Noether's theorem

Noether’s theorem states that every continuous symmetry corresponds to a conserved physical quantity. More precisely, if we have a family of unitary operators $U(\epsilon)$ such that $U(\epsilon) H U(\epsilon)\adj = H$ for small $\epsilon$ , then there exists a Hermitian operator $A$

\begin{align*} A = i\hbar \left. \frac{\partial U(\epsilon)}{\partial\epsilon} \right|_{\epsilon=0}. \end{align*}

Such that $[H,A] = 0$ , and $A$ is conserved with time.

In classical mechanics, consider a Lagrangian $L(x,\dot x,t)$ that is invariant under some coordinate transform $x(t) \to x'(t) = x(t) + \epsilon \, \delta x(t)$ . The variation of the Lagrangian is

\begin{align*} \delta L = \frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot x} \delta \dot x =: \frac{d F}{d t}. \end{align*}

For some function $F(t)$ . Then Noether’s theorem states that $Q$ is conserved

\begin{align*} Q = \frac{\partial L}{\partial \dot x} \delta x - F(t). \end{align*}

Intuitively, Nother’s theorem states that if some quantity is invariant under some transformation, that quantity is conserved. For example, momentum is invariant under translation in space, so momentum is conserved. Angular momentum is invariant under rotation, and so is conserved.