Sean Carroll: Hilbert Space and Infinity

What is Hilbert space and Euclidean space?

Yeah, you know, I think that people are very welcome to go through their lives not knowing what Hilbert space is. But if you, if you want to dig into a little bit more into quantum mechanics, it becomes necessary. You know, the English language was invented long before quantum mechanics or various forms of higher mathematics were invented. So we use the word space to mean different things. Of course, most of us think of space as this three dimensional world in which we live, right? I mean, some of us just think of it as outer space, okay? But space around us gives us a three dimensional location of things and objects. But mathematicians use any generic abstract collection of elements as a space, okay? A space of possibilities, you know, momentum, space, et cetera. So Hilbert space is the space of all possible quantum wave functions either for the universe or for some specific system. And it could be an infinite dimensional space, or it could be just really, really large dimensional, but finite. We don't know, because we don't know the final theory of everything. But this abstract Hilbert space is really, really, really big and has no immediate connection to the three dimensional space in which we live.

What do dimensions in Hilbert space mean?

You know, it's just a way of mathematically representing how much information is contained in the state of the system. How, how many numbers do you have to give me to specify what the thing is doing? So in classical mechanics, I give you the location of something by giving you three numbers, right up down XYZ coordinates. But then I might want to give you its entire state physical state, which means both its position and also its velocity. The velocity also has three components, so its state lives in something called phase space, which is six dimensional. Three dimensions of position, three dimensions of velocity. And then if it also has an orientation in space, that's another three dimensions and so forth. So as you describe more and more information about the system, you have an abstract mathematical space that has more and more numbers that you need to give. And each one of those numbers corresponds to a dimension in that space.

So in terms of the amount of information, what is entropy? This mystical word that's overused in math and physics, but has a very specific meaning in this context.

Sadly, it has more than one very specific meaning. This is the reason why it's hard. Entropy means different things, even to different physicists. But one way of thinking about it is a measure of how much we don't know about the state of a system. If I have a bottle of water molecules, and I know that, okay, there's a certain number of water molecules, I Could weigh it, figure out, figure out. I know the volume of it and I know the temperature and pressure and things like that. I certainly don't know the exact position and velocity of every water molecule. Right. So there's a certain amount of information I know, certain amount that I don't know that is, that is part of the complete state of the system. And that's what the entropy characterizes. How much unknown information there is, the difference between what I do know about the system and its full exact microscopic state.

So when we try to describe a quantum mechanical system, is it infinite or finite, but very large?

Yeah, we don't know. That depends on the system. You know, it's easy to mathematically write down a system that would have a potentially infinite entropy, an infinite dimensional Hilbert space. So let's go back a little bit. We said that the Hilbert space was the space in which quantum wave functions lived from. For different systems that will be different sizes, they could be infinite or finite. So that's the number of numbers, the number of pieces of information you could potentially give me about the system. So the bigger Hilbert space is, the bigger the entropy of that system could be, depending on what I know about it. If I don't know anything about it, then it has a huge entropy, but only up to the size of its Hilbert space. So we don't know in the real physical world whether or not this region of space that contains that water bott has potentially an infinite entropy or just a finite entropy. We have different arguments on different sides.

So if it's infinite, how do you think about infinity? Is this something you can, your cognitive abilities are able to process, or is it just a mathematical tool?

It's somewhere in between. Right? I mean, we can say things about it. We can use mathematical tools to manipulate infinity very, very accurately. We can define what we mean. You know, for any number N, there's a number bigger than it. So, so there's no biggest number. Right? So there's something called the total number of all numbers, and it's infinite. But it is hard to wrap your brain around that. And I think that gives people pause because we talk about infinity as if it's a number, but it has plenty of properties that real numbers don't have. You know, if you multiply infinity by two, you get infinity again. Right. It's a little bit different than what we're used to.

Okay, but are you comfortable with the idea that in thinking of what the real world actually is, that infinity could be part of that world? Are you comfortable that a world in some dimension.

I'm comfortable with lots of things. I mean, I don't want my level of comfort to affect what I think about the world. I'm pretty open minded about what the world could be at the fundamental level.

Yeah, but infinity is a tricky one. It's not almost a question of comfort, it's a question of is it an overreach of our intuition? Sort of. It could be a convenient. Almost like when you add a constant to an equation just because it'll help. It just feels like it's useful to at least be able to imagine a concept. Not directly, but in some kind of way that this feels like it's a description of the real world.

Think of it this way. There's only three numbers that are simple. There's zero, there's one, and there's infinity. A number like 318 is just bizarre. You need a lot of bits to give me what that number is. But 0 and 1 in infinity, like once you have 300 things, you might as well have infinity things, right? Otherwise you have to say when to stop making the things right. So there's a sense in which infinity is a very natural number of things to exist.

I was never comfortable with infinity because it's just such a. It was too good to be true. Because in math it just helps make things work out when things get very. It's when things get very large, close to infinity, things seem to work out nicely. It's kind of like. Because my. My deepest passion is probably psychology and I'm uncomfortable how in the average, the beauty of how much we vary is lost. In that same kind of sense, infinity seems like a convenient way to erase the details.

But the thing about infinity is it seems to pop up whether we like it or not, right? Like you're trying to be a computer scientist. You ask yourself, well, how long will it take this program to run? And you realize, well, for some of them the answer is infinitely long. It's not because you tried to get there. You wrote a five line computer program. It doesn't halt.