Path Integrals in Field Theory: An Introduction

ISBN-13: 9783540403821 / Angielski / Miękka / 2003 / 213 str.

Ulrich Mosel

Opis

This is an introductory book to path integral methods in ?eld theories. It is aimed at graduate students and physicists who need a working knowledge of ?eld theory and its methods for applications in hadron, particle and nuclear physics. While teaching ?eld theory courses over the years I have found that many books on ?eld theory present the path integral methods used in only a very recipe-like way. On the other hand, specialized books on path integrals often contain many more details than are actually needed by non-specialists. I hope that this book here ?lls the gap. It provides enough information to actually follow all the arguments necessary for ?eld theoretical developments without, however, elaborating on the method as such and its mathema- cal intricacies. This book is in a way a technical companion to Fields, Symmetries, and Quarks by the present author. The reader of this book should have some knowledge of the relativistic equations of motion of classical quantum theory, but no prior knowledge of ?eld theory is assumed. The material in this book can be covered in a o- semester course with 3 hrs/week. It has evolved in many years of teaching this subject. I am grateful to my students for many helpful questions and comments and, in particular, to Frank Froemel for his help in preparing the ?gures in this book."

Recenzje

From the reviews:

"Mosel's book, described as an introduction, is aimed at graduate students and research workers in particle physics. ... in about 200 pages the reader has the chance to learn in some detail about a most important area of modern physics. The subject is tough but the style is clear and pedagogic, results for the most part being derived explicitly. ... [It] is clearly the work of a man with considerable teaching experience and is recommended as a readable and helpful account of a rather non-trivial subject." (Lewis Ryder, Journal of Physics, Vol. 37(25), 2004)

"This book is an introduction to path integral methods in quantum theory. It is divided into three parts devoted correspondingly to nonrelativistic quantum theory, quantum field theory and gauge theory. ... in this book the author has achieved a reasonable compromise between compactness and profoundness. From one side, he avoided many technical details and concentrated on the main points of the path integral method. From the other side, the principal ideas and logical structure of the theory are made clear, contrary to the pragmatic, recipe-like style of the presentation pf path integral methods in many books on quantum field theory. The book is aimed at graduate students and physicists who need a working knowledge of field theory methods for applications in hadron, particle and nuclear physics." (Michael B. Mensky, Zentralblatt MATH, Vol. 1037(12), 2004)

"There is no shortage of books on field theory ... but in this book Ulrich Mosel aims to provide a gentler introduction, more accessible to beginning students. In this he is largely successful, the book is a valuable addition to the literature. ... The level of rigour is quite sufficient for the prospective readership. ... The book is clearly written, well laid out and generally student-friendly ... . It will certainly be valuable to students and researchers ... ." (Professor T. W. B. Kibble, Contemporary Physics, Vol. 46 (4), 2005)

"This short and concise textbook is intended as a primer on path integral formalism both in classical and quantum field theories ... . It is ideally suited as an intensive one-semester course, delivering the basics needed by readers to follow developments 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." (Revista Espanola de Fisica, Vol. 17 (6), 2003)

"This is an introductory book on path integral methods in field theories, aimed at graduate students and physicists with a working knowledge of field theory and its applications in particle and nuclear physics." (Lewis H. Ryder, Mathematical Reviews, Issue 2006 e)

Spis treści

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.

Blurb

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 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).