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The Conceptual Framework of Quantum Field Theory$
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Anthony Duncan

Print publication date: 2012

Print ISBN-13: 9780199573264

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199573264.001.0001

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Dynamics VIII: Interacting fields: perturbative aspects

Dynamics VIII: Interacting fields: perturbative aspects

(p.307) 10 Dynamics VIII: Interacting fields: perturbative aspects
The Conceptual Framework of Quantum Field Theory

Anthony Duncan

Oxford University Press

This chapter discusses the perturbative aspects of interacting field theory: namely, the techniques appropriate for studying those aspects of local quantum field theories which emerge from a formal asymptotic expansion in some parameter of the theory, both from an operatorial as well as a path-integral point of view. Perturbative expansions are shown to have a natural interpretation in terms of graphical objects (Feynman graphs), with simple rules (Feynman rules) allowing the evaluation of the amplitudes in terms of elementary algebraic expressions associated with each graphical element (line, vertex, etc.). This is done first using operatorial methods: the matrix elements of Heisenberg picture operators needed for the LSZ formula are expanded using the Gell–Mann–Low theorem, and the resultant expressions evaluated using Wick's theorem. The resultant graphical objects arise naturally in a path-integral formulation of the field theory. Finally, the significance of Haag's theorem is discussed to demonstrate the validity of results obtained in perturbation theory via interaction-picture methods.

Keywords:   quantum field theory, perturbation, interacting field theory, Feynman graphs, Gell–Mann–Low theorem, Wick's theorem, Haag's theorem

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