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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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Relativistic Fermions

Relativistic Fermions

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter discusses boson field theories, occasionally considering theories with fermions, in particular, when the fermion nature of fields plays an essential role. It analyzes the free action for Dirac fermions, and explains the relation between fields and particles. It then derives an expression of the scattering matrix and discusses the non-relativistic limit of a model of self-coupled massive Dirac fermions. It introduces the formalism of euclidean relativistic fermions. In the euclidean formalism fermions transform under the fundamental representation of the spin group Spin(d) associated with the SO(d) rotation group (spin 1/2 fermions for d = 4). Since Chapter 7 was already devoted to perturbation theory and functional methods, the chapter outlines here only the aspects that are specific to fermions. As for the scalar field theory, it first calculates the gaussian integral, which corresponds to a free field theory. Then adding a source term to the action, the generating functional of correlation functions are obtained. The functional integral corresponding to a general action with an interaction expandable in powers of the field can be expressed in terms of a series of gaussian integrals, which can be calculated for example with the help of Wick's theorem.

Keywords:   boson field theories, Dirac fermions, euclidean relativistic fermions, gaussian integral, functional integral

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