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Noether's theorem states that every
continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...
of the action of a physical system with
conservative force In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work don ...
s has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mathematician
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem applies to continuous and smooth symmetries of physical space. Noether's formulation is quite general and has been applied across classical mechanics, high energy physics, and recently
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. Noether's theorem is used in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
and the
calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on constants of motion in Lagrangian and
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
(developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.


Basic illustrations and background

As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it is invariant), its Lagrangian is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that the
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity â€“ the total ang ...
of the system be conserved, as a consequence of its laws of motion. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric. As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
within this system, respectively. Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system. As an illustration, suppose that a physical theory is proposed which conserves a quantity ''X''. A researcher can calculate the types of Lagrangians that conserve ''X'' through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory. There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the Ward–Takahashi identities. Generalizations of Noether's theorem to
superspace Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann num ...
s also exist.


Informal statement of the theorem

All fine technical points aside, Noether's theorem can be stated informally as: A more sophisticated version of the theorem involving fields states that: The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
of transformations satisfying certain technical criteria. The conservation law of a
physical quantity A physical quantity (or simply quantity) is a property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''nu ...
is usually expressed as a continuity equation. The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called the ''Noether charge'', while the flow carrying that charge is called the ''Noether current''. The Noether current is defined
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
a solenoidal (divergenceless) vector field. In the context of gravitation,
Felix Klein Felix Christian Klein (; ; 25 April 1849 â€“ 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
's statement of Noether's theorem for action ''I'' stipulates for the invariants:


Brief illustration and overview of the concept

The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate q and a continuous symmetry \varphi: q \mapsto q + \delta q (gray arrows on the diagram). Consider any trajectory q(t) (bold on the diagram) that satisfies the system's laws of motion. That is, the action S governing this system is stationary on this trajectory, i.e. does not change under any local variation of the trajectory. In particular it would not change under a variation that applies the symmetry flow \varphi on a time segment and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time \tau to transition between the segments gradually. The total change in the action S now comprises changes brought by every interval in play. Parts where variation itself vanishes, i.e outside _0,t_1/math>, bring no \Delta S. The middle part does not change the action either, because its transformation \varphi is a symmetry and thus preserves the Lagrangian L and the action S = \int L . The only remaining parts are the "buffering" pieces. In these regions both the coordinate q and velocity \dot change, but \dot changes by \delta q / \tau, and the change \delta q in the coordinate is negligible by comparison since the time span \tau of the buffering is small (taken to the limit of 0), so \delta q / \tau\gg \delta q. So the regions contribute mostly through their "slanting" \dot\rightarrow \dot\pm \delta q / \tau. That changes the Lagrangian by \Delta L \approx \bigl(\partial L/\partial \dot\bigr)\Delta \dot , which integrates to \Delta S = \int \Delta L \approx \int \frac\Delta \dot \approx \int \frac\left(\pm \frac\right) \approx \ \pm\frac \delta q = \pm\frac \varphi. These last terms, evaluated around the endpoints t_0 and t_1, should cancel each other in order to make the total change in the action \Delta S be zero, as would be expected if the trajectory is a solution. That is \left(\frac \varphi\right)(t_0) = \left(\frac \varphi\right)(t_1), meaning the quantity \left(\partial L /\partial \dot\right)\varphi is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of q by a constant are the symmetry, then the conserved quantity becomes just \left(\partial L/\partial \dot\right) = p, the canonical momentum. More general cases follow the same idea:


Historical context

A conservation law states that some quantity ''X'' in the mathematical description of a system's evolution remains constant throughout its motion – it is an invariant. Mathematically, the rate of change of ''X'' (its
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
with respect to
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
) is zero, :\frac = \dot = 0 ~. Such quantities are said to be conserved; they are often called constants of motion (although motion ''per se'' need not be involved, just evolution in time). For example, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it. Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws. The earliest constants of motion discovered were
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
and
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
, which were proposed in the 17th century by
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
and
Gottfried Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
on the basis of collision experiments, and refined by subsequent researchers.
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of
Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
. According to
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the stress–energy tensor (non-gravitational stress–energy) and the Landau–Lifshitz stress–energy–momentum pseudotensor (gravitational stress–energy). The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of the stress–energy tensor. Another important conserved quantity, discovered in studies of the
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
of astronomical bodies, is the Laplace–Runge–Lenz vector. In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering invariants. A major advance came in 1788 with the development of
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
, which is related to the principle of least action. In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, as was customary in Newtonian mechanics. The action is defined as the time integral ''I'' of a function known as the Lagrangian ''L'' :I = \int L(\mathbf, \dot, t) \, dt ~, where the dot over q signifies the rate of change of the coordinates q, :\dot = \frac ~. Hamilton's principle states that the physical path q(''t'')—the one actually taken by the system—is a path for which infinitesimal variations in that path cause no change in ''I'', at least up to first order. This principle results in the Euler–Lagrange equations, :\frac \left( \frac \right) = \frac ~. Thus, if one of the coordinates, say ''qk'', does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that :\frac \left( \frac \right) = \frac = 0~, where the momentum : p_k = \frac is conserved throughout the motion (on the physical path). Thus, the absence of the ignorable coordinate ''qk'' from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of ''qk''; the Lagrangian is invariant, and is said to exhibit a
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
under such transformations. This is the seed idea generalized in Noether's theorem. Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. For example, he developed a theory of
canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as ''form invariance''. Although Hamilton's equations are preserved, it need not ...
s which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the Hamilton–Jacobi equation. Emmy Noether's work on the invariance theorem began in 1915 when she was helping
Felix Klein Felix Christian Klein (; ; 25 April 1849 â€“ 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
and David Hilbert with their work related to
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's theory of general relativity By March 1918 she had most of the key ideas for the paper which would be published later in the year.


Mathematical expression


Simple form using perturbations

The essence of Noether's theorem is generalizing the notion of ignorable coordinates. One can assume that the Lagrangian ''L'' defined above is invariant under small perturbations (warpings) of the time variable ''t'' and the generalized coordinates q. One may write :\begin t &\rightarrow t^ = t + \delta t \\ \mathbf &\rightarrow \mathbf^ = \mathbf + \delta \mathbf ~, \end where the perturbations ''δt'' and ''δ''q are both small, but variable. For generality, assume there are (say) ''N'' such
symmetry transformations The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some Transformation (function), transformation. A family of particular transformations m ...
of the action, i.e. transformations leaving the action unchanged; labelled by an index ''r'' = 1, 2, 3, ..., ''N''. Then the resultant perturbation can be written as a linear sum of the individual types of perturbations, :\begin \delta t &= \sum_r \varepsilon_r T_r \\ \delta \mathbf &= \sum_r \varepsilon_r \mathbf_r ~, \end where ''ε''''r'' are infinitesimal parameter coefficients corresponding to each: * generator ''Tr'' of time evolution, and * generator Q''r'' of the generalized coordinates. For translations, Q''r'' is a constant with units of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
; for rotations, it is an expression linear in the components of q, and the parameters make up an
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
. Using these definitions, Noether showed that the ''N'' quantities :\left(\frac \cdot \dot - L \right) T_r - \frac \cdot \mathbf_r are conserved ( constants of motion).


Examples

I. Time invariance For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes ''t'' → ''t'' + δ''t'', without any change in the coordinates q. In this case, ''N'' = 1, ''T'' = 1 and Q = 0; the corresponding conserved quantity is the total
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
''H'' :H = \frac \cdot \dot - L. II. Translational invariance Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate ''q''''k''; so it is invariant (symmetric) under changes ''q''''k'' → ''q''''k'' + ''δq''''k''. In that case, ''N'' = 1, ''T'' = 0, and ''Q''''k'' = 1; the conserved quantity is the corresponding linear
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
''p''''k'' :p_k = \frac. In special and
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, these two conservation laws can be expressed either ''globally'' (as it is done above), or ''locally'' as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined ''locally'' at the space-time point: the stress–energy tensor(this will be derived in the next section). III. Rotational invariance The conservation of the
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity â€“ the total ang ...
L = r × p is analogous to its linear momentum counterpart. It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle ''δθ'' about an axis n; such a rotation transforms the
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
by the equation :\mathbf \rightarrow \mathbf + \delta\theta \, \mathbf \times \mathbf. Since time is not being transformed, ''T'' = 0, and ''N'' = 1. Taking ''δθ'' as the ''ε'' parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by :\mathbf = \mathbf \times \mathbf. Then Noether's theorem states that the following quantity is conserved, : \frac \cdot \mathbf = \mathbf \cdot \left( \mathbf \times \mathbf \right) = \mathbf \cdot \left( \mathbf \times \mathbf \right) = \mathbf \cdot \mathbf. In other words, the component of the angular momentum L along the n axis is conserved. And if n is arbitrary, i.e., if the system is insensitive to any rotation, then every component of L is conserved; in short,
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity â€“ the total ang ...
is conserved.


Field theory version

Although useful in its own right, the version of Noether's theorem just given is a special case of the general version derived in 1915. To give the flavor of the general theorem, a version of Noether's theorem for continuous fields in four-dimensional space–time is now given. Since field theory problems are more common in modern physics than
mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
problems, this field theory version is the most commonly used (or most often implemented) version of Noether's theorem. Let there be a set of differentiable fields \varphi defined over all space and time; for example, the temperature T(\mathbf, t) would be representative of such a field, being a number defined at every place and time. The principle of least action can be applied to such fields, but the action is now an integral over space and time :\mathcal = \int \mathcal \left(\varphi, \partial_\mu \varphi, x^\mu \right) \, d^4 x (the theorem can be further generalized to the case where the Lagrangian depends on up to the ''n''th derivative, and can also be formulated using jet bundles). A continuous transformation of the fields \varphi can be written infinitesimally as :\varphi \mapsto \varphi + \varepsilon \Psi, where \Psi is in general a function that may depend on both x^\mu and \varphi. The condition for \Psi to generate a physical symmetry is that the action \mathcal is left invariant. This will certainly be true if the Lagrangian density \mathcal is left invariant, but it will also be true if the Lagrangian changes by a divergence, :\mathcal \mapsto \mathcal + \varepsilon \partial_\mu \Lambda^\mu, since the integral of a divergence becomes a boundary term according to the divergence theorem. A system described by a given action might have multiple independent symmetries of this type, indexed by r = 1, 2, \ldots, N, so the most general symmetry transformation would be written as :\varphi \mapsto \varphi + \varepsilon_r \Psi_r, with the consequence :\mathcal \mapsto \mathcal + \varepsilon_r \partial_\mu \Lambda^\mu_r. For such systems, Noether's theorem states that there are N conserved current densities :j^\nu_r = \Lambda^\nu_r - \frac \cdot \Psi_r (where the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
is understood to contract the ''field'' indices, not the \nu index or r index). In such cases, the conservation law is expressed in a four-dimensional way :\partial_\nu j^\nu = 0, which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example,
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.


Examples

I. The stress–energy tensor For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, L \left(\boldsymbol\varphi, \partial_\mu, x^\mu \right) is constant in its third argument. In that case, ''N'' = 4, one for each dimension of space and time. An infinitesimal translation in space, x^\mu \mapsto x^\mu + \varepsilon_r \delta^\mu_r (with \delta denoting the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
), affects the fields as \varphi(x^\mu) \mapsto \varphi\left(x^\mu - \varepsilon_r \delta^\mu_r\right): that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point x^\mu with the value at the point x^\mu - \varepsilon X^\mu "behind" it which would be mapped onto x^\mu by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as :\Psi_r = -\delta^\mu_r \partial_\mu \varphi. The Lagrangian density transforms in the same way, \mathcal\left(x^\mu\right) \mapsto \mathcal\left(x^\mu - \varepsilon_r \delta^\mu_r\right), so :\Lambda^\mu_r = -\delta^\mu_r \mathcal and thus Noether's theorem corresponds to the conservation law for the stress–energy tensor ''T''''μ''''ν'', where we have used \mu in place of r. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each \mu) into a tensor T, Noether's theorem gives : T_\mu^\nu = -\delta^\nu_\mu \mathcal + \delta^\sigma_\mu \partial_\sigma \varphi \frac = \left(\frac\right) \cdot \varphi_ - \delta^\nu_\mu \mathcal with :T_\mu^\nu_ = 0 (we relabelled \mu as \sigma at an intermediate step to avoid conflict). (However, the T obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see Canonical stress–energy tensor.) I. The
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
The conservation of
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
, by contrast, can be derived by considering ''Ψ'' linear in the fields ''φ'' rather than in the derivatives. In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the probability amplitude ''ψ''(x) of finding a particle at a point x is a complex field ''φ'', because it ascribes a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability ''p'' = , ''ψ'', 2 can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the ''ψ'' field and its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
field ''ψ''* that leave , ''ψ'', 2 unchanged, such as :\psi \rightarrow e^ \psi\ ,\ \psi^ \rightarrow e^ \psi^~, a complex rotation. In the limit when the phase ''θ'' becomes infinitesimally small, ''δθ'', it may be taken as the parameter ''ε'', while the ''Ψ'' are equal to ''iψ'' and −''iψ''*, respectively. A specific example is the Klein–Gordon equation, the relativistically correct version of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
for spinless particles, which has the Lagrangian density :L = \partial_\psi \partial_\psi^ \eta^ + m^2 \psi \psi^. In this case, Noether's theorem states that the conserved (∂ â‹… ''j'' = 0) current equals :j^\nu = i \left( \frac \psi^ - \frac \psi \right) \eta^~, which, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle. This "gauge invariance" was first noted by
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
, and is one of the prototype gauge symmetries of physics.


Derivations


One independent variable

Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables q are such that the action integral I = \int_^ L mathbf [t \dot [t">.html" ;"title="mathbf [t">mathbf [t \dot [t t">">mathbf_[t<_a>_\dot_[t.html" ;"title=".html" ;"title="mathbf [t">mathbf [t \dot [t">.html" ;"title="mathbf [t">mathbf [t \dot [t t\, dt is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the Euler–Lagrange equations :\frac \frac = \frac [t]. And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a flow (mathematics), flow, φ, which acts on the variables as follows :\begin t &\rightarrow t' = t + \varepsilon T \\ \mathbf &\rightarrow \mathbf' '= \varphi mathbf [t \varepsilon">.html" ;"title="mathbf [t">mathbf [t \varepsilon= \varphi [\mathbf ' - \varepsilon T \varepsilon] \end where ''ε'' is a real variable indicating the amount of flow, and ''T'' is a real constant (which could be zero) indicating how much the flow shifts time. : \dot \rightarrow \dot' '= \frac \varphi mathbf [t \varepsilon">.html" ;"title="mathbf [t">mathbf [t \varepsilon= \frac [\mathbf ' - \varepsilon T \varepsilon] \dot [t' - \varepsilon T] . The action integral flows to : \begin I' varepsilon& = \int_^ L mathbf'[t' \dot' [t'">'.html" ;"title="mathbf'[t'">mathbf'[t' \dot' [t' t'">'">mathbf'[t'<_a>_\dot'_[t'.html" ;"title="'.html" ;"title="mathbf'[t'">mathbf'[t' \dot' [t'">'.html" ;"title="mathbf'[t'">mathbf'[t' \dot' [t' t'\, dt' \\ pt& = \int_^ L [\varphi [\mathbf ' - \varepsilon T \varepsilon], \frac [\mathbf ' - \varepsilon T \varepsilon] \dot ' - \varepsilon T t'] \, dt' \end which may be regarded as a function of ''ε''. Calculating the derivative at ''ε'' = 0 and using Leibniz's rule (derivatives and integrals), Leibniz's rule, we get : \begin 0 = \frac = & L mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2">_2">mathbf_[t_2<_a>_\dot_[t_2.html" ;"title="_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2T - L [\mathbf [t_1], \dot [t_1], t_1] T \\ pt& + \int_^ \frac \left( - \frac \dot T + \frac \right) + \frac \left( - \frac ^2 T + \frac \dot - \frac \ddot T \right) \, dt. \end Notice that the Euler–Lagrange equations imply : \begin \frac \left( \frac \frac \dot T \right) & = \left( \frac \frac \right) \frac \dot T + \frac \left( \frac \frac \right) \dot T + \frac \frac \ddot \, T \\ pt& = \frac \frac \dot T + \frac \left( \frac \dot \right) \dot T + \frac \frac \ddot \, T. \end Substituting this into the previous equation, one gets : \begin 0 = \frac = & L mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2">_2">mathbf_[t_2<_a>_\dot_[t_2.html" ;"title="_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2T - L [\mathbf [t_1], \dot [t_1], t_1] T - \frac \frac \dot [t_2] T + \frac \frac \dot [t_1] T \\ pt& + \int_^ \frac \frac + \frac \frac \dot \, dt. \end Again using the Euler–Lagrange equations we get : \frac \left( \frac \frac \right) = \left( \frac \frac \right) \frac + \frac \frac \dot = \frac \frac + \frac \frac \dot. Substituting this into the previous equation, one gets : \begin 0 = & L mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2">_2">mathbf_[t_2<_a>_\dot_[t_2.html" ;"title="_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2">_2.html" ;"title="mathbf [t_2">mathbf [t_2 \dot [t_2 t_2T - L [\mathbf [t_1], \dot [t_1], t_1] T - \frac \frac \dot [t_2] T + \frac \frac \dot [t_1] T \\ pt& + \frac \frac [t_2] - \frac \frac [t_1]. \end From which one can see that :\left( \frac \frac \dot - L \right) T - \frac \frac is a constant of the motion, i.e., it is a conserved quantity. Since φ ''q, 0= q, we get \frac = 1 and so the conserved quantity simplifies to :\left( \frac \dot - L \right) T - \frac \frac. To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes. The same result can be obtained in the more general case.


Geometric derivation

The Noether’s theorem can be seen as a consequence of the fundamental theorem of calculus (known by various names in physics such as the Generalized Stokes theorem or the Gradient theorem): for a function S analytical in a domain , \int_dS=0 where is a closed path in . Here, the ''function'' S(\mathbf,t) is the action ''function'' that is computed by the integration of the Lagrangian over optimal trajectories or equivalently obtained through the Hamilton-Jacobi equation. As \partial S/\partial\mathbf=\mathbf (where \mathbfis the momentum) and \partial S/\partial t=-H (where H is the Hamiltonian), the differential of this function is given by dS=\mathbfd\mathbf-Hdt. Using the geometrical approach, the conserved quantity for a symmetry in Noether’s sense can be derived. The symmetry is expressed as an infinitesimal transformation:\begin \mathbf & = & \mathbf+\epsilon\phi_(\mathbf,t)\\ t' & = & t+\epsilon\phi_(\mathbf,t) \end Let be an optimal trajectory and ' its image under the above transformation (\phi_,\phi_)^ (which is also an optimal trajectory). The closed path of integration is chosen as ABB'A', where the branches AB and A'B' are given and ' . By the hypothesis of Noether theorem, to the first order in \epsilon, \int_dS=\int_dS therefore, \int_^dS=\int_^dS By definition, on the AA' branch we have d\mathbf=\epsilon\phi_(\mathbf,t) and dt=\epsilon\phi_(\mathbf,t). Therefore, to the first order in \epsilon, the quantity I=\mathbf\phi_-H\phi_ is conserved along the trajectory.


Field-theoretic derivation

Noether's theorem may also be derived for tensor fields \varphi^A where the index ''A'' ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates ''x''μ where the index ''μ'' ranges over time (''μ'' = 0) and three spatial dimensions (''μ'' = 1, 2, 3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written :x^\mu \rightarrow \xi^\mu = x^\mu + \delta x^\mu whereas the transformation of the field variables is expressed as :\varphi^A \rightarrow \alpha^A \left(\xi^\mu\right) = \varphi^A \left(x^\mu\right) + \delta \varphi^A \left(x^\mu\right)\,. By this definition, the field variations \delta\varphi^A result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field ''α''''A'' depends on the transformed coordinates ξμ. To isolate the intrinsic changes, the field variation at a single point ''x''μ may be defined :\alpha^A \left(x^\mu\right) = \varphi^A \left(x^\mu\right) + \bar \varphi^A \left(x^\mu\right)\,. If the coordinates are changed, the boundary of the region of space–time over which the Lagrangian is being integrated also changes; the original boundary and its transformed version are denoted as Ω and Ω’, respectively. Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the action, which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as :\int_ L \left( \alpha^A, _, \xi^\mu \right) d^4\xi - \int_ L \left( \varphi^A, _, x^\mu \right) d^x = 0 where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g. :_ = \frac\,. Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form : \int_\Omega \left\ d^4 x = 0 \,. The difference in Lagrangians can be written to first-order in the infinitesimal variations as : \left L \left( \alpha^A, _, x^\mu \right) - L \left( \varphi^A, _, x^\mu \right) \right= \frac \bar \varphi^A + \frac \bar _ \,. However, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they commute : \bar _ = \bar \frac = \frac \left(\bar \varphi^A\right) \,. Using the Euler–Lagrange field equations : \frac \left( \frac \right) = \frac the difference in Lagrangians can be written neatly as :\begin &\left L \left( \alpha^A, _, x^\mu \right) - L \left( \varphi^A, _, x^\mu \right) \right\\ pt = &\frac \left( \frac \right) \bar \varphi^A + \frac \bar _ = \frac \left( \frac \bar \varphi^A \right). \end Thus, the change in the action can be written as : \int_\Omega \frac \left\ d^x = 0 \,. Since this holds for any region Ω, the integrand must be zero : \frac \left\ = 0 \,. For any combination of the various
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
transformations, the perturbation can be written :\begin \delta x^ &= \varepsilon X^\mu \\ \delta \varphi^A &= \varepsilon \Psi^A = \bar \varphi^A + \varepsilon \mathcal_X \varphi^A \end where \mathcal_X \varphi^A is the Lie derivative of \varphi^A in the ''X''''μ'' direction. When \varphi^A is a scalar or _ = 0 , :\mathcal_X \varphi^A = \frac X^\mu\,. These equations imply that the field variation taken at one point equals :\bar \varphi^A = \varepsilon \Psi^A - \varepsilon \mathcal_X \varphi^A\,. Differentiating the above divergence with respect to ''ε'' at ''ε'' = 0 and changing the sign yields the conservation law :\frac j^\sigma = 0 where the conserved current equals : j^\sigma = \left frac \mathcal_X \varphi^A - L \, X^\sigma\right - \left(\frac \right) \Psi^A\,.


Manifold/fiber bundle derivation

Suppose we have an ''n''-dimensional oriented
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, ''M'' and a target manifold ''T''. Let \mathcal be the configuration space of smooth functions from ''M'' to ''T''. (More generally, we can have smooth sections of a fiber bundle ''T'' over ''M''.) Examples of this ''M'' in physics include: * In
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, in the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
formulation, ''M'' is the one-dimensional manifold \mathbb, representing time and the target space is the cotangent bundle of
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
of generalized positions. * In field theory, ''M'' is the
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
manifold and the target space is the set of values the fields can take at any given point. For example, if there are ''m'' real-valued
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
s, \varphi_1,\ldots,\varphi_m, then the target manifold is \mathbb^. If the field is a real vector field, then the target manifold is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to \mathbb^. Now suppose there is a functional :\mathcal:\mathcal\rightarrow \mathbb, called the action. (It takes values into \mathbb, rather than \mathbb; this is for physical reasons, and is unimportant for this proof.) To get to the usual version of Noether's theorem, we need additional restrictions on the action. We assume \mathcal varphi/math> is the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
over ''M'' of a function :\mathcal(\varphi,\partial_\mu\varphi,x) called the Lagrangian density, depending on \varphi, its
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
and the position. In other words, for \varphi in \mathcal : \mathcal varphi,=\,\int_M \mathcal varphi(x),\partial_\mu\varphi(x),x\, d^x. Suppose we are given boundary conditions, i.e., a specification of the value of \varphi at the boundary if ''M'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, or some limit on \varphi as ''x'' approaches ∞. Then the subspace of \mathcal consisting of functions \varphi such that all functional derivatives of \mathcal at \varphi are zero, that is: :\frac\approx 0 and that \varphi satisfies the given boundary conditions, is the subspace of on shell solutions. (See principle of stationary action) Now, suppose we have an
infinitesimal transformation In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 ...
on \mathcal, generated by a functional derivation, ''Q'' such that :Q \left \int_N \mathcal \, \mathrm^n x \right\approx \int_ f^\mu varphi(x),\partial\varphi,\partial\partial\varphi,\ldots\, ds_\mu for all compact submanifolds ''N'' or in other words, :Q mathcal(x)approx\partial_\mu f^\mu(x) for all ''x'', where we set :\mathcal(x)=\mathcal varphi(x), \partial_\mu \varphi(x),x If this holds on shell and off shell, we say ''Q'' generates an off-shell symmetry. If this only holds on shell, we say ''Q'' generates an on-shell symmetry. Then, we say ''Q'' is a generator of a one parameter
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
. Now, for any ''N'', because of the Euler–Lagrange theorem, on shell (and only on-shell), we have : \begin Q\left int_N \mathcal \, \mathrm^nx \right& =\int_N \left frac - \partial_\mu \frac \right varphi\, \mathrm^nx + \int_ \fracQ varphi\, \mathrms_\mu \\ & \approx\int_ f^\mu \, \mathrms_\mu. \end Since this is true for any ''N'', we have :\partial_\mu\left fracQ[\varphif^\mu\right">varphi.html" ;"title="fracQ[\varphi">fracQ[\varphif^\mu\rightapprox 0. But this is the continuity equation for the current J^\mu defined by: :J^\mu\,=\,\fracQ varphif^\mu, which is called the Noether current associated with the
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
. The continuity equation tells us that if we integrate this current over a space-like slice, we get a conserved quantity called the Noether charge (provided, of course, if ''M'' is noncompact, the currents fall off sufficiently fast at infinity).


Comments

Noether's theorem is an on shell theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that : \int_ J^\mu ds_ \approx 0. The quantum analogs of Noether's theorem involving expectation values (e.g., \left\langle\int d^x~\partial \cdot \textbf \right\rangle = 0) probing off shell quantities as well are the Ward–Takahashi identities.


Generalization to Lie algebras

Suppose we have two symmetry derivations ''Q''1 and ''Q''2. Then, 'Q''1, ''Q''2is also a symmetry derivation. Let us see this explicitly. Let us say Q_1 mathcalapprox \partial_\mu f_1^\mu and Q_2 mathcalapprox \partial_\mu f_2^\mu Then, _1,Q_2\mathcal] = Q_1 _2[\mathcal-Q_2[Q_1[\mathcal">mathcal.html" ;"title="_2[\mathcal">_2[\mathcal-Q_2[Q_1[\mathcal\approx\partial_\mu f_^\mu where ''f''12 = ''Q''1[''f''2''μ''] âˆ’ ''Q''2[''f''1''μ'']. So, j_^\mu = \left(\frac \mathcal\right)(Q_1[Q_2[\varphi - Q_2[Q_1[\varphi)-f_^\mu. This shows we can extend Noether's theorem to larger Lie algebras in a natural way.


Generalization of the proof

This applies to ''any'' local symmetry derivation ''Q'' satisfying ''QS'' â‰ˆ 0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let ''ε'' be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ''ε'' is a
test function In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly suppor ...
. Then, because of the variational principle (which does ''not'' apply to the boundary, by the way), the derivation distribution q generated by ''q'' 'ε''Φ(''x'')] = ''ε''(''x'')''Q'' �(''x'')satisfies ''q'' 'ε''''S''] â‰ˆ 0 for every ''ε'', or more compactly, ''q''(''x'') 'S''nbsp;≈ 0 for all ''x'' not on the boundary (but remember that ''q''(''x'') is a shorthand for a derivation ''distribution'', not a derivation parametrized by ''x'' in general). This is the generalization of Noether's theorem. To see how the generalization is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on \varphi and its first derivatives. Also, assume :Q mathcalapprox\partial_\mu f^\mu Then, : \begin q varepsilon\mathcal] & = \int q varepsilon\mathcal] d^ x \\ pt& = \int \left\ d^ x \\ pt& = \int \left\ \, d^ x \\ pt& \approx \int \varepsilon \partial_\mu \left\ \, d^ x \end for all \varepsilon. More generally, if the Lagrangian depends on higher derivatives, then : \partial_\mu\left f^\mu - \left[\frac \mathcal \rightQ[\varphi">frac_\mathcal_\right.html" ;"title=" f^\mu - \left[\frac \mathcal \right"> f^\mu - \left[\frac \mathcal \rightQ[\varphi - 2\left[\frac \mathcal\right]\partial_\nu Q varphi + \partial_\nu\left[\left[\frac\mathcal\right] Q varphiright] - \,\dotsm \right] \approx 0.


Examples


Example 1: Conservation of energy

Looking at the specific case of a Newtonian particle of mass ''m'', coordinate ''x'', moving under the influence of a potential ''V'', coordinatized by time ''t''. The action, ''S'', is: :\begin \mathcal & = \int L\left (t),\dot(t)\right\, dt \\ & = \int \left(\frac m 2 \sum_^3\dot_i^2 - V(x(t))\right) \, dt. \end The first term in the brackets is the
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of the particle, while the second is its
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
. Consider the generator of time translations Q = \frac. In other words, Q (t)= \dot(t). The coordinate ''x'' has an explicit dependence on time, whilst ''V'' does not; consequently: :Q = \frac\left frac\sum_i\dot_i^2 - V(x)\right= m \sum_i\dot_i\ddot_i - \sum_i\frac\dot_i so we can set :L = \frac \sum_i\dot_i^2 - V(x). Then, :\begin j & = \sum_^3\fracQ _i- L \\ & = m \sum_i\dot_i^2 - \left frac\sum_i\dot_i^2 - V(x)\right\\ pt & = \frac\sum_i\dot_i^2 + V(x). \end The right hand side is the energy, and Noether's theorem states that dj/dt = 0 (i.e. the principle of conservation of energy is a consequence of invariance under time translations). More generally, if the Lagrangian does not depend explicitly on time, the quantity :\sum_^3 \frac\dot - L (called the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
) is conserved.


Example 2: Conservation of center of momentum

Still considering 1-dimensional time, let :\begin \mathcal\left vec\right & = \int \mathcal\left vec(t), \dot(t)\right dt \\ pt & = \int \left sum^N_ \frac\left(\dot_\alpha\right)^2 - \sum_ V_\left(\vec_\beta - \vec_\alpha\right)\rightdt, \end for N Newtonian particles where the potential only depends pairwise upon the relative displacement. For \vec, consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words, :Q_i\left ^j_\alpha(t)\right= t \delta^j_i. And :\begin Q_i mathcal & = \sum_\alpha m_\alpha \dot_\alpha^i - \sum_t \partial_i V_\left(\vec_\beta - \vec_\alpha\right) \\ & = \sum_\alpha m_\alpha \dot_\alpha^i. \end This has the form of \frac\sum_\alpha m_\alpha x^i_\alpha so we can set :\vec = \sum_\alpha m_\alpha \vec_\alpha. Then, :\begin \vec & = \sum_\alpha \left(\frac \mathcal\right)\cdot\vec\left vec_\alpha\right- \vec \\ pt & = \sum_\alpha \left(m_\alpha \dot_\alpha t - m_\alpha \vec_\alpha\right) \\ pt & = \vect - M\vec_ \end where \vec is the total momentum, ''M'' is the total mass and \vec_ is the center of mass. Noether's theorem states: :\frac = 0 \Rightarrow \vec - M \dot_ = 0.


Example 3: Conformal transformation

Both examples 1 and 2 are over a 1-dimensional manifold (time). An example involving spacetime is a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)- Minkowski spacetime. :\begin \mathcal varphi & = \int \mathcal\left varphi (x), \partial_\mu \varphi (x)\rightd^4 x \\ pt & = \int \left(\frac\partial^\mu \varphi \partial_\mu \varphi - \lambda \varphi^4\right) d^4 x \end For ''Q'', consider the generator of a spacetime rescaling. In other words, :Q varphi(x)= x^\mu\partial_\mu \varphi(x) + \varphi(x). The second term on the right hand side is due to the "conformal weight" of \varphi. And :Q mathcal= \partial^\mu\varphi\left(\partial_\mu\varphi + x^\nu\partial_\mu\partial_\nu\varphi + \partial_\mu\varphi\right) - 4\lambda\varphi^3\left(x^\mu\partial_\mu\varphi + \varphi\right). This has the form of :\partial_\mu\left fracx^\mu\partial^\nu\varphi\partial_\nu\varphi - \lambda x^\mu \varphi^4 \right= \partial_\mu\left(x^\mu\mathcal\right) (where we have performed a change of dummy indices) so set :f^\mu = x^\mu\mathcal. Then :\begin j^\mu & = \left frac\mathcal\right varphif^\mu \\ & = \partial^\mu\varphi\left(x^\nu\partial_\nu\varphi + \varphi\right) - x^\mu\left(\frac 1 2 \partial^\nu\varphi\partial_\nu\varphi - \lambda\varphi^4\right). \end Noether's theorem states that \partial_\mu j^\mu = 0 (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side). If one tries to find the Ward–Takahashi analog of this equation, one runs into a problem because of anomalies.


Applications

Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example: * Invariance of an isolated system with respect to spatial
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
(in other words, that the laws of physics are the same at all locations in space) gives the law of conservation of linear momentum (which states that the total linear momentum of an isolated system is constant) * Invariance of an isolated system with respect to
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
translation (i.e. that the laws of physics are the same at all points in time) gives the law of conservation of energy (which states that the total energy of an isolated system is constant) * Invariance of an isolated system with respect to
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
(i.e., that the laws of physics are the same with respect to all angular orientations in space) gives the law of conservation of
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity â€“ the total ang ...
(which states that the total angular momentum of an isolated system is constant) * Invariance of an isolated system with respect to Lorentz boosts (i.e., that the laws of physics are the same with respect to all inertial reference frames) gives the center-of-mass theorem (which states that the center-of-mass of an isolated system moves at a constant velocity). In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, the analog to Noether's theorem, the Ward–Takahashi identity, yields further conservation laws, such as the conservation of
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
from the invariance with respect to a change in the phase factor of the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
field of the charged particle and the associated gauge of the
electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
and vector potential. The Noether charge is also used in calculating the
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
of stationary black holes.


See also

* Conservation law *
Charge (physics) In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specificall ...
* Gauge symmetry * Gauge symmetry (mathematics) * Invariant (physics) * Goldstone boson *
Symmetry (physics) The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some Transformation (function), transformation. A family of particular transformations m ...


References


Further reading

* *
Online copy
* * *


External links

* * (Original in ''Gott. Nachr.'' 1918:235–257) * * * * *

at MathPages. * * * *{{cite journal, author1=Sardanashvily, journal= International Journal of Geometric Methods in Modern Physics, title=Gauge conservation laws in a general setting. Superpotential , volume=6 , pages=1047–1056 , year=2009 , arxiv=0906.1732, bibcode = 2009arXiv0906.1732S, doi=10.1142/S0219887809003862, issue=6 Articles containing proofs Calculus of variations Conservation laws Concepts in physics Eponymous theorems of physics Partial differential equations Physics theorems Quantum field theory Symmetry