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Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Quantum field theory
- 3 Basic examples
- 4 Dimensional regularization
- 5 Renormalization
- 6 Composite operators
- 7 Renormalization group
- 8 Large-mass expansion
- 9 Global symmetries
- 10 Operator-product expansion
- 11 Coordinate space
- 12 Renormalization of gauge theories
- 13 Anomalies
- 14 Deep-inelastic scattering
- References
- Index
4 - Dimensional regularization
Published online by Cambridge University Press: 10 March 2010
- Frontmatter
- Contents
- 1 Introduction
- 2 Quantum field theory
- 3 Basic examples
- 4 Dimensional regularization
- 5 Renormalization
- 6 Composite operators
- 7 Renormalization group
- 8 Large-mass expansion
- 9 Global symmetries
- 10 Operator-product expansion
- 11 Coordinate space
- 12 Renormalization of gauge theories
- 13 Anomalies
- 14 Deep-inelastic scattering
- References
- Index
Summary
We have seen how convenient it is to regulate the UV divergences of perturbation theory by continuation in the dimension of space-time. To date, no-one has shown how to use the method in the complete theory. But in perturbation theory, as we will now demonstrate, it is consistent and well-defined. Now all results obtained by this method can be obtained by other, more physical methods (say, a lattice regulator). But frequently much more labor is involved. This is not a triviality, for in complicated situations, especially in gauge theories, it enables us to handle the technicalities of renormalization in a simple way.
The idea of dimensional continuation has been used for a long time in statistical mechanics (see, for example, Fisher & Gaunt (1964)). It became very prominent when Wilson & Fisher (1972) discovered the ε-expansion and applied it to field-theoretic methods in statistical mechanics (Wilson (1973), Mack (1972), and Wilson & Kogut (1974)). In the ε-expansion one works in 4 — ε spatial dimensions, and expands in powers of ε. At the same time, in a purely field-theoretic context, a need arose to find a way of regulating non-abelian gauge theories that preserved gauge invariance and Poincaré invariance. This led to dimensional regularization ('t Hooft & Veltman (1972a), Bollini & Giambiagi (1972), Cicuta & Montaldi (1972), and Ashmore (1972)).
- Type
- Chapter
- Information
- RenormalizationAn Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion, pp. 62 - 87Publisher: Cambridge University PressPrint publication year: 1984