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First page of Emmy Noether's article "Invariante Variationsprobleme" (1918), where she proved her theorem.
As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is symmetric under continuous rotations: from this symmetry, Noether's theorem dictates that the angular momentum 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 within this system, respectively.
As a final example, if the behavior of a physical system does not change upon spatial or temporal reflection, then its Lagrangian has reflection symmetry and time reversal symmetry respectively: Noether's theorem says that these symmetries result in the conservation laws of parity and entropy,[discuss] 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. The original version applied only to ordinary differential equations (used for describing distinct particles) and not partial differential equations (used for describing fields). The original versions also assume that the Lagrangian depends only upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative.[disputed – discuss] There are natural quantum counterparts of this theorem, expressed in the Ward-Takahashi identities. Generalizations of Noether's theorem to superspaces also exist.
Informal statement of the theorem
All fine technical points aside, Noether's theorem can be stated informally
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.
A more sophisticated version of the theorem involving fields states that:
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 (since ca. 1980) 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 a solenoidal (divergenceless) vector field.
In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants:
If an integral I is invariant under a continuous group G? with ? parameters, then ? linearly independent combinations of the Lagrangian expressions are divergences.
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 with respect to time) is zero,
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.
where the dot over q signifies the rate of change of the coordinates q,
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,
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
where the momentum
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 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 transformations 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.
Simple form using perturbations
The essence of Noether's theorem is generalizing the ignorable coordinates outlined.
One can assume that the Lagrangian L defined above is invariant under small perturbations (warpings) of the time variable t and the generalized coordinatesq. One may write
where the perturbations ?t and ?q are both small, but variable. For generality, assume there are (say) N such symmetry transformations 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,
where ?r are infinitesimal parameter coefficients corresponding to each:
For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes t -> t + ?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 energyH
Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate qk; so it is invariant (symmetric) under changes qk -> qk + ?qk. In that case, N = 1, T = 0, and Qk = 1; the conserved quantity is the corresponding momentumpk
The conservation of the angular momentumL = 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 about an axis n; such a rotation transforms the Cartesian coordinates by the equation
Since time is not being transformed, T=0. Taking as the ? parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by
Then Noether's theorem states that the following quantity is conserved,
In other words, the component of the angular momentum L along the n axis is conserved.
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 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 the Noether theorem for continuous fields in four-dimensional space-time is now given. Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used version (or most often implemented) of Noether's theorem.
Let there be a set of differentiable fields defined over all space and time; for example, the temperature 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
(the theorem can actually be further generalized to the case where the Lagrangian depends on up to the nth derivative using jet bundles)
A continuous transformation of the fields can be written infinitesimally as
where is in general a function that may depend on both and . The condition for to generate a physical symmetry is that the action is left invariant. This will certainly be true if the Lagrangian density is left invariant, but it will also be true if the Lagrangian changes by a divergence,
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 , so the most general symmetry transformation would be written as
with the consequence
For such systems, Noether's theorem states that there are conserved current densities
(where the dot product is understood to contract the field indices, not the index or index)
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 is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.
For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, is constant in its third argument. In that case, N = 4, one for each dimension of space and time. An infinitesimal translation in space, (with denoting the Kronecker delta), affects the fields as : 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 with the value at the point "behind" it which would be mapped onto by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as
The Lagrangian density transforms in the same way, , so
and thus Noether's theorem corresponds to the conservation law for the stress-energy tensorT??, where we have used in place of . To wit, by using the expression given earlier, and collecting the four conserved currents (one for each ) into a tensor , Noether's theorem gives
(note that we relabelled as at an intermediate step to avoid conflict). (However, note that the obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see Canonical stress-energy tensor.)
The conservation of electric charge, by contrast, can be derived by considering ? linear in the fields ? rather than in the derivatives. In quantum mechanics, the probability amplitude?(x) of finding a particle at a point x is a complex field ?, because it ascribes a complex number to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability p = |?|2 can be inferred from a set of measurements. Therefore, the system is invariant under transformations of the ? field and its complex conjugate field ?* that leave |?|2 unchanged, such as
In this case, Noether's theorem states that the conserved (j = 0) current equals
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, and is one of the prototype gauge symmetries of physics.
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
is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the Euler-Lagrange equations
And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a flow, ?, which acts on the variables as follows
where ? 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.
The action integral flows to
which may be regarded as a function of ?. Calculating the derivative at ?' = 0 and using Leibniz's rule, we get
Notice that the Euler-Lagrange equations imply
Substituting this into the previous equation, one gets
Again using the Euler-Lagrange equations we get
Substituting this into the previous equation, one gets
From which one can see that
is a constant of the motion, i.e., it is a conserved quantity. Since ?[q, 0] = q, we get and so the conserved quantity simplifies to
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.
Noether's theorem may also be derived for tensor fields ?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? where the index ? ranges over time (? = 0) and three spatial dimensions (? = 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
whereas the transformation of the field variables is expressed as
By this definition, the field variations A result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field ?A depends on the transformed coordinates ??. To isolate the intrinsic changes, the field variation at a single point x? may be defined
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 ? and ?', 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
where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g.
Since ? is a dummy variable of integration, and since the change in the boundary ? is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the divergence theorem into the following form
The difference in Lagrangians can be written to first-order in the infinitesimal variations as
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
Using the Euler-Lagrange field equations
the difference in Lagrangians can be written neatly as
Thus, the change in the action can be written as
Since this holds for any region ?, the integrand must be zero
For any combination of the various symmetry transformations, the perturbation can be written
where is the Lie derivative of ?A in the X? direction. When ?A is a scalar or ,
These equations imply that the field variation taken at one point equals
Differentiating the above divergence with respect to ? at ? = 0 and changing the sign yields the conservation law
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 mreal-valued scalar fields, , then the target manifold is Rm. If the field is a real vector field, then the target manifold is isomorphic to R3.
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).
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
Suppose we have two symmetry derivations Q1 and Q2. Then, [Q1, Q2] is also a symmetry derivation. Let's see this explicitly. Let's say
where f12 = Q1[f2?] - Q2[f1?]. So,
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 ? 0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let ? be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ? 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[?][?(x)] = ?(x)Q[?(x)] satisfies q[?][S] ? 0 for every ?, or more compactly, q(x)[S] ? 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 ? and its first derivatives. Also, assume
for all ?.
More generally, if the Lagrangian depends on higher derivatives, then
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:
The first term in the brackets is the kinetic energy of the particle, whilst the second is its potential energy. Consider the generator of time translationsQ = d/dt. In other words, . Note that x has an explicit dependence on time, whilst V does not; consequently:
so we can set
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
For Q, 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 . Note that
This has the form of
(where we have performed a change of dummy indices) so set
Noether's theorem states that (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).
Note that if one tries to find the Ward-Takahashi analog of this equation, one runs into a problem because of anomalies.
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;
^The term "Noether charge" occurs in Seligman, Group theory and its applications in physics, 1980: Latin American School of Physics, Mexico City, American Institute of Physics, 1981. It comes enters wider use during the 1980s, e.g. by G. Takeda in: Errol Gotsman, Gerald Tauber (eds.) From SU(3) to Gravity: Festschrift in Honor of Yuval Ne'eman, 1985, p. 196.