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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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Symmetries I: Continuous spacetime symmetry: why we need Lagrangians in field theory

Symmetries I: Continuous spacetime symmetry: why we need Lagrangians in field theory

Chapter:
(p.414) 12 Symmetries I: Continuous spacetime symmetry: why we need Lagrangians in field theory
Source:
The Conceptual Framework of Quantum Field Theory
Author(s):

Anthony Duncan

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199573264.003.0012

This chapter analyzes the peculiar and indispensable utility of the Lagrangian approach to dynamics in relativistic quantum field theories. It underscores the facility with which a Lagrangian approach incorporates the desired — in fact, indispensable — spacetime symmetries of a relativistic field theory. It develops the canonical formalism of Lagrangian field theory as the natural solution to the problem of generating Hamiltonian energy densities that lead to a quantum field theory with fully Lorentz-invariant dynamics. The general connection between symmetries and conservation laws, expressed in the form most natural to field theory (Noether's theorem) is demonstrated along with its application to the case of Poincaré symmetry, conformal symmetry, and global internal symmetries. The chapter concludes with an introduction of the extension of Poincaré symmetry to the super-Poincaré algebra of supersymmetry.

Keywords:   Lagrangian approach, relativistic quantum field theory, Hamiltonian, Lorentz-invariant field theory, Noether's theorem, Poincaré symmetry

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