Editing Noether's theorem (section)

== 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 (physics)|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. 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).

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 number|complex]] field of the charged particle and the associated [[gauge invariance|gauge]] of the [[electric potential]] and [[vector potential]].

The Noether charge is also used in calculating the [[entropy]] of [[stationary black hole]]s.<ref>{{cite journal |last1=Iyer |first1=Vivek |last2=Wald |first2=Robert M. |author-link2=Robert Wald |date=15 October 1995 |title=A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes |journal=[[Physical Review D]] |volume=52 |issue=8 |pages=4430–4439 |arxiv=gr-qc/9503052 |bibcode=1995PhRvD..52.4430I |doi=10.1103/PhysRevD.52.4430 |pmid=10019667 |s2cid=2588285}}</ref>