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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 VII: Interacting fields: general aspects

Dynamics VII: Interacting fields: general aspects

(p.240) 9 Dynamics VII: Interacting fields: general aspects
The Conceptual Framework of Quantum Field Theory

Anthony Duncan

Oxford University Press

This chapter deals with the intricacies of interacting field theories, emphasizing very general aspects common to all local quantum field theories (LQFTs). The basic concept here is that of the interpolating Heisenberg field in terms of which the dynamics of the theory is specified, but which may be connected in various ways to the actual physical particle states. At this point, a characteristic feature of LQFTs becomes apparent: namely, the absence of any preferred, one-to-one connection between particles and fields. The discussion of field theory in the Heisenberg picture is first carried out in an ‘heuristic’ fashion, ignoring some important mathematical fine points; and then from a rigorous axiomatic point of view, starting with the Wightman axioms (spectral and field); and proceeding, via the Haag–Ruelle formulation of scattering theory, to the asymptotic formalism of Lehmann, Symanzik, and Zimmermann. The latter is treated in some detail, as it is central to subsequent discussions of the nature of the state space of field theory. The chapter also discusses the spectral properties of field theory, and the connection between the internal dynamics as specified by the interpolating fields and the phenomenological content of the theory as encapsulated in the asymptotic particle states and the S-matrix.

Keywords:   local quantum field theory, Heisenberg representation, Wightman axioms, Haag–Ruelle scattering theory, asymptotic formalism, Lehmann, Symanzik, and Zimmermann, spectral properties, particle states

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