Symmetry and conservation laws
The symmetries of the physical laws are very interesting at this level, but they turn out, in the end, to be even more interesting and exciting when we come to quantum mechanics. For a reason which we cannot make clear at the level of the present discussion—a fact that most physicists still find somewhat staggering, a most profound and beautiful thing, is that, in quantum mechanics, for each of the rules of symmetry there is a corresponding conservation law; there is a definite connection between the laws of conservation and the symmetries of physical laws. We can only state this at present, without any attempt at explanation. The fact, for example, that the laws are symmetrical for translation in space when we add the principles of quantum mechanics, turns out to mean that momentum is conserved. That the laws are symmetrical under translation in time means, in quantum mechanics, that energy is conserved. Invariance under rotation through a fixed angle in space corresponds to the conservation of angular momentum. These connections are very interesting and beautiful things, among the most beautiful and profound things in physics. Incidentally, there are a number of symmetries which appear in quantum mechanics which have no classical analog, which have no method of description in classical physics. One of these is as follows: If $\psi$ is the amplitude for some process or other, we know that the absolute square of $\psi$ is the probability that the process will occur. Now if someone else were to make his calculations, not with this $\psi$, but with a $\psi'$ which differs merely by a change in phase (let $\Delta$ be some constant, and multiply $e^{i\Delta}$ times the old $\psi$), the absolute square of $\psi'$, which is the probability of the event, is then equal to the absolute square of $\psi$: \begin{equation} \label{Eq:I:52:1} \psi' = \psi e^{i\Delta};\quad \abs{\psi'}^2 = \abs{\psi}^2. \end{equation} Therefore the physical laws are unchanged if the phase of the wave function is shifted by an arbitrary constant. That is another symmetry. Physical laws must be of such a nature that a shift in the quantum-mechanical phase makes no difference. As we have just mentioned, in quantum mechanics there is a conservation law for every symmetry. The conservation law which is connected with the quantum-mechanical phase seems to be the conservation of electrical charge. This is altogether a very interesting business!