We show that generalized symmetries cannot be charged under a continuous global symmetry having a Noether current. Further, only non-compact generalized symmetries can be charged under a continuous global symmetry. These results follow from a finer classification of twist operators, which naturally extends to finite group global symmetries. They unravel topological obstructions to the strong version of Noether's theorem in QFT, even if under general conditions a global symmetry can be implemented locally by twist operators (weak version). We use these results to rederive Weinberg-Witten's theorem within local QFT, generalizing it to massless particles in arbitrary dimensions and representations of the Lorentz group. Several examples with local twists but without Noether currents are described. We end up discussing the conditions for the strong version to hold, dynamical aspects of QFT's with non-compact generalized symmetries, scale vs conformal invariance in QFT, connections with the Coleman-Mandula theorem and aspects of global symmetries in quantum gravity.
For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.
We develop the general theory of Noether symmetries for constrained systems. In our derivation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of the conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fully analyzed, and we give a geometrical interpretation of the projectability conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We…
We present Noether's second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether's second theorem for Lagrangian systems on fiber bundles over X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved.
The aim of this paper is to present a new approach to construct constants of motion associated with scaling symmetries of dynamical systems. Scaling maps could be symmetries of the equations of motion but not of its associated Lagrangian action. We have constructed a Noether inspired theorem in a vertical extended space that can be used to obtain constants of motion for these symmetries. Noether theorem can be obtained as a particular case of our construction. To illustrate how the procedure works, we present two interesting examples, a) the Schwarzian Mechanics based on Schwarzian derivative …
We show that Belinfante construction of an improved energy-momentum tensor can be carried over to curved backgrounds, in analogy to the case of flat spacetime. The results hold irrespective of the background being dynamical or a fixed, non-backreacting one. It turns out that the analogous would-be canonical energy-momentum tensor is not covariantly conserved in general, but its Belinfante "improvement" is. We relate this last tensor with the Hilbert tensor obtained by functionally derivating the Lagrangian with respect to the metric. When the background in non-dynamical, we discuss some issues…
In this paper, we present a complete Noether Symmetry analysis in the framework of scalar-tensor cosmology. Specifically, we consider a non-minimally coupled scalar field action embedded in the FLRW spacetime and provide a full set of Noether symmetries for related minisuperspaces. The presence of symmetries implies that the dynamical system becomes integrable and then we can compute cosmological analytical solutions for specific functional forms of coupling and potential functions selected by the Noether Approach.
We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian then there exists a coordinate system in which a reversal symmetry exists. Moreover as far as concerns the scale-factor duality symmetry of the dilaton field, we show that it is related to the existence of a Noether symmetry for the field equations, and the reversal symmetry in the normal coordinates of the symmetry vector becomes scale-factor duality symmetry in the original coordinate…
In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. Will this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy-momentum tensor and a family of Hilbert energy-momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of …
We prove that, given a time-independent Lagrangian defined in the first tangent bundle of configuration space, every infinitesimal Noether symmetry that is defined in the $n$-tangent bundle and is not vanishing on-shell, can be written as a canonical symmetry in an enlarged phase space, up to constraints that vanish on-shell. The proof is performed by the implementation of a change of variables from the the $n$-tangent bundle of the Lagrangian theory to an extension of the Hamiltonian formalism which is particularly suited for the case when the Lagrangian is singular. This result proves the as…
The purpose of the current article is to present a brief albeit accurate presentation of the main tools used in the study of symmetries of Lagrange equations for holonomic systems and subsequently to show how these tools are applied in the major models of modern cosmology in order to derive exact solutions and deal with the problem of dark matter/energy. The key role in this approach are the first integrals of the field equations. We start with the Lie point symmetries and the first integrals defined by them, that is the Hojman integrals. Subsequently we discuss the Noether point symmetries an…
We discuss non-minimally coupled cosmologies involving different geometric invariants. Specifically, actions containing a non-minimally coupled scalar field to gravity described, in turn, by curvature, torsion and Gauss--Bonnet scalars are considered. We show that couplings, potentials and kinetic terms are determined by the existence of Noether symmetries which, moreover, allows to reduce and solve dynamics. The main finding of the paper is that different non-minimally coupled theories, presenting the same Noether symmetries, are dynamically equivalent. In other words, Noether symmetries are …
We apply the Noether Symmetry Approach to point-like teleparallel Lagrangians in view to derive minisuperspaces suitable for Quantum Cosmology. Adopting the Arnowitt-Deser-Misner formalism, we find out related Wave Functions of the Universe. Specifically, by means of appropriate changes of variables suggested by the existence of Noether symmetries, it is possible to obtain the cosmological Hamiltonians whose solutions are classical trajectories interpretable as observable universes.
The connection between symmetries and conservation laws as made by Noether's theorem is extended to the context of causal variational principles and causal fermion systems. Different notions of continuous symmetries are introduced. It is proven that these symmetries give rise to corresponding conserved quantities, expressed in terms of so-called surface layer integrals. In a suitable limiting case, the Noether-like theorems for causal fermion systems reproduce charge conservation and the conservation of energy and momentum in Minkowski space. Thus the conservation of charge and energy-momentum…
Spherical symmetry for f(R)-gravity is discussed by searching for Noether symmetries. The method consists in selecting conserved quantities in form of currents that reduce dynamics of f(R)-models compatible with symmetries. In this way we get a general method to obtain constants of motion without setting a priori the form of f(R). In this sense, the Noether symmetry results a physical criterium. Relevant cases are discussed.
A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether's Theorem to determine conservation laws for the field equations. In the Friedmann-Lemaître-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.
Non-local gravity cosmologies are considered under the standard of Noether Symmetry Approach. In particular, we focus on non-local theories whose gravitational actions depend on curvature and Gauss-Bonnet scalar invariants. Specific functional forms of the related point-like Lagrangians are selected by Noether symmetries and we solve the corresponding field equations finding out exact cosmological solutions.
We summarize our work on "hidden" Noether symmetries of multifield cosmological models and the classification of those two-field cosmological models which admit such symmetries.
Adopting Noether point symmetries, we classify and integrate dynamical systems coming from Horndeski cosmologies. The method is particularly effective both to select the form of Horndeski models and to derive exact cosmological solutions. Starting from the Lagrangians selected by the Noether symmetries, it is possible to derive several modified theories of gravity like $f(R)$ gravity, Brans-Dicke gravity, string inspired gravity and so on. In any case, exact solutions are found out.
We show that flavor 't Hooft anomalies automatically vanish in noncommutative field theories which are obtained from string theory in the decoupling limit. We claim that this is because the flavor symmetries are secretly local, because of coupling to closed string bulk modes. An example is the SU(4) R-symmetry of N=4 D=4 NCSYM. The gauge fields, along with all closed string bulk modes, are not on-shell external states but do appear as off-shell intermediate states in non-planar processes; these closed string modes are thereby holographically encoded in the NCFT.
The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select "observable" universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.
The Noether-Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether-Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar-tensor gravity with unknown s…
We provide a geometric extension of the generalized Noether theorem for scaling symmetries recently presented in \cite{zhang2020generalized}. Our version of the generalized Noether theorem has several positive features: it is constructed in the most natural extension of the phase space, allowing for the symmetries to be vector fields on such manifold and for the associated invariants to be first integrals of motion; it has a direct geometrical proof, paralleling the proof of the standard phase space version of Noether's theorem; it automatically yields an inverse Noether theorem; it applies al…
A geometric global formulation of the higher-order Lagrangian formalism for systems with finite number of degrees of freedom is provided. The formalism is applied to the study of systems with groups of Noetherian symmetries.
We consider the consequences of global higher-spin symmetries in quantum field theories on a fixed de Sitter background of spacetime dimension $D \ge 3$. These symmetries enhance the symmetry group associated with the isometries of the de Sitter background and thus strongly constrain the dynamics of the theory. In particular, we consider the case when a higher spin charge acts linearly on a scalar operator to leading order in a Fefferman-Graham expansion near the future/past conformal boundaries. We show that this implies that the expectation values of the operator inserted near the boundaries…
We develop a systematic algorithm, based on Noether's theorem, for defining the various currents in theories invariant under space dependent polynomial symmetries. A master equation is given that yields all the conservation laws corresponding to these currents. Explicit demonstration has been provided for dipole and quadrupole conservation symmetries.
We investigate the main features of the flat Friedmann-Lemaıtre-Robertson-Walker cosmological models in the f(T) teleparallel gravity. In particular, a general approach to find out exact cosmological solutions in f (T) gravity is discussed. Instead of taking into account phenomenological models, we consider as a selection criterion, the existence of Noether symmetries in the cosmological f(T) point-like Lagrangian. We find that only power-law models admit extra Noether symmetries. A complete analysis of such cosmological models is developed.
There has been a certain interest in some recent works in the derivation of Noether charges for Hopf-algebra space-time symmetries. Such analyses relied rather heavily on delicate manipulations of the fields of non-commuting coordinates whose charges were under study. Here we derive the same charges in a "coordinate-independent" symplectic-geometry type of approach and find results that are consistent with the ones of hep-th/0607221.
The Noether Symmetry Approach is applied to interacting quintessence cosmology with the aim to search for exact solutions and select scalar-field self-interaction potentials. It turns out that the solutions found are compatible with the accelerated expansion of the Universe and with observational dataset, as the SNeIa Pantheon data.
In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we prov…
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…
Spherically symmetric solutions for f(T) gravity models are derived by the so called Noether Symmetry Approach. First, we present a full set of Noether symmetries for some minisuperspace models. Then, we compute analytical solutions and find that spherically symmetric solutions in f(T) gravity can be recast in terms of Schwarzschild-like solutions modified by a distortion function depending on a characteristic radius. The obtained solutions are more general than those obtained by the usual solution methods.
We show that the recent results of \ [Int. J. Mod. Phys. D 25 (2016) 1650051] on the application of Lie/Noether symmetries in scalar field cosmology are well-known in the literature while the problem could have been solved easily under a coordinate transformation. That follows from the property, that the admitted group of invariant transformations of dynamical system is independent on the coordinate system.
We consider extensions of General Relativity based on the non-local function $f(R, \Box^{-1} R)$, where $R$ is the Ricci curvature scalar and the non-locality is due to the term $\Box^{-1} R$. We focus on cosmological minisuperspaces and select viable models by the Noether Symmetry Approach. Then we find viable exact solutions pointing out the role of non-locality in cosmology.
We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behaviors in cosmic evolution according to the so called Hartle criterion, that allows to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries…
A fundamental tenet of gauge theory is that physical quantities should be gauge-invariant. This prompts the question: can gauge symmetries have physical significance? On one hand, the Noether theorems relate conserved charges to symmetries, endowing the latter with physical significance, though this significance is sometimes taken as indirect. But for theories in spatially finite and bounded regions, the standard Noether charges are not gauge-invariant. I here argue that gauge-\emph{variance} of charges is tied to the nature of the non-locality within gauge theories. I will flesh out these lin…
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 is aim at students, not researchers. The main three topics discussed are (i) global symmetries and conserved charge…
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 …
It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topolog…
Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally important but much less well-known results. We present, in a general form, all the main results relating to the Noether variational problem for gauge theories, and we show the relationships between them. These results hold for both Abelian and non-Abelian gauge theories.
We exploit an ambiguity somewhat hidden in Noether's theorem to derive systematically, for relativistic field theories, the stress-energy tensor's improvement terms that are associated with additional spacetime symmetries beyond translations. We work out explicitly the cases of Lorentz invariance, scale invariance, and full conformal invariance. The main idea is to use, directly in the translation Noether theorem, the fact that these additional symmetries can be thought of as suitably modulated translations. Compared to more standard derivations of the improvement terms, ours (1) unifies all d…
The application of the Tomita-Takesaki modular theory to the Haag-Kastler net approach in QFT yields external (space-time) symmetries as well as internal ones (internal ``gauge para-groups") and their dual counterparts (the ``super selection para-group"). An attempt is made to develop a (speculative) picture on ``quantum symmetry" which links space-time symmetries in an inexorable way with internal symmetries. In the course of this attempt, we present several theorems and in particular derive the Kac-Wakimoto formula which links Jones inclusion indices with the asymptotics of expectation value…
Emmy Noether proved two deep theorems, and their converses, on the connection between symmetries and conservation laws. Because these theorems are not in the mainstream of her scholarly work, which was the development of modern abstract algebra, it is of some historical interest to examine how she came to make these discoveries. The present paper is an historical account of the circumstances in which she discovered and proved these theorems which physicists refer to collectively as Noether's Theorem. The work was done soon after Hilbert's discovery of the variational principle which gives the …
When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies,…
We prove Noether's direct and inverse second theorems for Lagrangian systems on fiber bundles in the case of gauge symmetries depending on derivatives of dynamic variables of an arbitrary order. The appropriate notions of reducible gauge symmetries and Noether's identities are formulated, and their equivalence by means of certain intertwining operator is proved.
Local symmetry transformations play an important role for establishing the existence and form of a conserved (Noether) current in systems with a global continuous symmetry. We explain how this fact leads to the existence of linear relations between Noether currents of distinct global symmetries that coincide on the local level, thus generalizing the well-known relationship $\vec L=\vec r\times\vec p$ between momentum $\vec p$ and angular momentum $\vec L$. As a byproduct, we find a natural interpretation for the discrepancy between the canonical and metric energy-momentum tensors in theories o…
We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar tensor, f(R)…
We perform a Noether analysis for a description of translation transformations in 4D kappa-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. Taking properly into account the properties of the differential calculus we arrive at an explicit formula for the conserved charges. We also propose a choice of basis for the 5D calculus which leads to an intuitive description of time derivatives.
A generalized teleparallel cosmological model, $f(T_\mathcal{G},T)$, containing the torsion scalar $T$ and the teleparallel counterpart of the Gauss-Bonnet topological invariant $T_{\mathcal{G}}$, is studied in the framework of the Noether Symmetry Approach. As $f(\mathcal{G}, R)$ gravity, where $\mathcal{G}$ is the Gauss-Bonnet topological invariant and $R$ is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, $f(T_\mathcal{G},T)$ contains all the possible information directly related to the torsion tensor. In th…
