Why is math a language critical to understanding quantum mechanics? Why can’t they just use good ol’ normal math? Excerpted from “Quantum Mechanics: The Theoretical Minimum,” May 22, 2014.

LEONARD SUSSKIND, Felix Bloch Professor of Theoretical Physics, Stanford University; Co-author, Quantum Mechanics: The Theoretical Minimum
ART FRIEDMAN, Data Engineer; Co-author, Quantum Mechanics: The Theoretical Minimum

ART FRIEDMAN: We try to convey the basic language of quantum mechanics, which is mathematics. If we think about languages, and about learning languages, languages are filled with idioms. They’re some of the things that make languages rich and interesting. Take for example idioms that carry the basic meaning “don’t bother me.” There are different ways to say that in different languages; there are different ways of saying that within the English language itself.

Just look at the English example, “get off my case.” Put yourself in the position of a person who is not a native English speaker, who is learning English and is trying to make sense out of that utterance, and they’ve never heard it before. They might go use dictionary definitions and simple grammar rules that they’ve been taught. If they do that, they’ll wonder, “Why does this person have a case? What in the world is inside the case? Why is this other person on top of the case? Why does the first person even care that that person is on the case and want them to get off?” In other words, you can go down this rabbit hole of very logical reasoning, reasoning that makes sense from your understanding of the way words connect to the world, but that really doesn’t get you anywhere in terms of understanding the meaning of that phrase. You have to connect those words to the world in a different way, and it might be even easier to see that in some other languages that are not so familiar to us.

Now, onto the language of physics. I think the process of learning quantum mechanics is tricky in a very similar fashion, because the connection between the math and the physical world is a different sort of connection. In physics, classical mechanics is your native language. It might surprise you that it’s your native language even if you’ve never studied it. It’s your native language in the sense that the connections between mathematics and physics are very straightforward.

[In quantum mechanics] we talk about the things that we talk about in classical mechanics or in any kind of physics: we talk about states and measurements. Those are two concepts that we have. In classical mechanics, there’s barely any difference between them. If you think about the state of a particle, just a little idealized lump of matter – it’s very tiny – you can describe it by giving its position and velocity. Let’s say its position is four and its velocity is five. That represents the state of that particle. That’s what you can know about it. On the other hand, suppose you don’t know what the state of the particle is and you want to find out. How do you find out? You find out by trying to measure the particle, and what do you measure? You measure those same things. You measure the position; you measure the velocity. In classical mechanics, we have this idea that in principle it’s possible to measure something as gently as you like. You can measure it so gently that, for all practical purposes, you’re not disturbing the thing that you’re measuring. So if you measure a position of four and a velocity of five, after the measurement takes place, the system that you’ve measured still has a position of four and a velocity of five. There’s no meaningful distinction, in other words, between the idea of a state and the idea of a measurement.

It’s different in this respect in quantum mechanics. What we have to do is separate the concept of a state from the concept of a measurement. In quantum mechanics, they become two very different things. A measurement in quantum mechanics is kind of similar to what a measurement is in classical mechanics. It’s not exactly the same thing, but it has the same feel. If you ask about what a measurement is, you’ll be looking for something like a number: “The particle is in position four.”

A state is a mathematical description that encapsulates everything that you can possibly know about that system. Having said that, it’s not the same as a state in classical mechanics. It’s a more abstract mathematical idea. You don’t find things like positions and velocities in a state description. You’ll find things like abstract state vectors that consist of complex value state vectors. They are more complicated abstract mathematical things that represent a state.

And by the way, now that [state and measurements] are separate, they get to have a relationship. They get to be related to each other in some specific way, and that relationship is mediated by another set of mathematical objects that are called linear operators. There’s nothing that mysterious or outlandish about linear operators. They come up all the time in applied physics and mathematics. They were not specially invented for quantum mechanics, I don’t think, but somehow they become this sort of mathematical pipeline that connects the two concepts of a state and a measurement.

Now this is something we don’t have in classical mechanics at all. So when you find this pipeline sitting there and these abstract state vectors in the state box, it’s a little surprising and you say to yourself, “Why don’t they behave?” Even if you don’t say this explicitly, you’re kind of expecting a model that resembles the classical model of physics, but that’s not what we get.

Now, as far as I can tell, the main obstacle in learning quantum mechanics is to get that idea. There are other things that have to be learned, of course; it’s a wide-ranging subject. But that’s really the main act.

Is it really that hard? It’s hard in the sense that breaking your old habits is hard to do. Is it worth the trouble? You have to answer that question for yourself. I obviously think it is. It’s not something that requires genius. It’s something that requires a roadmap of the kind that we’re trying to give you here.

Back to the reward: If you do decide to take this journey, you’ll be able to get some firsthand understanding of what I think is a rather amazing and beautiful body of work. It’s really one of the coolest things the human mind has ever come up with, as far as I can tell. Why wouldn’t you want to know about it? It’s surprising and counterintuitive. It will enable you to understand and think critically about some current discoveries and work that’s being done. It’s also fun.

LEONARD SUSSKIND: Art did say it exactly right. And I’m going to say it again in my own way. Why do we need mathematics to understand physics? Why can’t we understand physics just by pictures? Why can’t we understand it in words? Why is there this priesthood of theoretical physicists who speak in a language that nobody else can understand? The reason is, it’s not some willful nastiness; I don’t know what the right word is.

FRIEDMAN: Snobbery.

SUSSKIND: There’s nothing secretive about it. It is, as Art said, a language, but it is a necessary language. It became especially necessary around the turn of the last century when physics started dealing with phenomena that were extremely remote – extremely remote in the sense that they had to deal with things that were vastly smaller than you can see, vastly lighter than you could weigh, vastly faster than that with which you could possibly keep up. And suddenly you were in a world of experience in which you were never intended to be. When I say you were never intended to be in it, I mean to say that your facilities, your neural architecture – the thing that you were provided with by evolution – was not intended for these extremely remote kinds of phenomena.

What were you intended for? You were intended basically for classical mechanics – for how stones move, for how rocks move, for what happens if you fall off a cliff. You have an innate sense. I’m not a neural specialist, so I don’t really know what your neural network is intended for, but I have a feeling that, at least in part, some of us are very innate in the understanding of things like, “You better not fall off a cliff because if you do, you’re going to get crushed.” You have an innate sense of what a force is – a push. All sorts of things are the things that you can see with your eyes. When you think about a wave, you can think about a wave in the sea. It’s easy to see, easy to visualize.