We investigate generators of local transformations in the covariant canonical formalism (CCF). The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms $π_A$ are defined as derivatives of the Lagrangian $d$-form $L(ψ^A, dψ^A)$ with respect to $dψ^A$, namely $π_A := \partial L/\partial dψ^A$, where $ψ^A $ are $p$-form dynamical fields. The form-canonical equations are derived from the form-Legendre transformation of the Lagrangian form $H:=dψ^A \wedge π_A - L$. We show that the Noether current form is the generator of an infinitesimal transformation $ψ^A \to ψ^A + δψ^A$ if the transformation of the Lagrangian form is given by $δL=dl$ and $δψ^A$ and $l$ depend on only $ψ^A$ and the parameters. As an instance, we study the local gauge transformation for the gauge field and the local Lorentz transformation for the second order formalism of gravity.
We construct a manifestly covariant differential Noether charge for theories with Chern-Simons terms in higher dimensional spacetimes. This is in contrast to Tachikawa's extension of the standard Lee-Iyer-Wald formalism which results in a non-covariant differential Noether charge for Chern-Simons terms. On a bifurcation surface, our differential Noether charge integrates to the Wald-like entropy formula proposed by Tachikawa in arXiv:hep-th/0611141.
Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms to be projectable to phase-space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of General Relativity, or other generally covariant theories, only closes as a soft algebra and not a a Lie algebra. The deep relationship between these two issues is clarified. In particular, we see that the second one may be understo…
Covariant conservation laws, local invariance and Noether's second theorem
Universidade dos Açores（亚速尔群岛大学）；Faculdade de Ciências, Universidade de Lisboa（里斯本大学理学院）；Instituto de Astrofísica e Ciências do Espaço（天体物理学与空间科学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）；Centro de Física Teórica e Computacional（理论与计算物理中心）
Arbitrary diffeomorphically invariant metric-torsion theories of gravity are considered. It is assumed that Lagrangians of such theories contain derivatives of field variables (tensor densities of arbitrary ranks and weights) up to a second order only. The generalized Klein-Noether methods for constructing manifestly covariant identities and conserved quantities are developed. Manifestly covariant expressions are constructed without including auxiliary structures like a background metric. In the Riemann-Cartan space, the following \emph{manifestly generally covariant results} are presented: (a…
Double field theory is an approach for massless modes of string theory, unifying and geometrizing all gauge invariance in manifest $\mathbf{O}(D,D)$ covariant manner. In this approach, we derive off-shell conserved Noether current and corresponding Noether potential associated with unified gauge invariance. We add Wald-type counter two-form to the Noether potential and define conserved global charges as surface integral. We check our $\mathbf{O}(D,D)$ covariant formula against various string backgrounds, both geometric and non-geometric. In all cases we examined, we find perfect agreements wit…
The covariant Noether charge formalism (also known as the covariant phase method) of Wald and collaborators, including its cohomological extension, is a manifestly covariant Hamiltonian formalism that, in principle, allows one to define and compute the energy, angular momenta, and chemical potentials of generic solutions of gravitational theories. However, it has been observed that for some supergravity solutions the variation of the Noether charge is not (at least manifestably) integrable, and as a result it is unclear whether there are well-defined thermodynamic charges for these solutions. …
Noether's theorem is an elegant and powerful tool of classical mechanics, but it is of little to no consequence in discrete theories. Here we define and explore a discrete approach to covariant mechanics and show that within this framework a discrete version of Noether's theorem, completely analogous to the well-known continuum version with all its ramifications, remains valid. We also discuss why more traditional approaches to discretized mechanics violate certain conservation laws by construction.
We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the con…
It is well known that the Lagrangian and the Hamiltonian formalisms can be combined and lead to "covariant symplectic" methods. For that purpose a "pre-symplectic form" has been constructed from the Lagrangian using the so-called Noether form. However, analogously to the standard Noether currents, this symplectic form is only determined up to total divergences which are however essential ingredients in gauge theories. We propose a new definition of the symplectic form which is covariant and free of ambiguities in a general first order formulation. Indeed, our construction depends on the equati…
The Lagrangian proposed by York et al. and the covariant first order Lagrangian for General Relativity are introduced to deal with the (vacuum) gravitational field on a reference background. The two Lagrangians are compared and we show that the first one can be obtained from the latter under suitable hypotheses. The induced variational principles are also compared and discussed. A conditioned correspondence among Noether conserved quantities, quasilocal energy and the standard Hamiltonian obtained by 3+1 decomposition is also established. As a result, it turns out that the covariant first orde…
We review the Lagrangian formulation of Noether symmetries (as well as "generalized 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 and so on). It is discussed how the use of Poincare'-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether sy…
We investigate generators of local gauge transformations in the covariant canonical formalism (CCF) for matter fields, gauge fields and the second order formalism of gravity. The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms $π_A$ are defined as derivatives of the Lagrangian $d$-form $L(ψ^A, dψ^A)$ with respect to $dψ^A$, namely $π_A := \partial L/\partial dψ^A$, where $ψ^A $ are $p$-form dynamical fields. The form-canonical equations are derived from the form-Legendre transformation of the Lagrangian form $H:=dψ^A \w…
Motivated by the study of physical models associated with General Relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether's theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson-Hamiltonian framework. Also, by focusing on the space plus time decomposition…
We derive the Noether identities and the conservation laws for general gravitational models with arbitrarily interacting matter and gravitational fields. These conservation laws are used for the construction of the covariant equations of motion for test bodies with minimal and nonminimal coupling.
Noether's theorem is reviewed with a particular focus on an intermediate step between global and local gauge and coordinate transformations, namely linear transformations. We rederive the well known result that global symmetry leads to charge conservation (Noether's first theorem), and show that linear symmetry allows for the current to be expressed as a four divergence. Local symmetry leads to identical conservation of the current and allows for the expression of the charge as two dimensional surface integral (Noether's second theorem). In the context of coordinate transformations, an additio…
It is shown in the covariant phase space formalism that the Noether charges with respect to the diffeomorphism generated by vector fields and their horizontal variations in general relativity form a diffeomorphism algebra. It is also shown with the help of the null tetrad which is well defined everywhere that the central term of the reduced diffeomorphism algebra on the Killing horizon for a large class of vector fields vanishes.
A covariant formula for conserved currents of energy, momentum and angular-momentum is derived from a general form of Noethers theorem applied directly to the Einstein-Hilbert action of classical general relativity. Energy conservation in a closed big-bang cosmology is discussed as a special case. Special care is taken to distinguish between kinematic and dynamic expressions.
This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections between initial value constraints and gauge transformations in such theories (either relativistic or not). To do this, in the course of these four papers, we develop and use a number of tools from symplectic and multisymplectic geometry. Of central importance in our analysis is the notion of the ``energy-momentum map'' associated to the gauge group of a given classical field theory. We hope to demonst…
We study spacetime diffeomorphisms in Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of …
We consider a general, classical theory of gravity with arbitrary matter fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian, $\bL$. We first show that $\bL$ always can be written in a ``manifestly covariant" form. We then show that the symplectic potential current $(n-1)$-form, $þ$, and the symplectic current $(n-1)$-form, $\om$, for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current $(n-1)$-form, $\bJ$, and corresponding Noether charge $(n-2)$-form, $\bQ$. We derive a general ``decomp…
General matterless--theories in 1+1 dimensions include dilaton gravity, Yang--Mills theory as well as non--Einsteinian gravity with dynamical torsion and higher power gravity, and even models of spherically symmetric d = 4 General Relativity. Their recent identification as special cases of 'Poisson--sigma--models' with simple general solution in an arbitrary gauge, allows a comprehensive discussion of the relation between the known absolutely conserved quantities in all those cases and Noether charges, resp. notions of quasilocal 'energy--momentum'. In contrast to Noether like quantities, quas…
The first law of black hole mechanics was derived by Wald in a general covariant theory of gravity for stationary variations around a stationary black hole. It is formulated in terms of Noether charges, and has many advantages. In this paper several issues are discussed to strengthen the validity of the Noether charge form of the first law. In particular, a gauge condition used in the derivation is justified. After that, we justify the generalization to non-stationary variations done by Iyer-Wald.
The method of Lagrangians with covariant derivative (MLCD) is applied to a special type of Lagrangian density depending on scalar and vector fields as well as on their first covariant derivatives. The corresponding Euler-Lagrange's equations and energy-momentum tensors are found on the basis of the covariant Noether's identities.
We revisit the gauge-covariant canonical formalism by separating explicitly physical and gauge degrees of freedom. We show in particular that the gauge-invariant linear and angular momentum operators proposed by Chen et al. can consistently be derived from the standard procedure based on the Noether's theorem. Finally, we demonstrate that this approach is essentially equivalent to the gauge-invariant canonical formalism based on the concept of Dirac variables. Because of many similarities with the background field method, the formalism developed here should also be relevant to general relativi…
We derive the variational principle and Noether's theorem in generally covariant field theory in an explicitly coordinate-independent way by means of the exterior calculus over the space-time manifold. We then focus on the symmetry of active diffeomorphisms, that is, the pushforwards along the integral lines of any vector field, and its analogies with internal gauge symmetries. For instance, it is well known that a class of Noether currents associated to a gauge symmetry can be obtained by taking the partial derivative of the Lagrangian with respect to the corresponding gauge field. Here we sh…
The effective geometry and the gravitational coupling of nonabelian gauge and scalar fields on generic NC branes in Yang-Mills matrix models is determined. Covariant field equations are derived from the basic matrix equations of motions, known as Yang-Mills algebra. Remarkably, the equations of motion for the Poisson structure and for the nonabelian gauge fields follow from a matrix Noether theorem, and are therefore protected from quantum corrections. This provides a transparent derivation and generalization of the effective action governing the SU(n) gauge fields obtained in [1], including t…
The Noether symmetry of a generic $f(R)$ cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. We explicitly calculate the form of $f(R)$ for which such symmetries exist. It is shown that the resulting form of $f(R)$ yields a power law expansion for the cosmological scale factor. We also obtain the effective equation of state parameter for the corresponding cosmology and show that our model can provide a gravitational alternative to the quintessence.
We study the metric $f(R)$ cosmology using Noether symmetry approach by utilizing the behavior of the corresponding Lagrangian under infinitesimal generators of the desired symmetry. The existence of Noether symmetry of the cosmological $f(R)$ minisuperspace helps us to find out the form of $f(R)$ function for which such symmetry exist. It is shown that the resulting form for $f(R)$ yields a power law expansion for the cosmic scale factor. We also show that in the corresponding Noether symmetric quantum model, the solutions to the Wheeler-DeWitt equation can be expressed as a superposition of …
Noether's 2nd theorem applied to a total system states that a global symmetry which is a part of local symmetries does not provide a physically meaningful conserved charge but it instead leads to off-shell constraints as a form of conserved currents. In this paper, we propose a general method to derive a matter conserved current associated with a special global symmetry in the presence of local symmetries. While currents derived from local symmetries of a matter sector with a covariant background gauge field are not conserved in general, we show that the current associated with a special type …
In this paper, we present the Noether symmetries of a class of the Bianchi type I anisotropic model in the context of f(T) gravity. By solving the system of equations obtained from the Noether symmetry condition, we obtain the form of f(T) as a teleparallel form. This analysis shows that teleparallel gravity has the maximum number of Noether symmetries. We derive the symmetry generators and show that there are five kinds of symmetries, including time and scale invariance under metric coefficients. We classify the symmetries and we obtain the corresponding invariants.
Gravitational theories invariant under transverse diffeomorphisms and Weyl transformations have the same classical solutions as the corresponding fully diffeomorphism invariant theories. However, they solve some of the problems related to the cosmological constant and in principle allow local energy non-conservation. In the present work, we obtain the Noether charge formalism for these theories. We first derive expressions for the Noether currents and charges corresponding to transverse diffeomorphisms and Weyl transformations, showing that the latter vanish identically. We then use these resu…
The evolution of a generally covariant theory is under-determined. One hundred years ago such dynamics had never before been considered; its ramifications were perplexing, its future important role for all the fundamental interactions under the name gauge principle could not be foreseen. We recount some history regarding Einstein, Hilbert, Klein and Noether and the novel features of gravitational energy that led to Noether's two theorems. Under-determined evolution is best revealed in the Hamiltonian formulation. We developed a covariant Hamiltonian formulation. The Hamiltonian boundary term g…
The properties of the covariant quantum fields on de Sitter spacetimes are investigated focusing on the isometry generators and Casimir operators in order to establish the equivalence among the covariant representations and the unitary irreducible ones of the de Sitter isometry group. For the Dirac quantum field it is shown that the spinor covariant representation, transforming the Dirac field under de Sitter isometries, is equivalent with a direct sum of two unitary irreducible representations of the $Sp(2,2)$ group, transforming alike the particle and antiparticle field operators in momentum…
We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate $r$, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting $r$-Hamiltonian yields the correct classical solutions which can be identified with the Schwarzschild black hole. The Noether symmetry of model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noethe…
In Einstein-Cartan theory, by the use of the general Noether theorem, the general covariant angular-momentum conservation law is obtained with the respect to the local Lorentz transformations. The corresponding conservative Noether current is interpreted as the angular momentum tensor of the gravity-matter system including the spin density. It is pointed out that, assuming the tetrad transformation given by eq. (15), torsion tensor can not play a role in the conservation law of angular momentum.
In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group ${A}(4,R)$ and of its subgroup ${GL}(4,R)$ in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the $\overline{SL}(4,R)$ coveri…
In the framework of an arbitrary $D$-dimensional metric theory, perturbations are considered on arbitrary backgrounds that are however solutions of the theory. Conserved currents for perturbations are presented following two known prescriptions: canonical Nœther theorem and Belinfante symmetrization rule. Using generalized formulae, currents in the Einstein-Gauss-Bonnet (EGB) gravity for arbitrary types of perturbations on arbitrary curved backgrounds (not only vacuum) are constructed in an explicit covariant form. Special attention is paid to the energy-momentum tensors for perturbations whic…
The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and metrics [$(\bar{L}_n,g)$-spaces] is considered. The functional variation and the Lie variation of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of …
In the light of the local Lorentz transformations and the general Noether theorem, a new formulate of the general covariant angular momentum conservation law in Einstein-Cartan gravitation theory is obtained, which overcomes the critical difficulty in the other formulates that the conservation law depended on the coordinative choice.
