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Noether's theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. For example, the conservation of energy is a consequence of the fact that all laws of physics (including the values of the physical constants) are invariant under translation through time; they do not change as time passes. Image File history File links Broom_icon. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... This article or section does not cite its references or sources. ... Look up conservation of energy in Wiktionary, the free dictionary. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...

Noether's theorem, published in 1918, holds for all physical laws based upon the action principle. It is named after the early 20th century mathematician Emmy Noether. Noether's theorem is a relationship of classical mechanics between pairs of conjugate variables—if the action is invariant under a shift in one of the two physical variables, then the equations of motion resulting from holding that action stationary conserve the value of the other of the pair of variables. These conjugate pairs also play a crucial role in quantum theory—they are the pairs of variables that are related by the Heisenberg uncertainty principle (such as position and momentum, time and energy, angle and angular momentum, etc). For a list of set rules, see Laws of science. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physics, especially in quantum mechanics, conjugate variables are pairs of variables that share an uncertainty relation. ... Look up quantum in Wiktionary, the free dictionary. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... Look up position in Wiktionary, the free dictionary. ... This article is about momentum in physics. ... Look up time in Wiktionary, the free dictionary. ... This article is about angles in geometry. ... This gyroscope remains upright while spinning due to its angular momentum. ...

Mathematical statement of the theorem

Informally, Noether's theorem can be stated as (technical fine points aside):

To every differentiable symmetry generated by local actions, there corresponds a conserved current.

This article or section does not cite its references or sources. ... In physics, a conserved current, J, obeys a conservation law. ...

Explanation

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 of transformations satisfying certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity equation. It has been suggested that this article or section be merged with Covariant. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... A physical quantity is either a quantity within physics that can be measured (e. ... All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

The formal proof of the theorem uses only the condition of invariance to derive an expression for a current associated with a conserved physical quantity. The conserved quantity is called the Noether charge and the flow carrying that 'charge' is called the Noether current. The Noether current is defined up to a solenoidal vector field. This article is in need of attention. ...

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:

• the invariance of physical systems with respect to spatial translation (in other words, that the laws of physics do not vary with locations in space) gives the law of conservation of linear momentum;
• invariance with respect to rotation gives the law of conservation of angular momentum;
• invariance with respect to time translation gives the well known law of conservation of energy

In quantum field theory, the analog to Noether's theorem, the Ward-Takahashi identities, yields further conservation laws, such as the conservation of electric charge from the invariance with respect to a change in the phase factor of the complex field of the charged particle and the associated the gauge of the electric potential and vector potential. In physics, a translation is the operation changing the positions of all objects according to the formula where is a constant vector. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ... This article is about rotation as a movement of a physical body. ... This gyroscope remains upright while spinning due to its angular momentum. ... Look up time in Wiktionary, the free dictionary. ... Conservation of energy (the first law of thermodynamics) is one of several conservation laws. ... Quantum field theory (QFT) is the quantum theory of fields. ... This article is about a formulation of quantum mechanics. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... There are very few or no other articles that link to this one. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... This article does not cite any references or sources. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ...

The Noether charge is also used in calculating the entropy of stationary black holes^[1]. For a less technical and generally accessible introduction to the topic, see Introduction to entropy. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ...

Proof

Suppose we have an n-dimensional manifold, M and a target manifold T. Let be the configuration space of smooth functions from M to T. (More generally, we can have smooth sections of a fiber bundle over M) On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Examples of this "M" in physics include:

• In classical mechanics, in the Hamiltonian formulation, M is the one-dimensional manifold R, representing time and the target space is the cotangent bundle of space of generalized positions.
• In field theory, M is the spacetime 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 fields, φ₁,...,φ_m, then the target manifold is R^m. If the field is a real vector field, then the target manifold is isomorphic to R³.

Now suppose there is a functional Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... Space has been an interest for philosophers and scientists for much of human history. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... For other uses of this term, see Spacetime (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the term functional is applied to certain functions. ...

called the action. (Note that it takes values into , rather than ; this is for physical reasons, and doesn't really matter for this proof.) In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

To get to the usual version of Noether's theorem, we need additional restrictions on the action. We assume is the integral over M of a function In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... This article is about the concept of integrals in calculus. ...

called the Lagrangian, depending on φ, its derivative and the position. In other words, for φ in A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... For a non-technical overview of the subject, see Calculus. ...

Suppose we are given boundary conditions, ie., a specification of the value of φ at the boundary if M is compact, or some limit on φ as x approaches ∞. Then the subspace of consisting of functions φ such that all functional derivatives of at φ are zero, that is: In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... Screenshot (from SSCX Star Warzone). ... In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. ...

and that φ satisfies the given boundary conditions, is the subspace of on shell solutions. (See principle of stationary action) In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... The principle of stationary action for the Action (physics) S (a measure of the energy of the system under study) states that the variation in S is at an extremum, in symbols: where the independent variables are denoted by a set of acting at some time t. ...

Now, suppose we have an infinitesimal transformation on , generated by a functional derivation, Q such that To meet Wikipedias quality standards and make it more accessible, this article may require cleanup. ... In mathematics, the term functional is applied to certain functions. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ...

for all compact submanifolds N or in other words, Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...

for all x, where we set .

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 Lie group. In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is... Sphere symmetry group o. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Now, for any N, because of the Euler-Lagrange theorem, on shell (and only on-shell), we have In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...

Since this is true for any N, we have

But this is the continuity equation for the current defined by All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

which is called the Noether current associated with the symmetry. 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). Sphere symmetry group o. ... This article is about the concept of integrals in calculus. ... In the context of special relativity, space-like separated points (or events) in spacetime have a spacetime interval less than 0 (see sign convention). ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... In physics, a Noether charge is a physical quantity conserved as an effect of a continuous symmetry of the underlying system. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...

Comments

Noether's theorem is really a reflection of the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

Noether's theorem is an on shell theorem. The quantum analog of Noether's theorem are the Ward-Takahashi identities. In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... This article is about a formulation of quantum mechanics. ...

Suppose say we have two symmetry derivations Q₁ and Q₂. Then, [Q₁,Q₂] is also a symmetry derivation. Let's see this explicitly. Let's say

and

(it doesn't matter if this holds off shell or only on shell). Then, In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...

where f₁₂=Q₁[f₂^μ]-Q₂[f₁^μ]. So,

This shows we can (trivially) extend Noether's theorem to larger Lie algebras.

Generalisation of the proof

This applies to any derivation Q, not just symmetry derivations and also to more general functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields and nonlocal actions. 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. 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[φ(x)] satisfies q[ε][S]=0 for any ε on shell, or more compactly, q(x)[S] 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. This page deals with mathematical distributions. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...

To see how the generalization related to the version given above, assume that the action is the spacetime integral of a Lagrangian which only depends on φ and its first derivatives. Also, assume

(either off-shell or only on-shell is fine). Then,

for all ε.

More generally, if the Lagrangian depends on higher derivatives, then

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: In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

Consider the generator of time translations . In other words, . (Quantum field theoreticians would often put a factor of i on the right hand side.) Note that x has an explicit dependence on time, whilst V does not; consequently: In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

so we can set

Then,

The right hand side is the energy and Noether's theorem states that (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

(called the Hamiltonian) is conserved. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

Example 2: Conservation of center of momentum

Still considering 1-dimensional time, let

i.e. N Newtonian particles where the potential only depends pairwise upon the relative displacement.

For , let's consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,

Note that

This has the form of so we can set

Then,

where is the total momentum, M is the total mass and is the center of mass. Noether's theorem states:

.

Example 3: Conformal transformation

Both examples 1 and 2 are over a 1-dimensional manifold (time). For an example involving spacetime, let's work out the case of a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)-Minkowski spacetime. In mathematics and theoretical physics, a conformal transformation is a transformation of coordinates that preserves the angle. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

For Q, let's consider the generator of a spacetime rescaling. In other words,

The second term on the right hand side is due to the "conformal weight" of φ. Note that

This has the form of

(where we have performed a change of dummy indices) so we can set

Then,

Noether's theorem states that (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).

(Aside: If you try to find the Ward-Takahashi analog of this equation, you'd run into a problem because of anomalies.) This article is about a formulation of quantum mechanics. ... In physics, an anomaly is a classical symmetry — a symmetry of the Lagrangian — that is broken in quantum field theories. ...

See also

• Charge (physics)

In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. ...

References

• English translation of Noether's paper
• Article on Noether's theorem by John Baez
• E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws by Nina Byers
• Noether's Theorem at MathPages
• Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem

1. ^ Calculating the entropy of stationary black holes