Why is symmetry defined in Noether's theorem?

Noether's theorem establishes a profound connection between symmetries and conservation laws in physics. In this context, symmetry refers to the property that a physical system remains invariant under specific transformations, which can include rotations, translations, or time shifts. The theorem asserts that for every continuous symmetry of the action of a physical system, there corresponds a conserved quantity.

This relationship hinges on transformations satisfying certain technical prerequisites. These conditions ensure that the transformations applied do not alter the fundamental properties or the equations of motion governing the system. For example, if a system is symmetric under time translation, this implies conservation of energy; if symmetric under spatial translation, it implies conservation of momentum. Hence, the definition of symmetry is rooted in the requirement that these transformations uphold the valid mathematical structure laid out by the laws of physics.

The other options do not adequately capture the essence of how symmetry is defined in Noether's theorem. While covariance and randomness may play roles in broader discussions of physics, they do not directly relate to the precise requirements for symmetry as outlined in the theorem. Similarly, while arbitrary choices may exist in modeling, Noether's theorem is fundamentally grounded in the intrinsic properties of physical laws rather than subjective selections.