There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in the Ward–Takahashi identities. Generalizations of Noether's theorem to superspaces also exist. All fine technical points aside, Noether's theorem can be stated informally  A more sophisticated version of the theorem involving fields states that:  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. 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 c. 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:   The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate formula_1 and a continuous symmetry formula_2 (gray arrows on the diagram). Consider any trajectory formula_3 (bold on the diagram) that satisfies the system's laws of motion. That is, the action formula_4 governing this system is stationary on this trajectory, i.e. does not change under any local variation of the trajectory. In particular it would not change under a variation that applies the symmetry flow formula_5 on a time segment [] and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time formula_6 to transition between the segments gradually. The total change in the action formula_4 now comprises changes brought by every interval in play. Parts, where variation itself vanishes, bring no formula_8. The middle part doesn't change the action either, because its transformation formula_5 is a symmetry and thus preserves the Lagrangian formula_10 and the action formula_11. The only remaining parts are the "buffering" pieces. Roughly speaking, they contribute mostly through their "slanting" formula_12. That changes the Lagrangian by formula_13, which integrates to  These last terms, evaluated around the endpoints formula_14 and formula_15, should cancel each other in order to make the total change in the action formula_8 be zero, as would be expected if the trajectory is a solution.