This short and concise textbook is intended as a primer on path integral formalism both in classical and quantum field theories, although emphasis is on the latter. It is ideally suited as an intensive one-semester course, delivering the basics needed by readers to follow developments in field theory. Path Integrals in Field Theory seeks a path between recipe-like sketchy presentations and specialized monographs with unnecessary detail. It paves the way for both more rigorous studies in fundamental mathematical issues as well as for applications in hadron, particle and nuclear physics, thus addressing students in mathematical and theoretical physics alike. Assuming some background in relativistic quantum mechanics, it complements the author's monograph Fields, Symmetries, and Quarks (Springer, 1999).

Klappentext

Concise textbook intended as a primer on path integral formalism both in classical and quantum field theories, although emphasis is on the latter. It is ideally suited as an intensive one-semester course, delivering the basics needed by readers to follow developments in field theory. Path Integrals in Field Theory paves the way for both more rigorous studies in fundamental mathematical issues as well as for applications in hadron, particle and nuclear physics, thus addressing students in mathematical and theoretical physics alike. Assuming some background in relativistic quantum theory (but none in field theory), it complements the authors monograph Fields, Symmetries, and Quarks (Springer, 1999).

Inhalt

I Non-Relativistic Quantum Theory.- 1 The Path Integral in Quantum Theory.- 1.1 Propagator of the Schrödinger Equation.- 1.2 Propagator as Path Integral.- 1.3 Quadratic Hamiltonians.- 1.3.1 Cartesian Metric.- 1.3.2 Non-Cartesian Metric.- 1.4 Classical Interpretation.- 2 Perturbation Theory.- 2.1 Free Propagator.- 2.2 Perturbative Expansion.- 2.3 Application to Scattering.- 3 Generating Functionals.- 3.1 Groundstate-to-Groundstate Transitions.- 3.1.1 Generating Functional.- 3.2 Functional Derivatives of Gs-Gs Transition Amplitudes.- II Relativistic Quantum Field Theory.- 4 Relativistic Fields.- 4.1 Equations of Motion.- 4.1.1 Examples.- 4.2 Symmetries and Conservation Laws.- 4.2.1 Geometrical Space-Time Symmetries.- 4.2.2 Internal Symmetries.- 5 Path Integrals for Scalar Fields.- 5.1 Generating Functional for Fields.- 5.1.1 Euclidean Representation.- 6 Evaluation of Path Integrals.- 6.1 Free Scalar Fields.- 6.1.1 Generating Functional.- 6.1.2 Feynman Propagator.- 6.1.3 Gaussian Integration.- 6.2 Interacting Scalar Fields.- 6.2.1 Stationary Phase Approximation.- 6.2.2 Numerical Evaluation of Path Integrals.- 6.2.3 Real Time Formalism.- 7 Transition Rates and Green's Functions.- 7.1 Scattering Matrix.- 7.2 Reduction Theorem.- 7.2.1 Canonical Field Quantization.- 7.2.2 Derivation of the Reduction Theorem.- 8 Green's Functions.- 8.1 n-point Green's Functions.- 8.1.1 Momentum Representation.- 8.1.2 Operator Representations.- 8.2 Free Scalar Fields.- 8.2.1 Wick's Theorem.- 8.2.2 Feynman Rules.- 8.3 Interacting Scalar Fields.- 8.3.1 Perturbative Expansion.- 9 Perturbative ?4 Theory.- 9.1 Perturbative Expansion of the Generating Function.- 9.1.1 Generating Functional up to O(g).- 9.2 Two-Point Function.- 9.2.1 Terms up to O(g0).- 9.2.2 Terms up to O(g).- 9.2.3 Terms up to O(g2).- 9.3 Four-Point Function.- 9.3.1 Terms up to O(g).- 9.3.2 Terms up to O(g2).- 9.4 Divergences in n-Point Functions.- 9.4.1 Power Counting.- 9.4.2 Dimensional Regularization of ?4 Theory.- 9.4.3 Renormalization.- 10 Green's Functions for Fermions.- 10.1 Grassmann Algebra.- 10.1.1 Derivatives.- 10.1.2 Integration.- 10.2 Green's Functions for Fermions.- 10.2.1 Generating Functional for Fermions.- 10.2.2 Reduction Theorem for Fermions.- 10.2.3 Green's Functions.- 11 Interacting Fields.- 11.1 Feynman Rules.- 11.1.1 Fermion Loops.- 11.2 Wick's Theorem.- 11.3 Bosonization of Yukawa Theory.- 11.3.1 Perturbative Expansion.- III Gauge Field Theory.- 12 Path Integrals for QED.- 12.1 Gauge Invariance in Abelian Free Field Theories.- 12.2 Generating Functional.- 12.3 Gauge Invariance in QED.- 12.4 Feynman Rules of QED.- 13 Path Integrals for Gauge Fields.- 13.1 Non-Abelian Gauge Fields.- 13.2 Generating Functional.- 13.3 Gauge Fixing of L.- 13.4 Faddeev-Popov Determinant.- 13.4.1 Explicit Forms of the FP Determinant.- 13.4.2 Ghost Fields.- 13.5 Feynman Rules.- 14 Examples for Gauge Field Theories.- 14.1 Quantum Chromodynamics.- 14.2 Electroweak Interactions.- Units and Metric.- A.1 Units.- A.2 Metric and Notation.- Functionals.- B.1 Definition.- B.2 Functional Integration.- B.2.1 Gaussian Integrals.- B.3 Functional Derivatives.- Renormalization Integrals.- Gaussian Grassmann Integration.- References.