# Noether's Theorem

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Related to Emmy Noether's theorem: Noether charge, Noether current

## Noether’s Theorem

a fundamental theorem of physics that establishes the relation between the symmetry properties of a physical system and the laws of conservation. This theorem, formulated by Emmy Noether in 1918, asserts that, for a physical system whose equations of motion have the form of a system of differential equations and can be derived from a variational principle of mechanics, a conservation law corresponds to every transformation that is continuously dependent on one parameter and that leaves invariant a variational functional. The action S serves as the variational functional in the mechanics of particles or of fields. The equations of motion of the system are obtained when the variation of the action is set equal to zero: δS = 0 (the principle of least action). A differential conservation law corresponds to every transformation for which the action does not change. Integration of the equation that expresses such a law yields an integral conservation law.

Noether’s theorem gives the simplest universal method for deriving conservation laws in such areas as classical mechanics, quantum mechanics, and field theory.

Examples of continuous transformations in space-time that leave invariant the action and, consequently, the equations of motion are (1) a displacement of time, which expresses the physical property of the equivalence of all instants of time (the homogeneity of time), (2) a displacement of space, which expresses the property of the equivalence of all points in space (the homogeneity of space), (3) a three-dimensional spatial rotation, which expresses the property of the equivalence of all directions in space (the isotropy of space), and (4) four-dimensional rotations in space-time, in particular, the Lorentz transformations, which express the principle of relativity. According to Noether’s theorem, the law of conservation of energy follows from invariance with respect to a displacement of time; the law of conservation of momentum follows from invariance with respect to three-dimensional displacements; the law of conservation of angular momentum follows from invariance with respect to three-dimensional rotation; and the law of conservation of the Lorentz moment, or the generalized law of motion of the center of mass (the center of mass of a relativistic system moves uniformly and rectilinearly), follows from invariance with respect to the Lorentz transformations.

The applications of Noether’s theorem are not limited to space-time symmetries. For example, the law of conservation of electric charge follows from the independence of the dynamics of charged particles in electromagnetic fields with respect to gauge transformations of the first kind; in such transformations, the complex functions of the field ϕ(x) and ϕ*(x) are multiplied, respectively, by the factors eɑ and e~’a, where α is a continuous real parameter. Noether’s theorem is of particular importance in quantum field theory, where the conservation laws that follow from the existence of a certain symmetry group often are the main source of information on the properties of the objects under study.

### REFERENCES

Polak, L. S. Variatsionnye printsipy mekhaniki, ikh razvitie i primeneniia v fizike. Moscow, 1960.
Pauli, W. Reliativistskaia teoriia elementarnykh chastits. Moscow, 1947. (Translated from English.)
Bogoliubov, N. N., and D. V. Shirkov. Vvedenie v teoriiu kvantovannykh polei, 2nd ed. Moscow, 1973.
Matthews, P. T. Reliativistskaia kvantovaia teoriia vzaimodeistvii elementarnykh chastits. Moscow, 1959. (Translated from English.)

D. N. ZUBAREV

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