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

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

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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