I know about Noether's theorem, but I don't actually know how to use it myself. Suppose our universe were symmetric with respect to reflections about planes. What conserved quantity would then exist by Noether's theorem?
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Noether's theorem states that there exists a conservation law for every continuous (in fact, differentiable) symmetry. Reflection is a discrete symmetry, so the theorem is not applicable here.
But, in quantum mechanics, you have the parity operator $P$, that reflects the coordinates $$P\psi(\vec{r}) = \psi(-\vec{r})$$ Since $P^2 = I$, the operator $P$ has eigenvalues $\pm 1$. If the hamiltonian is invariant under reflection, i.e. $[H,P]=0$, every eigenfunction of the hamiltonian is also a eigenfunction of the parity operator with eigenvalue $p$ (called intrinsic parity). Then, the intrinsic parity is conserved.
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