When the fall quarter started, there were six papers that I absolutely had to finish by the end of the term. Three have been completed, two are very close, and the last one — sadly, I think the deadline has irrevocably passed, and it’s not going to make it. So here’s the upshot.
About a year ago I gave a talk at the Philosophy of Science Association annual meeting in Austin. The topic of the session was “The Dimensions of Space,” and my talk was on “Why Three Spatial Dimensions Just Aren’t Enough” (pdf slides). I gave an overview of the idea of extra dimensions, how they arose historically and the role they currently play in string theory.
But in retrospect, I didn’t do a very good job with one of the most basic questions: how many dimensions does spacetime really have, according to string theory? The answer used to be easy: ten, with six of them curled up into a tiny manifold that we couldn’t see. But in the 1990’s we saw the “Second Superstring Revolution,” featuring ideas about D-branes, duality, and the unification of what used to be thought of as five distinct versions of string theory.
One of the most important ideas in the second revolution came from Ed Witten. Ordinarily, we like to examine field theories and string theories at weak coupling, where perturbation theory works well (QED, for example, is well-described by perturbation theory because the fine-structure constant α = 1/137 is a small number). Witten figured out that when you take the strong-coupling limit of certain ten-dimensional string theories, new degrees of freedom begin to show up (or more accurately, begin to become light, in the sense of having a low mass). Some of these degrees of freedom form a series of states with increasing masses. This is precisely what happens when you have an extra dimension: modes of ordinary fields that wrap around the extra dimension will have a tower of increasing masses, known as Kaluza-Klein modes.
In other words: the strong-coupling limit of certain ten-dimensional string theories is an eleven-dimensional theory! In fact, at low energies, it’s eleven-dimensional supergravity, which had been studied for years, but whose connection to string theory had been kind of murky. Now we know that 11-d supergravity and the five ten-dimensional string theories are just six different low-energy weakly-coupled limits of some single big theory, which we call M-theory even though we don’t know what it really is. (Even though the 11-d theory can arise as the strong-coupling limit of a 10-d string theory, it is itself weakly coupled in its own right; this is an example of strong-weak coupling duality.)
So … how many dimensions are there really? If one limit of the theory is 11-dimensional, and others are 10-dimensional, which is right?
I’ve heard respected string theorists come down on different sides of the question: it’s really ten-dimensional, it’s really eleven. (Some have plumped for twelve, but that’s obviously crazy.) But it’s more accurate just to say that there is no unique answer to this question. “The dimensionality of spacetime” is not something that has a well-defined value in string theory; it’s an approximate notion that is more or less useful in different circumstances. If you look at spacetime a certain way, it can look ten-dimensional, and another way it can look like eleven. In yet other configurations, thank goodness, it looks like four!
And it only gets worse. According to Juan Maldacena’s famous gravity-gauge theory correspondence (AdS/CFT), we can have a theory that is equally well described as a ten-dimensional theory of gravity, or a four-dimensional gauge theory without any gravity at all. It might sound like the degrees of freedom don’t match up, but ultimately infinity=infinity, so a lot of surprising things can happen.
This story is one of the reasons for both optimism and pessimism about the prospects for connecting string theory to the real world. On the one hand, string theory keeps leading us to discover amazing new things: it wasn’t as if anyone guessed ahead of time that there should be dualities between theories in different dimensions, it was forced on us by pushing the equations as far as they would go. On the other, it’s hard to tell how many more counterintuitive breakthroughs will be required before we can figure out how our four-dimensional observed universe fits into the picture (if ever). But it’s nice to know that the best answer to a seemingly-profound question is sometimes to unask it.
There are ten dimensions. There are twelve dimensions. Yes?
Matt B.: Sorry for taking so long to reply. I’ve been away from the computer for a while. Introducing the observer doesn’t change the system being studied because he isn’t introduced into the system. He’s outside it, making occasional measurements and then updating his mathematical description of the system to take into account the results of his measurements.
Moshe: I can try to explain any part that was confusing; I probably made statements that seem alien to people who don’t already see things my way.
Moshe and Aaron: The S-matrix is nice if you can indeed sit at infinity and throw strings and branes and so on into the middle and see what comes out. You can do this in terms of calculating scattering amplitudes in a fixed background. Then the S-matrix elements are observables (in the sense that you can experimentally observe something about them) if we live in that background and can produce the ingoing states and can detect the outgoing states. But do we, and can we? The dualities make the problem more explicit. If I live inside the boundary CFT, then the scattering amplitudes inside the bulk of AdS are not what I’m going to see if I do a scattering experiment. The best you can say is that they’re honest experimental observables for hypothetical creatures that live inside the bulk.
Or take T-duality. A nine-dimensional creature living in R^9 x S^1, where the radius of the S^1 is tiny would say the world is nine-dimensional, but a ten-dimensional creature living in the T-dual space, where the radius of S^1 is much larger would say that the world is ten-dimensional. From the point of view of string theory, those two worlds are “the same”, and before you can answer the question of how many large dimensions there are you have to first know whether it’s a creature in one space or a creature in the T-dual space asking the question.
This isn’t a criticism of the theory; as I said before, it just underlines the importance of making contact with experiment.
JC, normally everything has 4dim features and also extra-dimensional features, there is no such thing as 4dim sector, no more than an “up and down” sector in more familiar physics problems.
rof, there seem to be a few issues conflated here, but let me take the T-duality picture and phrase it as I would Lorentz invariance; a 9-dimensional observer cannot make any measurments that will distinguish a very small extra circle from a very large one. They can say the total space is mostly 9dim (if they choose the small circle “frame”) or almost 10dim (if they choose the large circle picture). Both are right, and therefore the question “how many dimensions are there *really*?” is a bad question to ask. Similar to an observer that insists on knowing if they are really at rest or not. As Sean said “the best answer to a seemingly-profound question is sometimes to unask it”.
While the discussion is on going here, I just wanted to interject this leading statement, to help those who are coming from the outside, looking in. 🙂
Lubos Motl
The S-matrix is contact with (hypothetical) experiments. Most of the things we compute in QFT are S-matrix elements. The fact that we’re not really living in a region with free |in> and |out> states doesn’t stop us from figuring out what happens in a collider.
The same with T-duality. We can compute scattering. Those are the physical observables. You can interpret them however you like.
How many dimensions are there? Well, it depends when you look. Just now, the cosmicvariance sidebar answered as follows:
So, as of Dec 10, 2005, 11:16AM US EST, there are 67 dimensions. I note also that Aaron Bergman’s post is numbered 55. It must mean that there are 12 hidden dimensions.
I wonder how many dimensions we’ll end up with?
Moshe: I’m not proposing that either of the hypothetical creatures should ask how many dimensions there “really” are. I’m just saying that each of them will be in no doubt about how many independent directions he can move his arms around in. R^9xS^1 is compatible with the answer to this being both eight or nine, if the S^1 is small/big enough.
Aaron; I agree fully. In fact I was going to write almost exactly what you wrote in my previous comment. Asymptotic string states can be identified with results of gedankenexperiments which we do not know how to perform but we can hypothesize that they might be possible in principle (if string theory is correct). That says it all, really.
rof, both descriptions of the same model have precisely the same set of answers to precsicely the same sets of questions. The set of things “observers” can do on a small or large circles is precisely identical, they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics. To me that suggests that focusing solely on the geometry is the wrong language to use.
And back to the observables issue you are discussing with Aaron, the situation is that there are set of observables that are well-defined and are associated with actual measurments that could be performed in principle. Those are S-matrix elements and they depends on having an asymptotic region where the physics simplifies. As we never performed experiments inside a strongly gravitating regions this includes all the experiments we ever performed, and as far as we know all the experiments that could be performed in principle.
Now there could be other things, and if they make sense (are gauge invariant) and are associated in a clean way with measurments that could be performed in principle, a theory of quantum gravity such as string theory ought to be able to calculate them. Once such things are discovered I’ll be interested to learn about them.
I was thinking of Moshe’s relation to circles large and small. The “complexity” of the situation in distinquishing which is which?
Difficulties in inner and outer?
It reminded me of Greene’s statements
Of course I am in a state of confusion 🙂
Plato, you’re in a state of confusion because almost all physicists are in a state of confusion. The problem is that the notion of “reality” has been completely reversed in the minds of the theoretical physics community over the last century, so that now they consider the most abstract things (such as quantum fields, elementary particles and so forth) to be the most real, while they view the concrete things (such as experimental phenomena, and objects like cups and bananas, along with concrete itself) as dependent for their existence on the abstract ones. The problems that I’m having explaining what I’m saying to Moshe and Aaron can be traced back to this essential point.
Moshe, you say: they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics.
This is precisely the point that I wanted to make, except expressed in a way which supposes that the “real world” is the string-theoretic representation of the world, and that the world in which we wave our arms around is “up to us”, that is, arbitrary. You agree with me that a certain type of observer in the world described by string theory on R^9xS^1 might see nine spatial dimensions around him while another sees eight, and you correctly (from the point of view of string theory) attribute these differences to differing characteristics of the observers, but incorrectly suppose that it is merely a choice that they made.
There are two types of physical theory, or rather, two attitudes that one can take towards theoretical physics. The first is the “ontological” theory, which is a theory which attempts to say how the world looks “to God”. That is, such theories attempt to describe how the world actually is in itself and considered in isolation from the manner in which we observe it. The attitude associated with those who construct and prefer these types of theories is that the physicist’s job is to describe nature, to say what the universe “really” is, and that the observer should be just a part of the description, since he is just another part of the material world, and the observer certainly shouldn’t play any privileged role in the formulation of the theory. It is this attitude, which I now give the title “physics fundamentalism”, which requires one to regard the most abstract objects as the most real.
Fundamentalist physicists have produced numerous attempts to remove the observer from quantum mechanics, because of their insistence that measurement is just an interaction like any other and should thus play no special role in the formulation. These attempts have, predictably, always led to assertions about the “real” existence of supersensible undetectable objects, even parallel worlds. For fundamentalist physicists, the idea that quantum mechanics deals with the observer’s knowledge about the system rather than the system itself is not allowed, because it conflicts with the basic tenets of physics fundamentalism. Hence the discontinuous change of the wavefunction, which happens because our knowledge changes discontinuously when we make a measurement, is regarded by them as a problem, which they call the measurement problem.
The problem with ontological theories in practice is that, since the observer is now supposed to be just a regular object inside the theory, we cannot say, on the basis of the theory, what it is that the observer is going to see, because that depends very much on the structure of the observer. Hence you say that whether he will “describe geometrically” some part of the “real world” is up to him, meaning it depends on his structure.
The other type of theory, or attitude towards theoretical physics, is one that could be called epistemological, and these theories attempt to describe how the world looks to us (rather than to God). This attitude regards physics as the process of establishing relationships between the results of experiments, or identifying the regularities in the results of experiments. Quantum mechanics is one of these theories, which makes statistical assertions about the results of measurements. It gives a procedure, which the observer can follow, in order to assign probabilities to the results of experiments. To follow the procedure one needs to actually make measurements, construct a set of formal sums of measurement results, fix parameters involved in the mathematical representation by referring to empirical data, and so on.
This is why I said that string theory is not a quantum theory in the usual sense. It does not construct a set of formal sums of results of actual measurements. It is not an application of the quantum procedure, which is a procedure that can only be applied to empirical data. It is essentially an ontological theory, which is why it suffers from the problem that it cannot predict what the observer will see, because what the observer sees depends on the structure of the observer.
An example of a genuine quantum theory is the theory of spin measurements with Stern-Gerlach devices. We find that the particle gets deflected either up or down, so we construct the set of expressions b|up>+c|down>, and follow the quantum procedure. When using this theory, we will not find that the predictions about what will be seen depend on the structure of the observer, and we do not need to assign a structure to the observer in order to use the theory to make predictions.
In string theory, on the other hand, we find that it is insisted that S-matrix elements are observables, regardless of the fact that the structure of the observer needs to be specified before it can even be said how many dimensions he will find himself in, or rather, how many dimensions he will “describe geometrically”. If you will excuse the metaphorical language, as fundamentalist physicists engaged in a war on reality, string theorists have moved beyond the original idea of changing the meaning of “reality” and applying it to abstract mathematical entities, and have now changed the meaning of “observable” so that it applies to merely mathematical entities as well.
Also, I like your remark that “all the experiments we ever performed, and as far as we know all the experiments that could be performed in principle” are observations of string-theoretic observables. It reminds me of how frequently religious types claim that the size, complexity, or mere existence of the universe provides a proof of the existence of God.
Moshe said:
“rof, both descriptions of the same model have precisely the same set of answers to precsicely the same sets of questions. The set of things “observers” can do on a small or large circles is precisely identical, they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics.”
Perhaps you can justify this apparently strange assertion?
Let me explain. If you look at the history [say 1930] of Kaluza Klein theory, you will find that there were two schools of thought. One said that the 5th dimension was real, the other that it was just a mathematical formalism. Of course, nobody disputed that the KK equations were *exactly equivalent* mathematically to the Einstein-Maxwell system, but nobody assumed that *exact mathematical equivalence* was the same thing as “equally real”. Similarly, string theorists circa 1985 surely knew that a purely formal interpretation of Calabi-Yau compactifications was possible, but evidently nobody felt moved to attach any importance to this observation.
Now you want to assert that mathematically indistinguishable situations are *equally real*. Maybe you can persuade us, but you won’t do it by means of analogies with special relativity. You will do it by describing *precisely* how an observer who waves his arms about can possibly describe his experiences in terms of two different dimensionalities. More generally, a string-theoretic analysis of the concept of “reality” would do the trick nicely.
Guys, there are too many high horses around for my comfort. I believe I stated precisely what I mean, let me try to get over some confusions I can identify, and maybe for the rest you can try to read what I actually write.
First, I emphasize I did not mention string theory at all when talking about observables of quantum gravity, whether in 4 dimensions or in higher dimensions. If you define “observables” as those things that can be observed, where observation includes and actual observer and a measuring device, the only things that we know exist are S-matrix elements. Those include everything we ever measured. Now this was not a statement about string theory at all, and please try to contain your passion and wait for me to actually make an arrogant and ill-informed statement. Yes “string theory observables” is a term you made up yourself rof.
Second, since the beginning of quantum theory physics is defined as the attempt to describe and predict results of measurments, I have no interest in “reality” beyond that, whatever that may mean. That is precisely why I caution talking about all kinds of comfortable notions one may have, and try to concentrate on precise statements about results of measurments.
All I had to say about T-duality was in the context of string theory, where this thing exist. It was not a statement about the real world where we don’t have experimental evidence for T-duality. If string thoery describes the world and it has a compact circle, there are no measurements that will distinguish a small circle from a large one. Since I am only interested in results of measurments there is no reason for me to choose.
I think I will leave it at that, there is too much war on reality to do this week.
IMO, there are no dimensions at all. My existance, along with all of yours, is a virtual one. At some time, me and everybody else will die. From your own point of veiw at that moment, the universe, space-time, everything will come to an end and this will mean that; you will, never have existed.
My virtual existance, one day reached a singularity. To become real, meant I had to escape from the edge of an horizon and fall into a black hole, at the same time. This was not to live forever, I did this… Just to be born. When I was born, you all! Was already dead. For you all to have existed, would mean I had no choice, other than to escape, I did and I did become real and I was born.
Teddy bears dont work, Quantum physics is nonsense, there are no magic boxes, CATS SEE WITH TWO EYES and one day I will die!!
Fyodor says: Now you want to assert that mathematically indistinguishable situations are *equally real*. Maybe you can persuade us, but you won’t do it by means of analogies with special relativity. You will do it by describing *precisely* how an observer who waves his arms about can possibly describe his experiences in terms of two different dimensionalities.
Exactly.
Moshe: I’m sorry if my tone was too aggressive. I did get carried away on my high horse. Still, I stand by everything I said. Fyodor has expressed it perhaps more succinctly than I did. Good luck with the war on reality.
Fyodor: I believe that the answer from string theory would be the following. The observer would wave his arms around and count a certain number of dimensions. Then he would learn string theory and do experiments with an LHC and conclude from the particle spectrum that there are extra dimensions that he didn’t count while waving his arms around. Some observers would be able to count all the dimensions by waving their arms and others would have a very difficult time finding them. What is indistinguishable is the final string-theoretic description of the world that the two would reach (which is what string theorists would call the real world), and the differences in the respective worlds that they see around them would be attributed to the fact that the observers would regard themselves as embedded in the stringy world in different ways. An observer imprisoned in a D-brane would be able to wave his arms in fewer directions than an observer who wasn’t so imprisoned.
when speaking of physics inventions
there is none like the extra dimensions
but how many you query
well there’s more than one theory
and discussion increases the tension
….oh wait this was supposed to go over on the poetry thread…. 😉
Elliot
In the perfect world, I like circles that are smoothly expressed, but it is not always like that?
There are 129600 dimensions. Don’t ask why…
I know I am Physic-ally challenged sometimes, but I can’t just take that for granted Claire. 🙂
It would seem that there should be as many dimensions as there are numbers to denote them. An infinite number that can go on forever in a “positive field”, as well as an infinite number in a “negative” field.
I see no reason that there need be a finite number of dimensions anymore than there can be a finite numeric system.
Perhaps “zero” and “infinity” are actually one and the same.
I’ve heard of the higher dimensions in string theory being spoken of as “existing within a realm that’s smaller than we can measure”…which just sounds like a complicated explanation of describing the attributes of the “infinitely small”.
Any reason this assumption should be considered “in error”??