Green's Functions and Ordered Exponentials

by H. M. Fried

• ISBN13: 9780521448628
• ISBN10: 052144862X
• Format: Paperback
• Copyright: 2005-08-22
• Publisher: Cambridge University Press

Summary

This book presents a functional approach to the construction, use and approximation of Green's functions and their associated ordered exponentials. After a brief historical introduction, the author discusses new solutions to problems involving particle production in crossed laser fields and non-constant electric fields. Applications to problems in potential theory and quantum field theory are covered, along with approximations for the treatment of color fluctuations in high-energy QCD scattering, and a model for summing classes of eikonal graphs in high-energy scattering problems. The book also presents a variant of the Fradkin representation which suggests a new non-perturbative approximation scheme, and provides a qualitative measure of the error involved in each such approximation. Covering the basics as well as more advanced applications, this book is suitable for graduate students and researchers in a wide range of fields, including quantum field theory, fluid dynamics and applied mathematics.

Table of Contents

Preface	ix
List of abbreviations	xi
1 Introduction	1	(16)
1.1 Historical remarks	1	(2)
1.2 Linear Physics	3	(10)
1.3 Ordered exponentials	13	(3)
Notes	16	(1)
2 Elementary functional methods	17	(16)
2.1 Functional differentiation	17	(1)
2.2 Linear translation	18	(2)
2.3 Quadratic (Gaussian) translation	20	(3)
2.4 Functional integration	23	(4)
2.5 Examples drawn from quantum field theory	27	(3)
2.6 Cluster decomposition	30	(2)
Notes	32	(1)
3 Schwinger–Fradkin methods	33	(18)
3.1 Proper-time representations of Schwinger and Fradkin	33	(4)
3.2 Fradkin representations for QED and QCD	37	(3)
3.3 Gauge structure in QED and QCD	40	(3)
3.4 Soluble examples: quadratic forms and perturbative approximations	43	(2)
3.5 Pair production in generalized electric fields	45	(5)
Notes	50	(1)
4 Lasers and crossed lasers	51	(24)
4.1 Classical charged-particle propagation in a laser (epw) field	51	(3)
4.2 The "scalar" laser solution for Gc[A]	54	(2)
4.3 The QED laser solutions for Gc[A] and L[A]	56	(6)
4.4 Pair production via crossed lasers	62	(10)
Notes	72	(3)
5 Special variants of the Fradkin representation	75	(18)
5.1 Exact representations for scalar interactions	75	(7)
5.2 Finite-quadrature approximations	82	(5)
5.3 Exact and approximate vectorial interactions	87	(3)
5.4 The Stojkov variation	90	(2)
Notes	92	(1)
6 Quantum chaos and vectorial interactions	93	(14)
6.1 First-quantization chaos	93	(5)
6.2 Chaos suppression in second quantization	98	(3)
6.3 Fluctuation-induced chaos suppression	101	(5)
Notes	106	(1)
7 Infrared approximations	107	(18)
7.1 The Bloch–Nordsieck approximation	108	(2)
7.2 IR damping at large momentum transfers	110	(4)
7.3 Eikonal scattering amplitudes in particle physics	114	(5)
7.4 IR approximations and resealing corrections to non-linear ODEs	119	(4)
Notes	123	(2)
8 Models of high-energy. non-Abelian scattering	125	(24)
8.1 An Abelian separation	126	(2)
8.2 The quasi-Abelian limit	128	(5)
8.3 Loop, ladder and crossed-ladder approximations	133	(9)
8.4 Summing all the eikonal graphs	142	(5)
Notes	147	(2)
9 Unitary ordered exponentials	149	(19)
9.1 Algebraic and differential structure	149	(1)
9.2 The SU(2) adiabatic limit	150	(4)
9.3 The stochastic limit	154	(9)
9.4 Functional integration over the stochastic limit	163	(4)
Notes	167	(1)
Index	168