Gauge invariance, charge conservation, and variational principles

Symmetry, 2010

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).

International Journal of Geometric Methods in Modern Physics, 2005

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities that measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well-known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms that have recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental physics are considered in detail, starting from classical mechanics.

2010

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).

Classical and Quantum Gravity, 2003

We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles). The contraction of the Euler-Lagrange equations with Lie derivatives of the dynamical fields allows one to derive a variational Lagrangian for any given set of Lagrangian equations. A two steps algorithmical procedure can be thence applied to the variational Lagrangian in order to produce a general expression for the variation of all quantities which are (covariantly) conserved along the given dynamics. As a concrete example we test this new formalism on Einstein's equations: well known and widely accepted formulae for the variation of the Hamiltonian and the variation of Energy for General Relativity are recovered. We also consider the Einstein-Cartan (Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new insight on the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.

International Journal of Modern Physics D, 2016

This review is dedicated to some modern applications of the remarkable paper written in 1918 by E. Noether. On a single paper, Noether discovered the crucial relation between symmetries and conserved charges as well as the impact of gauge symmetries on the equations of motion. Almost a century has gone since the publication of this work and its applications have permeated modern physics. Our focus will be on some examples that have appeared recently in the literature. This review aims at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charges (ii) local symmetries and gauge structure of a theory (iii) boundary conditions and algebra of asymptotic symmetries. All three topics are discussed through examples.

Fluid Dynamics Research, 2007

Following the gauge principle in the field theory of physics, a new variational formulation is presented for flows of an ideal fluid. In the present gauge-theoretical analysis, it is assumed that the field of fluid flow is characterized by a translation symmetry (group) and in addition that the fluid itself is a material in motion characterized thermodynamically by mass density and entropy (per unit mass). Local gauge transformation in the present case is local Galilean transformation (without rotation) which is a subgroup of a generalized local Galilean transformation group between non-inertial frames. In complying with the requirement of local gauge invariance of Lagrangians, a gauge-covariant derivative with respect to time is defined by introducing a gauge term. Galilean invariance requires that the covariant derivative should be the convective derivative, i.e. the so-called Lagrange derivative. Using this gauge-covariant operator, a free-field Lagrangian and Lagrangians associated with gauge fields are defined under the gauge symmetry. Euler's equation of motion is derived from the action principle. Simutaneously, the equation of continuity and equation of entropy conservation are derived from the variational principle. It is found that general solution thus obtained is equivalent to the classical Clebsch solution. If entropy of the fluid is non-uniform, the flow will be rotational. However, if the entropy is uniform throughout the space (i.e. homentropic), then the flow field reduces to that of a potential flow. Discussions are given on the issue. From the gauge invariance with respect to translational transformations, a differential conservation law of momentum is deduced as Noether's theorem.

Annales De La Fondation Louis De Broglie, 2005

"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The physically meaningful degrees of freedom then reemerge as being those invariant under a transformation connecting the variables (gauge transformation). Thus, one introduces extra variables to make the description more transparent and brings in at the same time a gauge symmetry to extract the physically relevant content. It is a remarkable occurrence that the road to progress has invariably been towards enlarging the number of variables and introducing a more powerful symmetry rather than conversely aiming at reducing the number of variables and eliminating the symmetry" [1]. We claim that the potentials of Classical Electromagnetism are not indetermined with respect to the so-called gauge transformations. Indeed, these transformations raise paradoxes that imply their rejection. Nevertheless, the potentials are still indetermined up to a constant.

2021

Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches.

Physics Letters B, 1995

It is shown that in the absence of free abelian gauge fields, the conserved currents of (classical) Yang-Mills gauge models coupled to matter fields can be always redefined so as to be gauge invariant. This is a direct consequence of the general analysis of the Wess-Zumino consistency condition for Yang-Mills theory that we have provided recently.

Revista Mexicana de Fisica

Noether's theorem relating continuous symmetries of a Lagrangian system to the existence of conserved quantities is shown to be valid at the level of the variational equations of the system. This result can be helpful in the study of perturbations and of integrability in various areas of current interest. As examples, we derive conserved quatities in linearized general relativity and obtain conserved quantities valid in perturbed classical dynamics.