Abstract
This chapter is intended to remind the basic notions of the Lagrangian and Hamiltonian formalisms as well as Noether’s theorem. We shall first start with a discrete system with N degrees of freedom, state and prove Noether’s theorem. Afterwards we shall generalize all the previously introduced notions to continuous systems and prove the generic formulation of Noether’s Theorem. Finally we will reproduce a few well known results in Quantum Field Theory.
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Notes
- 1.
This is not the most general case, of course, but as we are interested in applying field theory to Special Relativity we shall only restrict our study to this case.
- 2.
See Chap. 1 for details.
- 3.
See Chap. 1 for more details.
- 4.
See Chap. 5 for details on spinor algebra and for the proof of this statement.
- 5.
I am calling it pseudo tensor because it is obviously not invariant under translations!.
Further Reading
A. Pich, Class Notes on Quantum Field Theory. http://eeemaster.uv.es/course/view.php?id=6
W. Greiner, J. Reinhardt, D.A. Bromley (Foreword), Field Quantization
E.L. Hill, Hamilton’s principle and the conservation theorems of mathematical physics. Rev. Mod. Phys. 23, 253
J.A. de Azcárraga, J.M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics. Cambridge Monographs in Mathematical Physics
J.A. Oller, Mecnica Terica, http://www.um.es/oller/docencia/versionmteor.pdf
M. Kaksu, Quantum Field Theory: A Modern Introduction
M. Srednicki, Quantum Field Theory
D.E. Soper, Classical Field Theory
D.V. Galtsov, Iu.V. Grats, Ch. Zhukovski, Campos Clásicos
S. Noguera, Class Notes
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Ilisie, V. (2016). Lagrangians, Hamiltonians and Noether’s Theorem. In: Concepts in Quantum Field Theory. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22966-9_2
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DOI: https://doi.org/10.1007/978-3-319-22966-9_2
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