Someone asked a question the other day that made me realise that there seems to be a parallel between the Heisenberg uncertainty principle (HUP) and Noether's theorem.
For example, one conjugate pair of variables in the HUP is energy and time. While in Noether's theorem, the conservation of energy is related to time symmetry.
Another conjugate pair is momentum and position. And the conservation of momentum is related to spatial translation symmetry.
I assume this is not just a coincidence. So is there a deeper reason? Is Noether's theorem the reason why these are conjugate pairs?