Noether's Theorem for a Fixed Region

The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies

The C*-algebraic formulation of generic interacting quantum field theories, recently presented by Detlev Buchholz and one of the authors (KF), is enriched by a unitary version of the Master Ward Identity, which was postulated some time ago by Franz Marc Boas, Ferdinand Brennecke and two of us (MD,KF). It is shown that the corresponding axiom implies the validity of the time slice axiom. Moreover, it opens the way for a new approach to Noether's Theorem where it yields directly the unitaries implementing the symmetries. It also unravels interesting aspects of the role of anomalies in quantum fi…

Non-local Lagrangian fields: Noether's theorem and Hamiltonian formalism

This article aims to study non-local Lagrangians with an infinite number of degrees of freedom. We obtain an extension of Noether's theorem and Noether's identities for such Lagrangians. We then set up a Hamiltonian formalism for them. In addition, we show that $n$-order local Lagrangians can be treated as a particular case and the standard results can be recovered. Finally, this formalism is applied to the case of $p$-adic open string field.

A Time scales Noether's theorem

We prove a time scales version of the Noether's theorem relating group of symmetries and conservation laws. Our result extends the continuous version of the Noether's theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres in \cite{BT}.

Noether's theorem in non-local field theories

Explicit expressions are constructed for a locally conserved vector current associated with a continuous internal symmetry and for energy-momentum and angular-momentum density tensors associated with the Poincaré group in field theories with higher-order derivatives and in non-local field theories. An example of non-local charged scalar field equations with broken C and CPT symmetries is considered. For this case, we find simple analytical expressions for the conserved currents.

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether's Theorem. We state and prove the di…

Noether theorems and quantum anomalies

In this communication, we show that both infinite-dimensional versions of Noether's theorems, and the explanation of quantum anomalies can be obtained using similar formulas for the derivatives of functions whose values are measures (Smolyanov and von Weizsaecker, 1995) or pseudomeasures (Gough, Ratiu and Smolyanov, 2015). In particular, we improve son these results.

A geometric approach to the generalized Noether theorem

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…

Noether-Like Theorems for Causal Variational Principles

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…

Extensions of Noether's Second Theorem: from continuous to discrete systems

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler--Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.

A generalization of Noether's theorem based on the virtual work functional

In a series of previous articles by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems, as well as ones that subject to non-holonomic constraints by starting with the virtual work functional instead of an action functional. In this article, it is shown that when one starts with a virtual work functional that is not exact, so it does not admit an action functional, instead of conservation laws the extension of Noether's theorem gives balance principles for non-conserved currents. Examples of this generalized Noether identity…

Noether's theorem for the variational equations

We introduce an generalized action functional describing the equations of motion and the variational equations for any Lagrangian system. Using this novel scheme we are able to generalize Noether's theorem in such a way that to any $n$-parameter continuous symmetry group of the Lagrangian there exist 1) the usual $n$ constants of motion and 2) $n$ extra constants valid in the variational equations. The new constants are related to the infinitesimal generators of the symmetry transformation by relations similar to the ones that stem from the `nonextended' Noether theorem.

Second Noether theorem for quasi-Noether systems

Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this paper, we discuss quasi-Noether systems that possess infinite-dimensional (infinite) symmetries involving arbitrary functions of independent variables. For quasi-Noether systems admitting infinite symmetries with arbitrary functions of all independent variables, we state and prove an extension of the Second Noether Theorem. In addition, we prove that infinite …

Lie Groupoids in Classical Field Theory I: Noether's Theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether's theorem about the connection between symmetries and conservation laws in this approach.

A formulation of Noether's theorem for fractional classical fields

This paper presents a formulation of Noether's theorem for fractional classical fields. We extend the variational formulations for fractional discrete systems to fractional field systems. By applying the variational principle to a fractional action $S$, we obtain the fractional Euler-Lagrange equations of motion. Considerations of the Noether's variational problem for discrete systems whose action is invariant under gauge transformations will be extended to fractional variational problems for classical fields. The conservation laws associated with fractional classical fields are derived. As an…

Noether's inverse second theorem in homology terms

A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-trivial higher-stage Noether identities are ill defined, unless a certain homology condition holds. We show that, under this condition, there exists the exact Koszul-Tate chain complex whose boundary operator produces all non-trivial Noether and higher-stage Noether identities of an…

Noether theorem for mu-symmetries

We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagr…

Fractional Noether's theorem in the Riesz-Caputo sense

We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.

Pseudotensor problem of gravitational energy-momentum and Noether's theorem revisited

Based on a general variational principle, Noether's theorem is revisited. It is shown that the so called pseudotensor problem of the gravitational energy-momentum is a result of mis-reading Noether's theorem, and in fact, all the Noether's conserved quantities are scalars. It is also shown, by using a counter-example, that the non-localizability of gravitational energy-momentum can not be attributed to the equivalence principle. As a direct consequence of variational principle, a generalized Hamilton-Jacobi equation for the Hamilton's principal functional is obtained.

Vertical extension of Noether Theorem for Scaling Symmetries

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 …

Non-conservative Noether's theorem for fractional action-like variational problems with intrinsic and observer times

We extend Noether's symmetry theorem to fractional action-like variational problems with higher-order derivatives.

A Noether Theorem for Markov Processes

Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer holds, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are constant in time for every state.

二阶延迟常微分方程的拉格朗日形式体系与 Noether 型定理

Lagrangian formalism and Noether-type theorems for second-order delay ODEs

该研究构建二阶延迟常微分方程变分问题的拉格朗日框架，提出相应 Noether 型算子恒等式与定理，并以示例说明如何利用对称性代数构造积分。

Comment on "Noether's-type theorems on time scales" [J. Math. Phys. 61, 113502 (2020)]

We comment on the validity of Noether's theorem and on the conclusions of [J. Math. Phys. 61 (2020), no. 11, 113502].

A modern view of the classical Herglotz-Noether theorem

In this paper we give a new proof, valid for all dimensions, of the classical Herglotz-Noether theorem that all rotational shear-free and expansion-free flows (rotational Born-rigid flows) in Minkowski spacetime are generated by Killing vector fields (isometric flows). This is aimed as an illustration of a general framework for working with problems that can be described as a structure-preserving submersion, which we will describe in a subsequent paper.

Noether theorems in a general setting

The first and second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of non-trivial higher-stage Noether identities and the corresponding higher-stage gauge symmetries which are described in the homology terms. In these terms, the second Noether theorems associate to the Koszul - Tate chain complex of higher-stage Noether identities the gauge cochain sequence whose ascent operator provides higher-order gauge symmetries of Lagrangian th…

Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs

Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabelling symmetries are derived using Noether's first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noether's theorem, in conjunction with a fluid relabelling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same met…

Noether's Second Theorem on natural bundles

Following work by I. Anderson, in this note we present a formulation of Noether's Second Theorem that is valid on any natural bundle.

Dirac-Bergmann Constraints in Physics: Singular Lagrangians, Hamiltonian Constraints and the Second Noether Theorem

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac-Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed: i) the connection of the rank and eigenvalues of the Hessian matrix in the Euler-Lagrange equations with the chains of first and second class constraints; ii) the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints; iii) the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dir…

Generalized symmetries and Noether's theorem in QFT

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 theo…

Noether Theorems and Reality of Motion

We will read, through the Emmy Noether paper and the two concepts of `proper' and `improper' conservation laws, the problem, posed by Hilbert, of the nature of the law of conservation of energy in the theory of General Relativity. Epistemological issues involved with the two kind of conservation laws will be enucleate.

多接触场论的对称性与 Noether 定理

Symmetries and Noether's theorem for multicontact field theories

本文利用多接触几何框架分析作用量依赖的Lagrangian和Hamiltonian场论中的对称性及耗散定律，定义了守恒量与耗散量，并推导出适用于该设定的Noether定理。

Noether's Theorem Under the Legendre Transform

In this paper we demonstrate how the Legendre transform connects the statements of Noether's theorem in Hamiltonian and Lagrangian mechanics. We give precise definitions of symmetries and conserved quantities in both the Hamiltonian and Lagrangian frameworks and discuss why these notions in the Hamiltonian framework are somewhat less rigid. We explore conditions which, when put on these definitions, allow the Legendre transform to set up a one-to-one correspondence between them. We also discuss how to preserve this correspondence when the definitions of symmetries and conserved quantities are …

Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal's necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using Agrawal's Euler-Lagrange equation and the Lagrange multiplier technique, we obtain here a Noether-like theorem for fractional optimal control problems in the sense of Caputo.

On second Noether's theorem and gauge symmetries in Mechanics

We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to the process of stabilization of constraints, both in Lagrangian and Hamiltonian formalisms. The connections between both formalisms are established by means of the time-evolution operator.

Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems

We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.

The Noether theorems in context

I shall sketch the contents of Noether's 1918 article, "Invariante Variationsprobleme", in the context of the debate on the conservation of energy that had arisen from Einstein's general theory of relativity. How original was Noether's "Invariante Variationsprobleme"? how modern were her use of Lie groups and her introduction of generalized vector fields? and how influential was her article? I shall sketch the curious transmission of her results from 1918 to 1970, the history of the later developments of her theory in mathematics and in physics, and the ultimate recognition of the wide applica…

Noether's first theorem in Hamiltonian mechanics

Non-autonomous non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noether's first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noether's first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian mechanics. The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. W…

A Proper Extension of Noether's Symmetry Theorem for Nonsmooth Extremals of the Calculus of Variations

For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which g…

Noether's Theorem for Fractional Optimal Control Problems

We begin by reporting on some recent results of the authors (Frederico and Torres, 2006), concerning the use of the fractional Euler-Lagrange notion to prove a Noether-like theorem for the problems of the calculus of variations with fractional derivatives. We then obtain, following the Lagrange multiplier technique used in (Agrawal, 2004), a new version of Noether's theorem to fractional optimal control systems.

Matrix Gauge Fields and Noether's theorem

These notes have been triggered by physicist's considerations on (1)'Pre-quantized wave equations' for matter fields, (2) Conservation laws, (3) Gauge Field extensions. All starting from variational principles. These notes do not contain functional analysis or differential geometry. Only bare elementary considerations on matrix-valued functions. As a motivation there is a lengthy preface/summary.

Extension the Noether's theorem to Lagrangian formulation with nonlocality

A Lagrangian formulation with nonlocality is investigated in this paper. The nonlocality of the Lagrangian is introduced by a new nonlocal argument that is defined as a nonlocal residual satisfying the zero mean condition. The nonlocal Euler-Lagrangian equation is derived from the Hamilton's principle. The Noether's theorem is extended to this Lagrangian formulation with nonlocality. With the help of the extended Noether's theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby tensor are determined in the nonlocal elasticity associated with the mec…

The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.

Noether's Theorem for Dissipative Quantum Dynamical Semigroups

Noether's Theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) by Baez and Fong. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we construct a mapping from observables to CP maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semi-groups, we are able to cha…

A Noether Theorem on Unimprovable Conservation Laws for Vector-Valued Optimization Problems in Control Theory

We obtain a version of Noether's invariance theorem for optimal control problems with a finite number of cost functionals. The result is obtained by formulating E. Noether's result to optimal control problems subject to isoperimetric constraints, and then using the unimprovable (Pareto) notion of optimality. A result of this kind was posed to the author, as a mathematical open question, of great interest in applications of control engineering, by A. Gugushvili.

Nonconservative Noether's Theorem in Optimal Control

We extend Noether's theorem to dynamical optimal control systems being under the action of nonconservative forces. A systematic way of calculating conservation laws for nonconservative optimal control problems is given. As a corollary, the conserved quantities previously obtained in the literature for nonconservative problems of mechanics and the calculus of variations are derived.

Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem

The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques …

Noether's theorem for fractional variational problems of variable order

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

Fractional Noether's Theorem with Classical and Caputo Derivatives: constants of motion for non-conservative systems

Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether's theorem can not yields informations about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagragians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians dep…

Hight-order Noether's Theorem for Nonsmooth Extremals of Isoperimetric Variational Problems with Time Delay

We obtain a nonsmooth higher-order extension of Noether's symmetry theorem for variational isoperimetric problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed higher-order Euler--Lagrange extremals are restricted to those that satisfy the delayed higher-order DuBois--Reymond necessary optimality condition. The important case of delayed isoperimetric optimal control problems is considered as well.

Noether's first theorem and the energy-momentum tensor ambiguity problem

Noether's theorems are widely praised as some of the most beautiful and useful results in physics. However, if one reads the majority of standard texts and literature on the application of Noether's first theorem to field theory, one immediately finds that the ``canonical Noether energy-momentum tensor" derived from the 4-parameter translation of the Poincaré group does not correspond to what's widely accepted as the ``physical'' energy-momentum tensor for central theories such as electrodynamics. This gives the impression that Noether's first theorem is in some sense not working. In recogniti…

Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems

In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, \textrm{e.g.}, conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus …

Noether's Theorem for Nonsmooth Extremals of Variational Problems with Time Delay

We obtain a nonsmooth extension of Noether's symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler-Lagrange extremals are restricted to those that satisfy the DuBois-Reymond necessary optimality condition. The important case of delayed variational problems with higher-order derivatives is considered as well.

Noether定理的协变Hamiltonian表示及其在SU(N)规范理论中的应用

Covariant Hamiltonian representation of Noether's theorem and its application to SU(N) gauge theories

本文提出Noether定理的协变Hamiltonian表示，重新区分容许正则变换与Hamiltonian对称性，并基于无穷小生成函数导出U(1)、U(N)及SU(N)规范理论的Noether流与守恒关系。

Noether's Theorem in Multisymplectic Geometry

We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multi-Hamiltonian system. We show that a homotopy co-momentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. We also apply our results to manifolds with a torsion-free $G_2$ structure.