String theory used to get everyone all tied up in knots. Even its practitioners fretted about how complicated it was, while other physicists mocked its lack of experimental predictions. The rest of the world was largely oblivious. Scientists could scarcely communicate just why string theory was so exciting why it could fulfill Albert Einstein's dream of the ultimate unified theory, how it could give insight into such deep questions as why the universe exists at all. But in the mid-1990s the theory started to click together conceptually. Researchers came up with ways it might be tested experimentally. The outside world began to pay attention. Woody Allen satirized the theory in a New Yorker column in July 2003—probably the first time anyone has used Calabi-Yau spaces to tell a story about interoffice romance.

Few People can take more credit for demystifying string theory than Brian Greene, a Columbia University physics professor and a major contributor to the theory. His 1999 book The Elegant Universe reached number four on the New York Times best-seller list and was a finalist for the Pulitzer Prize. In 2003 Greene hosted a three-part Nova series on PBS based on that book and in 2004 published The Fabric of the Cosmos, a best-seller on the nature of space and time. SCIENTIFIC AMERICAN staff editor George Musser spoke with him over a plate of stringy spaghetti. Here is an abridged, edited version of that conversation.

SCIENTIFIC AMERICAN: Sometimes when our readers hear the words “string theory” or “cosmology,” they throw up their hands and say, “I'll never understand it.”

BRIAN GREENE: I've definitely encountered a certain amount of intimidation at the outset when it comes to ideas like string theory or cosmology. But what I have found is that the basic interest is so widespread and so deep in most people that I've spoken with, that there is a willingness to go a little bit further than you might with other subjects that are more easily taken in.

SA: I noticed that at several points in The Elegant Universe, you first gave a rough idea of the physics concepts and then the detailed version.

BG: I found that to be a useful way of going about it, especially in the harder parts. It gives the reader permission: If the rough idea is the level at which you want to take it in, that's great; feel free to skip this next stuff. If not, go for it. I like to say things more than one way. I just think that when it comes to abstract ideas, you need many roads into them. From the scientific point of view, if you stick with one road, I think you really compromise your ability to make breakthroughs. I think that's really what breakthroughs are about. Everybody's looking at a problem one way, and you come at it from the back. That different way of getting there somehow reveals things that the other approach didn't.

SA: What are some examples of that back-door approach?

BG: Well, probably the biggest ones are Ed Witten's breakthroughs. Ed [of the Institute for Advanced Study in Princeton, N.J.] just walked up the mountain and looked down and saw the connections that nobody else saw and in that way united the five string theories that previously were thought to be completely distinct. It was all out there; he just took a different perspective, and bang, it all came together. And that's genius.

To me that suggests what a fundamental discovery is. The universe in a sense guides us toward truths, because those truths are the things that govern what we see. If we're all being governed by what we see, we're all being steered in the same direction. Therefore, the difference between making a breakthrough and not can often be just a small element of perception, either true perception or mathematical perception, that puts things together in a different way.

SA: Do you think that these discoveries would have been made without the intervention of genius?

BG: Well, it's tough to say. In the case of string theory, I think so, because the pieces of the puzzle were really becoming clearer and clearer. It may have been five or 10 years later, but I suspect it would have happened. But with general relativity, I don't know. General relativity is such a leap, such a monumental rethinking of space, time and gravity, that it's not obvious to me how and when that would have happened without Einstein.

SA: Are there examples in string theory that you think are analogous to that huge leap?

BG: I think we're still waiting for a leap of that magnitude. String theory has been built up out of a lot of smaller ideas that a lot of people have contributed and been slowly stitching together into an ever more impressive theoretical edifice. But what idea sits at the top of that edifice, we still don't really know. When we do have that idea, I believe that it will be like a beacon shining down; it will illuminate the edifice, and it will also, I believe, give answers to critical questions that remain unresolved.

SA: In the case of relativity, you had the equivalence principle and general covariance in that beacon role.

In the Standard Model, it's gauge invariance. In The Elegant Universe, you suggested the holographic principle could be that principle for string theory [see also “Information in the Holographic Universe,” by Jacob D. Bekenstein, on page 74]. What's your thinking on that now?

BG: Well, the past few years have only seen the holographic principle rise to a yet greater prominence and believability. Back in the mid-1990s, shortly after the holographic ideas were suggested, the supporting ideas were rather abstract and vague, all based on features of black holes: Black hole entropy resides on the surface; therefore, maybe the degrees of freedom reside on the surface; therefore, maybe that's true of all regions that have a horizon; maybe it's true of cosmological horizons; maybe we're living within a cosmological region that has its true degrees of freedom far away. Wonderfully strange ideas, but the supporting evidence was meager.

But that changed with the work of Juan Maldacena [of the Institute for Advanced Study], in which he found an explicit example within string theory, where physics in the bulk—that is, in the arena that we consider to be real—would be exactly mirrored by physics taking place on a bounding surface. There'd be no difference in terms of the ability of either description to truly describe what's going on, yet in detail the descriptions would be vastly different. One would be in five dimensions, the other in four. So even the number of dimensions seems not to be something that you can count on, because there can be alternative descriptions that would accurately reflect the physics you're observing.

So to my mind, that makes the abstract ideas now concrete; it makes you believe the abstract ideas. And even if the details of string theory change, I think, as many others do—not everyone, though—that the holographic idea will persist and will guide us. Whether it truly is the idea, I don't know. I don't think so. But I think that it could well be one of the key stepping-stones toward finding the essential ideas of the theory. It steps outside the details of the theory and just says, Here's a very general feature of a world that has quantum mechanics and gravity.

SA: Let's talk a bit about loop quantum gravity and some of the other approaches. You've always described string theory as the only game in town when it comes to quantum gravity. Do you still feel that way?

BG: I think it's the most fun game in town! But to be fair, the loop quantum gravity community has made tremendous progress. There are still many very basic questions that I don't feel have been answered, not to my satisfaction. But it's a viable approach, and it's great there are such large numbers of extremely talented people working on it. My hope—and it has been one that Lee Smolin has championed—is that ultimately we're developing the same theory from different angles [see “Atoms of Space and Time,” by Lee Smolin, on page 56]. It's far from impossible that we're going down our route to quantum gravity, they're going down their route to quantum gravity, and we're going to meet someplace. Because it turns out that many of their strengths are our weaknesses. Many of our strengths are their weaknesses.

One weakness of string theory is that it's so-called background dependent. We need to assume an existing spacetime within which the strings move. You'd hope, though, that a true quantum theory of gravity would have spacetime emerge from its fundamental equations. They [the loop quantum gravity researchers], however, do have a background-independent formulation in their approach, where spacetime does emerge more fundamentally from the theory itself. On the other hand, we are able to make very direct contact with Einstein's general relativity on large scales. We see it in our equations. They have some difficulty making contact with ordinary gravity. So naturally, you'd think maybe one could put together the strengths of each.

SA: Has that effort been made?

BG: Slowly. There are very few people who are really well versed in both theories. These are both two huge subjects, and you can spend your whole life, every moment of your working day, just in your own subject, and you still won't know everything that's going on. But many people are heading down that path and starting to think along those lines, and there have been some joint meetings.

SA: If you have this background dependence, what hope is there to really understand, in a deep sense, what space and time are?

BG: Well, you can chip away at the problem. For instance, even with background dependence, we've learned things like mirror symmetry—there can be two spacetimes, one physics. We've learned topology change—that space can evolve in ways that we wouldn't have thought possible before. We've learned that the micro world might be governed by noncommutative geometry, where the coordinates, unlike real numbers, depend on the order in which you multiply them. So you can get hints. You can get isolated glimpses of what's truly going on down there. But I think without the background-independent formalism, it's going to be hard to put the pieces together on their own.

SA: The mirror symmetry is incredibly profound, because it divorces spacetime geometry from physics. The connection between the two was always the Einsteinian program.

BG: That's right. Now, it doesn't divorce them completely. It simply says that you're missing half of the story. Geometry is tightly tied to physics, but it's a two-to-one map. It's not physics and geometry. It's physics and geometry-geometry, and which geometry you want to pick is up to you. Sometimes using one geometry gives you more insight than the other. Again, different ways of looking at one and the same physical system: two different geometries and one physics. And people have found there are mathematical questions about certain physical and geometrical systems that people couldn't answer using the one geometry. Bring in the mirror geometry that had previously gone unrealized, and, all of a sudden, profoundly difficult questions, when translated, were mind-bogglingly simple.

SA: Can you describe noncommutative geometry?

BG: Since the time of Descartes, we've found it very powerful to label points by their coordinates, either on Earth by their latitude and longitude or in three-space by the three Cartesian coordinates, x, y and z, that you learn in high school. And we've always imagined that those numbers are like ordinary numbers, which have the property that, when you multiply them together—which is often an operation you need to do in physics—the answer doesn't depend on the order of operation: 3 times 5 is 5 times 3. What we seem to be finding is that when you coordinatize space on very small scales, the numbers involved are not like 3's and 5's, which don't depend on the order in which they're multiplied. There's a new class of numbers that do depend on the order of multiplication.

They're actually not that new, because for a long time we have known of an entity called the matrix. Sure as shooting, matrix multiplication depends on the order of multiplication. A times B does not equal B times A if A and B are matrices. String theory seems to indicate that points described by single numbers are replaced by geometrical objects described by matrices. On big scales, it turns out that these matrices become more and more diagonal, and diagonal matrices do have the property that they commute when you multiply. It doesn't matter how you multiply A times B if they're diagonal matrices. But then if you venture into the microworld, the off-diagonal entries in the matrices get bigger and bigger and bigger until way down in the depths, they are playing a significant part.

Noncommutative geometry is a whole new field of geometry that some people have been developing for years without necessarily an application of physics in mind. French mathematician Alain Connes has this big thick book called Noncommutative Geometry. Euclid and Gauss and Riemann and all those wonderful geometers were working in the context of commutative geometry, and now Connes and others are taking off and developing the newer structure of noncommutative geometry.

SA: It is baffling to me—maybe it should be baffling—that you would have to label points with a matrix or some nonpure number. What does that mean?

BG: The way to think about it is: There is no notion of a point. A point is an approximation. If there is a point, you should label it by a number. But the claim is that, on sufficiently small scales, that language of points becomes such a poor approximation that it just isn't relevant. When we talk about points in geometry, we really talk about how something can move through points. It's the motion of objects that ultimately is what's relevant. Their motion, it turns out, can be more complicated than just sliding back and forth. All those motions are captured by a matrix. So rather than labeling an object by what point it's passing through, you need to label its motion by this matrix of degrees of freedom.

SA: What is your current thinking on anthropic and multiverse-type ideas? You talked about it in The Elegant Universe in the context of whether there is some limit to the explanatory power of string theory.

BG: I and many others have never been too happy with any of these anthropic ideas, largely because it seems to me that at any point in the history of science, you can say, “Okay, we're done, we can't go any further, and the final answer to every currently unsolved question is: ‘Things are the way they are because had they not been this way, we wouldn't have been here to ask the question.’” So it sort of feels like a cop-out. Maybe that's the wrong word. Not necessarily like a cop-out; it feels a little dangerous to me, because maybe you just needed five more years of hard work and you would have answered those unresolved questions, rather than just chalking them up to, “That's just how it is.” So that's my concern: that one doesn't stop looking by virtue of having this fallback position.

But you know, it's definitely the case that the anthropic ideas have become more developed. They're now real proposals whereby you would have many universes, and those many universes could all have different properties, and it very well could be that we're simply in this one because the properties are right for us to be here, and we're not in those others because we couldn't survive there. It's less of just a mental exercise.

SA: String theory, and modern physics generally, seems to be approaching a single logical structure that had to be the way it is; the theory is the way it is because there's no other way it could be. On the one hand, that would argue against an anthropic direction. But on the other hand, there's a flexibility in the theory that leads you to an anthropic direction.

BG: The flexibility may or may not truly be there. That really could be an artifact of our lack of full understanding. But were I to go by what we understand today, the theory seems to be able to give rise to many different worlds, of which ours seems to be potentially one, but not even necessarily a very special one. So yes, there is a tension with the goal of absolute, rigid inflexibility.

SA: If you had other grad students waiting in the wings, what would you steer them to?

BG: Well, the big questions are, I think, the ones that we've discussed. Can we understand where space and time come from? Can we figure out the fundamental ideas of string theory? Can we show that this fundamental idea yields a unique theory with a unique solution, which happens to be the world as we know it? Is it possible to test these ideas through astronomical observations or through accelerator-based experiment?

Can we even take a step further back and understand why quantum mechanics had to be part and parcel of the world as we know it? How many of the things that we rely on at a very deep level in any physical theory that has a chance of being right—such as space, time, quantum mechanics—are truly essential, and how many of them can be relaxed and potentially still yield the world that appears close to ours?

Could physics have taken a different path that would have been experimentally as successful but completely different? I don't know. But I think it's a real interesting question to ask. How much of what we believe is truly fundamentally driven in a unique way by data and mathematical consistency, and how much of it could have gone one way or another, and we just happened to go down one path because that's what we happened to discover? Could beings on another planet have completely different sets of laws that somehow work just as well as ours?