Terence Tao

One involves particles, one involves waves, and so forth.

But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

For example, in classical mechanics, there's this thing called Noether's theorem.

Every time there's a symmetry in a physical system, there is a conservation law.

So the laws of physics are translation invariant.

Like if I move 10 steps to the left, I experience the same laws of physics as if I was here.

And that corresponds to conservation momentum.

If I turn around by some angle, again, I experience the same laws of physics.

This corresponds to the conservation of angular momentum.

If I wait for 10 minutes, I still have the same laws of physics.

So this time transition invariance, this corresponds to the law of conservation of energy.

So there's this fundamental connection between symmetry and conservation.

And that's also true in quantum mechanics, even though the equations are completely different.

But because they're both coming from the Hamiltonian, the Hamiltonian controls everything.

Every time the Hamiltonian has a symmetry, the equations will have a conservation law.

So once you have the right language, it actually makes things a lot cleaner.

One of the problems why we can't unify quantum mechanics and general relativity yet

We haven't figured out what the fundamental objects are.

For example, we have to give up the notion of space and time being these almost Euclidean-type spaces.

And we kind of know that at very tiny scales, there's going to be corner fluctuations, there's space-time foam.