Sean Carroll: Hilbert Space and Infinity

00:00:00.000 | >>Kaiper What is Hilbert space and Euclidean space?

00:00:06.480 | >>John Yeah, you know, I think that people are very

00:00:09.240 | welcome to go through their lives not knowing what Hilbert space is. But if you want to

00:00:13.120 | dig into a little bit more into quantum mechanics, it becomes necessary. You know, the English

00:00:17.720 | language was invented long before quantum mechanics or various forms of higher mathematics

00:00:23.000 | were invented. So we use the word space to mean different things. Of course, most of

00:00:28.280 | us think of space as this three dimensional world in which we live, right? I mean, some

00:00:31.640 | of us just think of it as outer space. Okay, but space around us, it gives us the three

00:00:36.560 | dimensional location of things and objects. But mathematicians use any generic abstract

00:00:44.800 | collection of elements as a space, okay, a space of possibilities, you know, momentum

00:00:50.880 | space, etc. So Hilbert space is the space of all possible quantum wave functions, either

00:00:56.240 | for the universe or for some specific system. And it could be an infinite dimensional space,

00:01:01.760 | or it could be just really, really large dimensional, but finite, we don't know, because we don't

00:01:05.320 | know the final theory of everything. But this abstract Hilbert space is really, really,

00:01:09.720 | really big and has no immediate connection to the three dimensional space in which we

00:01:13.360 | live.

00:01:14.360 | >>Kaiper What do dimensions in Hilbert space mean?

00:01:17.160 | >>John You know, it's just a way of mathematically

00:01:20.240 | representing how much information is contained in the state of the system. How many numbers

00:01:24.800 | do you have to give me to specify what the thing is doing? So in classical mechanics,

00:01:29.840 | I give you the location of something by giving you three numbers, right up, down, left, X,

00:01:35.280 | Y, Z coordinates. But then I might want to give you its entire state, physical state,

00:01:41.760 | which means both its position and also its velocity. The velocity also has three components.

00:01:46.940 | So its state lives in something called phase space, which is six dimensional, three dimensions

00:01:52.280 | of position, three dimensions of velocity. And then if it also has an orientation in

00:01:57.000 | space, that's another three dimensions and so forth. So as you describe more and more

00:02:01.480 | information about the system, you have an abstract mathematical space that has more

00:02:06.720 | and more numbers that you need to give and each one of those numbers corresponds to a

00:02:10.320 | dimension in that space.

00:02:11.680 | >>Kaiper So in terms of the amount of information,

00:02:15.580 | what is entropy? This mystical word that's overused in math and physics, but has a very

00:02:22.300 | specific meaning in this context.

00:02:24.220 | >>Zakaria Sadly, it has more than one very specific

00:02:26.460 | meaning. This is the reason why it's hard. Entropy means different things even to different

00:02:30.620 | physicists. But one way of thinking about it is a measure of how much we don't know

00:02:35.940 | about the state of a system, right? So if I have a bottle of water molecules, and I

00:02:41.300 | know that, okay, there's a certain number of water molecules, I could weigh it, right,

00:02:44.700 | and figure out, I know the volume of it, and I know the temperature and pressure and things

00:02:48.340 | like that, I certainly don't know the exact position and velocity of every water molecule,

00:02:54.020 | right? So there's a certain amount of information I know, certain amount that I don't know that

00:02:58.660 | is part of the complete state of the system. And that's what the entropy characterizes,

00:03:03.040 | how much unknown information there is, the difference between what I do know about the

00:03:07.340 | system and its full exact microscopic state.

00:03:09.860 | >>Kaiper So when we try to describe a quantum mechanical

00:03:13.860 | system, is it infinite or finite but very large?

00:03:19.260 | >>Zakarai Yeah, we don't know. That depends on the

00:03:21.260 | system. You know, it's easy to mathematically write down a system that would have a potentially

00:03:26.740 | infinite entropy, an infinite dimensional Hilbert space. So let's go back a little bit.

00:03:32.540 | We said that the Hilbert space was the space in which quantum wave functions lived. For

00:03:36.780 | different systems, that will be different sizes. They could be infinite or finite. So

00:03:41.300 | that's the number of numbers, the number of pieces of information you could potentially

00:03:45.820 | give me about the system. So the bigger Hilbert space is, the bigger the entropy of that system

00:03:51.900 | could be, depending on what I know about it. If I don't know anything about it, then it

00:03:56.260 | has a huge entropy, right? But only up to the size of its Hilbert space. So we don't

00:04:01.180 | know in the real physical world whether or not this region of space that contains that

00:04:08.020 | water bottle has potentially an infinite entropy or just a finite entropy. We have different

00:04:12.340 | arguments on different sides.

00:04:13.700 | >>Steve So if it's infinite, how do you think about

00:04:16.660 | infinity? Is this something you can, your cognitive abilities are able to process? Or

00:04:23.900 | is it just a mathematical tool?

00:04:25.740 | >>Zakarai It's somewhere in between, right? I mean,

00:04:27.740 | we can say things about it. We can use mathematical tools to manipulate infinity very, very accurately.

00:04:34.060 | We can define what we mean. For any number n, there's a number bigger than it. So there's

00:04:38.820 | no biggest number, right? So there's something called the total number of all numbers, that's

00:04:43.020 | infinite. But it is hard to wrap your brain around that. And I think that gives people

00:04:47.620 | pause because we talk about infinity as if it's a number, but it has plenty of properties

00:04:53.520 | that real numbers don't have. If you multiply infinity by two, you get infinity again, right?

00:04:58.300 | That's a little bit different than what we're used to.

00:05:01.140 | >>Steve Okay, but are you comfortable with the idea

00:05:03.980 | that in thinking of what the real world actually is, that infinity could be part of that world?

00:05:11.260 | Are you comfortable that a world in some dimension, in some aspect-

00:05:13.860 | >>Zakarai I'm comfortable with lots of things. I mean,

00:05:16.100 | you know, I don't want my level of comfort to affect what I think about the world. You

00:05:23.220 | know, I'm pretty open-minded about what the world could be at the fundamental level.

00:05:26.300 | >>Steve Yeah, but infinity is a tricky one. It's

00:05:30.380 | not almost a question of comfort. It's a question of, is it an overreach of our intuition? It

00:05:40.260 | could be a convenient, almost like when you add a constant to an equation just because

00:05:44.180 | it'll help. It just feels like it's useful to at least be able to imagine a concept,

00:05:50.420 | not directly, but in some kind of way that this feels like it's a description of the

00:05:56.060 | real world.

00:05:57.060 | >>Zakarai Think of it this way. There's only three numbers

00:05:59.780 | that are simple. There's zero, there's one, and there's infinity. A number like 318 is

00:06:10.060 | just bizarre. You need a lot of bits to give me what that number is. But zero and one and

00:06:16.820 | infinity, once you have 300 things, you might as well have infinity things, right? Otherwise,

00:06:19.860 | you have to say when to stop making the things, right? So there's a sense in which infinity

00:06:24.540 | is a very natural number of things to exist.

00:06:27.100 | >>Kamala It was never comfortable with infinity because

00:06:30.540 | it was too good to be true. Because in math, it just helps make things work out. When things

00:06:40.420 | get very large, close to infinity, things seem to work out nicely. It's kind of like,

00:06:46.540 | because my deepest passion is probably psychology. And I'm uncomfortable how in the average,

00:06:54.540 | the beauty of how much we vary is lost. In that same kind of sense, infinity seems like

00:07:03.300 | a convenient way to erase the details.

00:07:06.140 | >>Zakarai But the thing about infinity is it seems to

00:07:09.500 | pop up whether we like it or not, right? Like you're trying to be a computer scientist,

00:07:14.060 | you ask yourself, well, how long will it take this program to run? And you realize, well,

00:07:17.740 | for some of them, the answer is infinitely long. It's not because you tried to get there.

00:07:22.000 | You wrote a five-line computer program. It doesn't halt.

00:07:24.940 | and I think I'll see you in the next video.

00:07:26.700 | Bye.

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Sean Carroll: Hilbert Space and Infinity

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