Feynman's Operational Calculus and Beyond: Noncommutativity and Time-Ordering

ISBN : 9780198702498

Gerald W. Johnson; Michel L. Lapidus; Lance Nielsen
384 ページ
165 x 236 mm

This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book. The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections with certain analytic Feynman integrals are noted. This volume is essentially self-contained and we only assume that the reader has a reasonable, graduate level, background in analysis, measure theory and functional analysis or operator theory. Much of the necessary remaining background is supplied in the text itself.


1. Introduction
2. Disentangling: Definitions, Properties and Elementary Examples
3. Disentangling via Tensor Products and Ordered Supports
4. Extraction of Multilinear Factors and Iterative Disentangling
5. Auxiliary Operations and Disentangling Algebras
6. Time-Dependent Feynman's Operational Calculus and Evolution Equations
7. Stability Properties of Feynman's Operational Calculi
8. Disentangling via Continuous and Discrete Measures
9. Derivational Derivatives and Feynman's Operational Calculi
10. Spectral Theory for Noncommuting Operators
11. Epilogue: Miscellaneous Topics and Possible Extensions


Gerald W. Johnson is Emeritus Professor of Mathematics University of Nebraska, Lincoln. ; Michel Lapidus is Professor of Mathematics at the University of California, Riverside; Lance Nielsen is Professor of Mathematics in the Department of Mathematics, Creighton University.