[pa7x1]

> Specifically, I'd love the intuition behind why it must be so that if the laws of physics are time/position invariant, it must be impossible to create or destroy energy/momentum.

The geometrical intuition is that the momentum operators are the generator of spatial translations (i.e. acting with the momentum operator P_x on a system is the same as applying an infinitesimal displacement in the direction x). https://en.wikipedia.org/wiki/Translation_operator_(quantum_...

And the Hamiltonian is the generator of time evolution (i.e. acting with the Hamiltonian on a system shifts it in time an infinitesimal amount). This is quite literally what the Schrödinger equation says, btw. H |Psi> ~ d/dt |Psi>

If the physics of a system (which are given by its Hamiltonian) are invariant under translations then it must be the case that a shift in time (Hamiltonian generates shifts in time) of the momentum (generates shifts in space) of a system is 0.

As the Hamiltonian gives us the energy of the system, if the system is invariant under time translations then its energy is conserved. Using the previous argument.

Rinse and repeat for any symmetry of the system. For instance, angular momentum operators are the generators of rotations. If your system is invariant under rotations, then it conserves angular momentum. Invariance under relative movement (relativistic invariance) gives conservation of center of mass. Etc... https://en.wikipedia.org/wiki/Symmetry_(physics)#Conservatio...