Sean Carroll

For black holes, there are ways to change the coordinate system so that it looks like the event horizon is just a boundary that you just can't get past it, right?

And it took a lot of work for people to figure out how to disentangle, as it were, what is real, what is physically there from what are merely coordinate implications.

So the coordinates are not the territory, but that's a very simple motto to repeat a very difficult lesson to truly internalize.

And the reason why I'm diverting here into Einstein and coordinates is because I think, and here I'm in a tiny idiosyncratic minority, so you're absolutely empowered to disbelieve me if you want, but I think that we are facing a very similar kind of problem in quantum mechanics.

The choice of position as something you could measure in a way of expressing the wave function, or the choice of momentum as an equally good way to represent the wave function, these are just choices of coordinates.

We call the set of all possible wave functions or quantum states, we call it Hilbert space, as you've heard me talk about before, a big old vector space.

Vector space because you can add wave functions together, scale them by numbers and things like that.

So mathematically, we understand what Hilbert space is very, very well.

And position of momentum, as we alluded to with the simple example of the two-dimensional plane, are two different choices of basis vectors in this vector space, which is just a particular kind of coordinate system on the vector space.

And this idea that coordinates aren't what is real, they are simply convenient ways of talking, okay?

If you take that seriously, which you were taking it seriously two minutes ago when I said it, that means that position and momentum are not real, okay?

Those are not in quantum mechanics, they're not.

They're just choices of coordinates on Hilbert space.

They're choices of bases.

They are certain things you are able to observe.

but they're not fundamental to the description of the theory.

And this is another stop at which people get off the bus in certain ways because they say, but I can, the fact that I can observe and measure position and momentum relatively straightforwardly to me does give them

a sort of privileged status in the description of the theory.

Well, it's absolutely true that there is something very, very convenient about position and momentum.

They're easier to observe than other things.