Classical and quantum dynamics of constrained Hamiltonian systems
This book is an introduction to the field of constrained Hamiltonian systems and their quantization, a topic which is of central interest to theoretical physicists who wish to obtain a deeper understanding of the quantization of gauge theories, such as describing the fundamental interactions in natu...
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| Main Authors: | , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
New Jersey
World Scientific Publishing Co. Pte. Ltd
2010
World Scientific World Scientific Publishing Company WORLD SCIENTIFIC World Scientific Publishing |
| Edition: | 1 |
| Series: | World Scientific lecture notes in physics |
| Subjects: | |
| ISBN: | 9814299642, 9789814299640, 9789814299657, 9814299650 |
| Online Access: | Get full text |
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Table of Contents:
- Classical and quantum dynamics of constrained Hamiltonian systems -- Preface -- Notation -- Contents -- Chapter 1: Introduction -- Chapter 2: Singular Lagrangians and Local Symmetries -- Chapter 3: Hamiltonian Approach. The Dirac Formalism -- Chapter 4: Symplectic Approach to Constrained Systems -- Chapter 5: Local Symmetries within the Dirac Formalism -- Chapter 6: The Dirac Conjecture -- Chapter 7: BFT Embedding of Second Class Systems -- Chapter 8: Hamilton-Jacobi Theory of Constrained Systems -- Chapter 9: Operator Quantization of Second Class Systems -- Chapter 10: Functional Quantization of Second Class Systems -- Chapter 11: Dynamical Gauges. BFV Functional Quantization -- Chapter 12: Field-Antifield Quantization -- Appendix A: Local Symmetries and Singular Lagrangians -- Appendix B: The BRST Charge of Rank One -- Appendix C: BRST Hamiltonian of Rank One -- Appendix D: The FV Principal Theorem -- Appendix E: BRST Quantization of SU(3) Yang-Mills Theory in α-gauges -- Bibliography -- Index
- Intro -- Contents -- Preface -- Notation -- 1 Introduction -- 2 Singular Lagrangians and Local Symmetries -- 2.1 Introduction -- 2.2 Singular Lagrangians -- 2.3 Algorithm for detecting local symmetries on Lagrangian level -- 2.4 Examples -- 2.5 Generator of gauge transformations and Noether identities -- 3 Hamiltonian Approach. The Dirac Formalism -- 3.1 Introduction -- 3.2 Primary constraints -- 3.3 The Hamilton equations of motion -- 3.3.1 Streamlining the Hamilton equations of motion -- 3.3.2 Alternative derivation of the Hamilton equations -- 3.3.3 Examples -- 3.4 Iterative procedure for generating the constraints -- 3.4.1 Particular algorithm for generating the constraints -- 3.5 First and second class constraints. Dirac brackets -- 4 Symplectic Approach to Constrained Systems -- 4.1 Introduction -- 4.2 The case fab singular -- 4.2.1 Example: particle on a hypersphere -- 4.3 Interpretation of W(L) and F -- 4.4 The Faddeev-Jackiw reduction -- 5 Local Symmetries within the Dirac Formalism -- 5.1 Introduction -- 5.2 Local symmetries and canonical transformations -- 5.3 Local symmetries of the Hamilton equations of motion -- 5.4 Local symmetries of the total and extended action -- 5.5 Local symmetries of the Lagrangian action -- 5.6 Solution of the recursive relations -- 5.7 Reparametrization invariant approach -- 6 The Dirac Conjecture -- 6.1 Introduction -- 6.2 Gauge identities and Dirac's conjecture -- 6.3 General system with two primaries and one secondary constraint -- 6.4 Counterexamples to Dirac's conjecture? -- 7 BFT Embedding of Second Class Systems -- 7.1 Introduction -- 7.2 Summary of the BFT-procedure -- 7.3 The BFT construction -- 7.4 Examples of BFT embedding -- 7.4.1 The multidimensional rotator -- 7.4.2 The Abelian self-dual model -- 7.4.3 Abelian self-dual model and Maxwell-Chern-Simons theory -- 7.4.4 The non-abelian SD model
- 8 Hamilton-Jacobi Theory of Constrained Systems -- 8.1 Introduction -- 8.1.1 Caratheodory's integrability conditions -- 8.1.2 Characteristic curves of the HJ-equations -- 8.2 HJ equations for first class systems -- 8.3 HJ equations for second class systems -- 8.3.1 HPF for reduced second class systems -- 8.3.2 Examples -- 8.3.3 HJ equations for second class systems via BFT embedding -- 8.3.4 Examples -- 9 Operator Quantization of Second Class Systems -- 9.1 Introduction -- 9.2 Systems with only second class constraints -- 9.3 Systems with first and second class constraints -- 9.3.1 Example: the free Maxwell field in the Coulomb gauge -- 9.3.2 Concluding remark -- 10 Functional Quantization of Second Class Systems -- 10.1 Introduction -- 10.2 Partition function for second class systems -- 11 Dynamical Gauges. BFV Functional Quantization -- 11.1 Introduction -- 11.2 Grassmann variables -- 11.3 BFV quantization of a quantum mechanical model -- 11.3.1 The gauge-fixed effective Lagrangian -- 11.3.2 The conserved BRST charge in configuration space -- 11.3.3 The gauge fixed effective Hamiltonian -- 11.3.4 The BRST charge in phase space -- 11.4 Quantization of Yang-Mills theory in the Lorentz gauge -- 11.5 Axiomatic BRST approach -- 11.5.1 The BRST charge and Hamiltonian for rank one theories -- 11.5.2 FV Principal Theorem -- 11.5.3 A large class of gauges -- 11.5.4 Connecting Z with the quantum partition function in a physical gauge. The SU(N) Yang-Mills theory -- 11.6 Equivalence of the SD and MCS models -- 11.7 The physical Hilbert space. Some remarks -- 12 Field-Antifield Quantization -- 12.1 Introduction -- 12.2 Axiomatic field-antifield formalism -- 12.3 Constructive proof of the field-antifield formalism for a restricted class of theories -- 12.3.1 From the FV-phase-space action to the Hamiltonian master equation
- 12.3.2 Transition to configuration space -- 12.4 The Lagrangian master equation -- 12.5 The quantum master equation -- 12.5.1 An alternative derivation of the quantum master equation -- 12.5.2 Gauge invariant correlation functions -- 12.6 Anomalous gauge theories. The chiral Schwinger model -- 12.6.1 Quantum Master equation and the anomaly -- A Local Symmetries and Singular Lagrangians -- A.1 Local symmetry transformations -- A.2 Bianchi identities and singular Lagrangians -- B The BRST Charge of Rank One -- C BRST Hamiltonian of Rank One -- D The FV Principal Theorem -- E BRST Quantization of SU(3) Yang-Mills Theory in -gauges -- Bibliography -- Index