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Symmetry 2010, 2(2), 970-998; doi:10.3390/sym2020970

Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics

1,2,* , 1,2,3 and 1,2
1 Department of Mathematics, University of Torino, via C. Alberto 10, Torino, Italy 2 INFN, Sezione di Torino, Iniziativa Specifica Na12, Torino 10125, Italy 3 LCS, University of Calabria, Cosenza, Italy
* Author to whom correspondence should be addressed.
Received: 12 January 2010 / Revised: 13 April 2010 / Accepted: 21 April 2010 / Published: 29 April 2010
(This article belongs to the Special Issue Entropy, Order and Symmetry)
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We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called “Natural Theories” and “Gauge-Natural Theories” that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincar´e–Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as “generators of Noether symmetries”, energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-calledADMlaws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation “à la Palatini” and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero–Immirzi connections).
Keywords: Noether symmetries; (gauge)-natural theories Noether symmetries; (gauge)-natural theories
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Fatibene, L.; Francaviglia, M.; Mercadante, S. Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics. Symmetry 2010, 2, 970-998.

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