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

Abstract

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 Poincaré-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-called ADM laws 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).

The emergent gravity’s entropy Ansatz from an augmented variational principle

“…In Section 3 we shall show the relation between the normal unit vectors n, the normal vector ζ (used in AVP) and Killing vectors ξ. Such a relation is not necessary (since Killing vectors do not play a special role in conservation laws for covariant theories; [13]). However, it is relevant to render explicit the relations with Wald and Iyer results.…”

Covariant differential identities and conservation laws in metric-torsion theories of gravitation. I. General consideration

“…In the most of the present-day works related to analyzing general gauge theories (see, e.g., works by Julia and Silva 32,33 95,96,[110][111][112] ), unfortunately, authors frequently use rarely known formalisms for physicists, such as the aforementioned variational bi-complex, jet stratification, and also differential form technique, etc. Unlike them, we perform all the calculations and present the final results in the usual tensorial language.…”

Quantum cosmology for non-minimally coupled scalar field in FLRW space–time: A symmetry analysis

“…In particular, symmetries of dynamical systems are related to their first integrals for the physical quantities which do not evolve with the system and are related to the fundamental physical quantities, namely energy, linear momentum, angular momentum, etc. [39], [40]. Further, in the context of quantum cosmology, Noether symmetries help one to obtain a typical subset of the general solution of the Wheeler-DeWitt equation having oscillating behaviors [41], [42], [28].…”