I don't know about quantum mechanics, but when we talk about space we should be free to add a quantity to the whole universe (like adding 1 to the x coordinate

I don't know about quantum mechanics, but when we talk about space we should be free to add a quantity to the whole universe (like adding 1 to the x coordinate of everything) because this just shifts the whole universe, or accordingly, shifts the origin - the (0, 0, 0) point - in the opposite direction. The origin is set at an arbitrary point so this "space shift invariance" is saying that it doesn't matter what point we set for the origin (and mathematically this corresponds to the conservation of momentum - see Noether's theorem[0]) Hmm maybe the "zero" for the quantum states is arbitrary, so you should be able to add anything to it for the whole universe, and this merely changes the zero state in the opposite direction.. and since this should be a conversation law, pretty sure this is equivalent to the conservation of electric charge https://en.wikipedia.org/wiki/Noether's_theorem https://en.wikipedia.org/wiki/Noether's_theorem