%0 Journal Article
%T Noether Symmetries and Covariant Conservation Laws in Classical, Relativistic and Quantum Physics
%A Lorenzo Fatibene
%A Mauro Francaviglia
%A Silvio Mercadante
%J Symmetry
%D 2010
%I MDPI AG
%R 10.3390/sym2020970
%X 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¡äe¨CCartan 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 ¡°¨¤ 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¨CImmirzi connections).
%K Noether symmetries
%K (gauge)-natural theories
%U http://www.mdpi.com/2073-8994/2/2/970