Noethertétel

Noethertétel, the Hungarian term for Noether's theorem, is a foundational result in theoretical physics and mathematics that links continuous symmetries to conserved quantities. It was formulated by Emmy Noether in 1915 and published in 1918, and it has two related statements: the first theorem for finite-dimensional systems and the second theorem for gauge or local symmetries.

The first theorem states that if a system described by a Lagrangian is invariant under a continuous group of transformations depending on constant parameters, there exists a corresponding conserved current or charge. In mechanics, this explains why time translation symmetry leads to energy conservation, spatial translation symmetry to linear momentum conservation, and rotational symmetry to angular momentum conservation. In field theory, these ideas appear as conserved currents with vanishing divergence, yielding conserved quantities.

Applications and examples of the first theorem are widespread. Time invariance yields conserved energy; spatial translations yield conserved momentum; rotational symmetry yields conserved angular momentum. In quantum field theory, Noether’s theorem underpins conserved charges associated with global symmetries, such as electric charge, and more generally the quantities tied to internal symmetries.

The second theorem addresses theories with gauge or local symmetries. For these, invariance under an infinite-dimensional group leads to identities among the equations of motion rather than independent conserved currents. This reflects the redundancy of gauge degrees of freedom and gives rise to gauge identities and constraints in the formalism.

Impact: Noether’s theorem provides a unifying framework that connects symmetries to conservation laws across classical mechanics, electromagnetism, general relativity, and quantum field theory, making it a central tool in both physics and mathematics.