*Gerald W Johnson, Michel L. Lapidus, and Lance Nielsen*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198702498
- eISBN:
- 9780191772160
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198702498.001.0001
- Subject:
- Mathematics, Mathematical Physics, Numerical Analysis

This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and ...
More

This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and time-ordering (or disentangling) rules in his seminal 1951 paper, as is made 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. In particular, the work presented in this volume is oriented towards dealing with abstract and (typically) noncommuting linear operators acting on some Banach space, rather than operators arising from some variety of path integration. Some of the key structures developed in this volume enable us to obtain, in some sense, an appropriate abstract substitute for a generalized functional integral associated with the Feynman operational calculus attached to a given n-tuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators Aj and probability measures μj, for j = 1, …, n and n ≥ 2.Less

This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and time-ordering (or *disentangling*) rules in his seminal 1951 paper, as is made 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. In particular, the work presented in this volume is oriented towards dealing with abstract and (typically) noncommuting linear operators acting on some Banach space, rather than operators arising from some variety of path integration. Some of the key structures developed in this volume enable us to obtain, in some sense, an appropriate abstract substitute for a generalized functional integral associated with the Feynman operational calculus attached to a given *n*-tuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators *A*_{j} and probability measures *μ*_{j}, for *j* = 1, …, *n* and *n* ≥ 2.

*Tom Lancaster and Stephen J. Blundell*

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0018
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter connects Green’s functions to propagators in a quantum field picture and introduce the Feynman propagator, which involves time-ordering of operators. This is applied to Yukawa’s model of ...
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This chapter connects Green’s functions to propagators in a quantum field picture and introduce the Feynman propagator, which involves time-ordering of operators. This is applied to Yukawa’s model of virtual particle exchange.Less

This chapter connects Green’s functions to propagators in a quantum field picture and introduce the Feynman propagator, which involves time-ordering of operators. This is applied to Yukawa’s model of virtual particle exchange.

*Gerald W. Johnson, Michel L. Lapidus, and Lance Nielsen*

- Published in print:
- 2015
- Published Online:
- September 2015
- ISBN:
- 9780198702498
- eISBN:
- 9780191772160
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198702498.003.0001
- Subject:
- Mathematics, Mathematical Physics, Numerical Analysis

The first part of the introduction provides a (short) historical overview of Feynman’s development of his operational calculus, i.e. the formation of functions of several not necessarily commuting ...
More

The first part of the introduction provides a (short) historical overview of Feynman’s development of his operational calculus, i.e. the formation of functions of several not necessarily commuting operators. This is done using quotes from Feynman’s Nobel lecture, as well as from biographical information and parts of Feynman’s early papers. This portion of the introduction will aid the reader not familiar with Feynman’s ideas to gain some insight into their intellectual and physical origins. After this historical sketch of the origins of the operational calculus, we take the time to introduce, via simple examples, Feynman’s time-ordering convention and Feynman’s heuristic “rules.” The remainder of the introduction is devoted to a rather detailed chapter-by-chapter description of the contents of this volume.Less

The first part of the introduction provides a (short) historical overview of Feynman’s development of his operational calculus, i.e. the formation of functions of several not necessarily commuting operators. This is done using quotes from Feynman’s Nobel lecture, as well as from biographical information and parts of Feynman’s early papers. This portion of the introduction will aid the reader not familiar with Feynman’s ideas to gain some insight into their intellectual and physical origins. After this historical sketch of the origins of the operational calculus, we take the time to introduce, via simple examples, Feynman’s time-ordering convention and Feynman’s heuristic “rules.” The remainder of the introduction is devoted to a rather detailed chapter-by-chapter description of the contents of this volume.