Book contents
- Frontmatter
- Contents
- PREFACE TO VOLUME II
- NOTATION
- 15 NON-ABELIAN GAUGE THEORIES
- 16 EXTERNAL FIELD METHODS
- 17 RENORMALIZATION OF GAUGE THEORIES
- 18 RENORMALIZATION GROUP METHODS
- 19 SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES
- 20 OPERATOR PRODUCT EXPANSIONS
- 21 SPONTANEOUSLY BROKEN GAUGE SYMMETRIES
- 22 ANOMALIES
- 23 EXTENDED FIELD CONFIGURATIONS
- AUTHOR INDEX
- SUBJECT INDEX
18 - RENORMALIZATION GROUP METHODS
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- PREFACE TO VOLUME II
- NOTATION
- 15 NON-ABELIAN GAUGE THEORIES
- 16 EXTERNAL FIELD METHODS
- 17 RENORMALIZATION OF GAUGE THEORIES
- 18 RENORMALIZATION GROUP METHODS
- 19 SPONTANEOUSLY BROKEN GLOBAL SYMMETRIES
- 20 OPERATOR PRODUCT EXPANSIONS
- 21 SPONTANEOUSLY BROKEN GAUGE SYMMETRIES
- 22 ANOMALIES
- 23 EXTENDED FIELD CONFIGURATIONS
- AUTHOR INDEX
- SUBJECT INDEX
Summary
The method of the renormalization group was originally introduced by Gell-Mann and Low as a means of dealing with the failure of perturbation theory at very high energies in quantum electrodynamics. An n-loop contribution to an amplitude involving momenta of order q, such as the vacuum polarization Πμν(q), is found to contain up to n factors of In() as well as a factor αn, so perturbation theory will break down when is large, even though the fine structure constant a is small. Even in a massless theory like a non-Abelian gauge theory we must introduce some scale μ to specify a renormalization point at which the renormalized coupling constants are to be defined, and in this case we encounter logarithms In(E/μ), so that perturbation theory may break down if E ≫ μ or E ≪ μ, even if the coupling constant is small.
Fortunately, there is a modified version of perturbation theory that can often be used in such cases. The key idea of this approach consists in the introduction of coupling constants gμ defined at a sliding renormalization scale μ — that is, a scale that is not related to particle masses in any fixed way. By then choosing μ to be of the same order of magnitude as the energy E that is typical of the process in question, the factors In(E/μ) are rendered harmless. We can then do perturbation theory as long as gμ remains small.
- Type
- Chapter
- Information
- The Quantum Theory of Fields , pp. 111 - 162Publisher: Cambridge University PressPrint publication year: 1996