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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Quantum Field Theory: Divergences and Regularization

Quantum Field Theory: Divergences and Regularization

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter considers specifically local relativistic field theories: the action is the integral of the lagrangian density that is a function of fields and their derivatives. It shows that as a consequence, infinities appear in perturbative calculations due to severe short distance singularities, or after Fourier transformation, to integrals diverging at large momenta: one speaks of UV divergences. These divergences are peculiar to local Quantum Field Theory: in contrast with Classical Mechanics or non-relativistic Quantum Mechanics with a finite number of particles, a straightforward construction of a quantum field theory of point-like objects is impossible. The chapter frst displays the problem, using the example of the φ3 field theory at one-loop order. It then systematically characterizes divergences by power counting: this leads to a classification of local field theories into three families: super-renormalizable, renormalizable, and non-renormalizable. It shows that divergences of correlation functions involving composite operators require a separate analysis.

Keywords:   local relativistic field theories, quantum field theory, super-renormalizable, renormalizable, non-renormalizable

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