Abstract
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).
Highlights
Symmetries have acquired a central role in Physics
Theoretical Relativistic Cosmology is entirely based on the symmetry ansatz of homogeneity and isotropy
Most applications of Quantum Gravity to Cosmology are entirely based on symmetries since the approach in full generality is still hindered by massive technical difficulties
Summary
Symmetries have acquired a central role in Physics. In Theoretical Physics discrete symmetries encode most of the intriguing structure of the Standard Model for particles, in Chemistry they encode for spectroscopic and physical properties of molecules. Infinite dimensionality enters strategically to require new techniques to extend the procedures which are standard in a finite dimensional arena Another interesting setting where symmetries play their role is the framework of Lagrangian field theories, which are the current basis for any approach to fundamental interactions in Physics. This corresponds, loosely speaking, to consider a dynamical system with a continuous infinity of degrees of freedom. The continuity equation holding for the Noether current relates the changes of conserved quantities to the flows at the boundary of the region (something entering or escaping the region) and some residual at singularities This setting is suitable for physical interpretation; it was developed, e.g., to define electric charges in Electromagnetism. Hereafter we shall show some of the relations among symmetries and conservation laws in different areas, preferring a unifying languages that stress similarities
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