantum metric and Barnes-Rivers projectors 18.5 Gravitational waves, quantization and gravitons 425 425 427 434 438 443
x Contents 18.6 Gauge-invariant renormalization in quantum gravity 18.7 Power counting, and classification of quantum gravity models 446 447 19 Massive ghosts in higher-derivative models 19.1 How to meet a massive ghost 19.2 The dangers of having a ghost or a tachyon 19.3 Massive ghosts in polynomial models 19.4 Complex poles and the unitarity of the 5-matrix 19.5 Ghosts in nonlocal models 19.6 Effective approach to the problem of ghosts 19.7 Stable solutions in the presence of massive ghosts 455 455 457 460 462 463 465 467 20 One-loop renormalization in quantum gravity 20.1 Preliminary considerations 20.2 Gauge-fixing dependence in quantum GR 20.3 Gauge-fixing dependence in higher-derivative models 20.4 One-loop divergences in quantum general relativity 20.5 One-loop divergences in a fourth-derivative model 20.6 One-loop divergences in superrenormalizable models 474 474 475 476 478 481 486 21 The 21.1 21.2 21.3 renormalization group in perturbative quantum gravity On-shell renormalization group in quantum GR The renormalization group in fourth-derivative gravity The renormalization group in superrenormalizable models 488 489 490 492 22 The 22.1 22.2 22.3 induced gravity approach Gravity induced from the cut-off Gravity induced from phase transitions Once again on the cosmological constant problem 494 495 498 500 23 Final remarks on Part II 503 References 504 Index 522
Applications of quantum field theoretical methods to gravitational physics, in both the semidassical and the full quantum frameworks, require a careful formulation of the fundamental basis of quantum theory, with special attention to such important issues as renormalization, the quantum theory of gauge theories and, especially, effective action formalism. The first part of this graduate textbook provides both a conceptual and a technical introduction to the theory of quantum fields. The presentation is consistent, starting from elements of group theory and classical fields and then moving on to effective action formalism in general gauge theories. Compared to other books on these topics, this book describes the general formalism of renormalization in more detail, with spe cial attention paid to gauge theories. Part I of this book can serve as a textbook for a one-semester introductory course in quantum field theory, In Part II, we discuss the basic aspects of quantum field theory in curved space, and per turbative quantum gravity. More than half of this part is written with a full exposition of details and includes elaborated examples of the simplest calculations. All chapters include exercises ranging from very simple ones to those requiring small original investigations. The selection of material in the second part was done using the "must-know" principle: this means we included detailed expo sitions of relatively simple techniques and calculations, expecting that the interested reader will be able to learn more advanced issues independently after working through the
basic material and completing the exercises. "It is a good moment for summarizing the various advances, and these two authors are among the best experts in the specific field of quantum effective actions in gravity." Michele Maggiore, University of Geneva "The subject of the book is timely, especially since many results of quantum field theory are actively used in modern cosmology, and both authors have long-time experience of teaching courses on the subject at several universities." Valeri Frolov, University of Alberta OXFORD UNIVERSITY PRESS www.oup.com ISBN 978-0-19-883831-9 9780198838319 |
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| record_format | marc |
| series2 | Oxford graduate texts |
| spelling | Buchbinder, Iosif L. Verfasser (DE-588)1229205225 aut Introduction to quantum field theory with applications to quantum gravity Iosif L. Buchbinder, Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia, Ilya L. Shapiro, Departamento de Física - Instituto Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330,MG, Brazil First edition Oxford Oxford University Press 2021 © 2021 x, 525 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts Gekrümmte Raumzeit (DE-588)4322953-0 gnd rswk-swf Quantengravitation (DE-588)4124012-1 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 s Gekrümmte Raumzeit (DE-588)4322953-0 s DE-604 Quantengravitation (DE-588)4124012-1 s Shapiro, Ilya L. Verfasser (DE-588)104184493X aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032527566&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032527566&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Buchbinder, Iosif L. Shapiro, Ilya L. Introduction to quantum field theory with applications to quantum gravity Gekrümmte Raumzeit (DE-588)4322953-0 gnd Quantengravitation (DE-588)4124012-1 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4322953-0 (DE-588)4124012-1 (DE-588)4047984-5 |
| title | Introduction to quantum field theory with applications to quantum gravity |
| title_auth | Introduction to quantum field theory with applications to quantum gravity |
| title_exact_search | Introduction to quantum field theory with applications to quantum gravity |
| title_exact_search_txtP | Introduction to quantum field theory with applications to quantum gravity |
| title_full | Introduction to quantum field theory with applications to quantum gravity Iosif L. Buchbinder, Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia, Ilya L. Shapiro, Departamento de Física - Instituto Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330,MG, Brazil |
| title_fullStr | Introduction to quantum field theory with applications to quantum gravity Iosif L. Buchbinder, Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia, Ilya L. Shapiro, Departamento de Física - Instituto Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330,MG, Brazil |
| title_full_unstemmed | Introduction to quantum field theory with applications to quantum gravity Iosif L. Buchbinder, Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk, 634061, Russia, Ilya L. Shapiro, Departamento de Física - Instituto Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, CEP 36036-330,MG, Brazil |
| title_short | Introduction to quantum field theory with applications to quantum gravity |
| title_sort | introduction to quantum field theory with applications to quantum gravity |
| topic | Gekrümmte Raumzeit (DE-588)4322953-0 gnd Quantengravitation (DE-588)4124012-1 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Gekrümmte Raumzeit Quantengravitation Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032527566&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032527566&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT buchbinderiosifl introductiontoquantumfieldtheorywithapplicationstoquantumgravity AT shapiroilyal introductiontoquantumfieldtheorywithapplicationstoquantumgravity |