noether’s theorem


  • 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 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 Ω, the integrand must be zero For any combination of the various symmetry transformations, the perturbation can be written where is the Lie derivative of in the Xμ direction.

  • 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.

  • Generalization of the proof[edit] This applies to any local symmetry derivation Q satisfying , and also to more general local functional differentiable actions, including
    ones where the Lagrangian depends on higher derivatives of the fields.

  • 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 be further generalized to the case where
    the Lagrangian depends on up to the nth derivative, and can also be formulated using jet bundles).

  • The action is defined as the time integral I of a function known as the Lagrangian L 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.

  • 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 tensor Tμν,[13] where we have used in place of .

  • For example: • Invariance of an isolated system with respect to spatial translation (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 translation (i.e.

  • 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 result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field αA depends on the transformed coordinates ξμ.

  • 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.

  • 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).

  • [8] A more sophisticated version of the theorem involving fields states that: To every differentiable symmetry generated by local actions there corresponds a conserved current.

  • [7] 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.

  • In that case, ; the conserved quantity is the corresponding linear momentum pk[12] In special and general relativity, these two conservation laws can be expressed either globally
    (as it is done above), or locally as a continuity equation.

  • 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.

  • 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).

  • Also, assume Then, for all .More generally, if the Lagrangian depends on higher derivatives, then Examples Example 1: Conservation of energy[edit] 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, while the second is its potential

  • 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 (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 (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).

  • When is a scalar or , These equations imply that the field variation taken at one point equals Differentiating the above divergence with respect to ε at ε = 0 and changing
    the sign yields the conservation law where the conserved current equals Manifold/fiber bundle derivation[edit] Suppose we have an n-dimensional oriented Riemannian 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.) Examples of this M in physics include: • In classical mechanics, in the Hamiltonian formulation, M is the one-dimensional manifold , representing
    time and the target space is the cotangent bundle of space of generalized positions.

  • To wit, by using the expression given earlier, and collecting the four conserved currents (one for each ) into a tensor , Noether’s theorem gives with (we relabelled as at
    an intermediate step to avoid conflict).

  • Example 2: Conservation of center of momentum[edit] Still considering 1-dimensional time, let for Newtonian particles where the potential only depends pairwise upon the relative

  • To isolate the intrinsic changes, the field variation at a single point xμ 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 Ω and Ω’, respectively.

  • 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).

  • 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

  • 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.

  • (However, the obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see Canonical stress–energy tensor.)

  • In quantum field theory, the analog to Noether’s theorem, the Ward–Takahashi identity, 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 gauge of the electric potential and vector potential.

  • 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.

  • 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.

  • Mathematically such a symmetry is represented as a flow, φ, which acts on the variables as follows 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.

  • [2] 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.

  • [15] In quantum mechanics, the probability amplitude ψ(x) of finding a particle at a point x is a complex field φ, because it ascribes a complex number to every point in space
    and 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.

  • A specific example is the Klein–Gordon equation, the relativistically correct version of the Schrödinger equation for spinless particles, which has the Lagrangian density
    In this case, Noether’s theorem states that the conserved current equals which, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle.

  • Assuming no boundary terms in the action, Noether’s theorem implies that The quantum analogs of Noether’s theorem involving expectation values (e.g., ) probing off shell quantities
    as well are the Ward–Takahashi identities.

  • 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.

  • More general cases follow the same idea: • When more coordinates undergo a symmetry transformation , their effects add up by linearity to a conserved quantity .

  • • Finally, when instead of a trajectory entire fields are considered, the argument replaces o the interval with a bounded region of the -domain, o the endpoints and with the
    boundary of the region, o and its contribution to is interpreted as a flux of a conserved current , that is built in a way analogous to the prior definition of a conserved quantity.

  • Time invariance For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes , without any change in the coordinates

  • The middle part does not change the action either, because its transformation is a symmetry and thus preserves the Lagrangian and the action .

  • 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).

  • 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

  • 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[13] (this will be derived in the next section).

  • A continuous transformation of the fields can be written infinitesimally as where is in general a function that may depend on both and .

  • • 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.

  • Noether’s theorem or Noether’s first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation

  • Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.

  • In such cases, the conservation law is expressed in a four-dimensional way 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.

  • 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.

  • 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.

  • 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’s invariant), its Lagrangian
    is symmetric under continuous rotation: from this symmetry, Noether’s theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.

  • And This has the form of (where we have performed a change of dummy indices) so set Then Noether’s theorem states that (as one may explicitly check by substituting the Euler–Lagrange
    equations into the left hand side).

  • In other words, for in Suppose we are given boundary conditions, i.e., a specification of the value of at the boundary if M is compact, or some limit on as x approaches ∞.

  • 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,
    as was customary in Newtonian mechanics.

  • (See principle of stationary action) Now, suppose we have an infinitesimal transformation on , generated by a functional derivation, Q such that for all compact submanifolds
    N or in other words, for all x, where we set If this holds on shell and off shell, we say Q generates an off-shell symmetry.

  • Brief illustration and overview of the concept The main idea behind Noether’s theorem is most easily illustrated by a system with one coordinate and a continuous symmetry
    (gray arrows on the diagram).

  • The conservation law of a physical quantity is usually expressed as a continuity equation.

  • 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 coordinate x has an explicit dependence on time, whilst V does not; consequently: so we can set Then, The right hand side is the energy, and Noether’s theorem states that

  • • When there are time transformations , they cause the “buffering” segments to contribute the two following terms to : first term being due to stretching in temporal dimension
    of the “buffering” segment (that changes the size of the domain of integration), and the second is due to its “slanting” just as in the exemplar case.

  • In particular it would not change under a variation that applies the symmetry flow on a time segment [t0, t1] and is motionless outside that segment.

  • These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates xμ where the index μ ranges over time and three spatial

  • Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used (or most often implemented) version
    of Noether’s theorem.

  • 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 of the stress–energy

  • In modern (since c. 1980[9]) terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current.

  • In these regions both the coordinate and velocity change, but changes by , and the change in the coordinate is negligeable by comparison since the time span of the buffering
    is small (taken to the limit of 0), so .

  • Then, because of the variational principle (which does not apply to the boundary, by the way), the derivation distribution q generated by satisfies for every ε, or more compactly,
    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).

  • For translations, Qr is a constant with units of length; for rotations, it is an expression linear in the components of q, and the parameters make up an angle.


Works Cited

[‘This is sometimes referred to as Noether’s first theorem, see Noether’s second theorem.
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