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Predicting the spacetime dimensionality

In this text, I would like to expand Moshe Rozali's comments about string gas cosmology and clarify the difference between legitimate speculative work in physics and hopeless charlatanism. We will talk about the attempts to theoretically justify why the dimensionality of our spacetime is 3+1 and I will compare the following two papers:

Brandenberger & Vafa: Superstrings in the early Universe (1988)
Ambjorn, Jurkiewicz, Loll: Emergence of a 4D world from causal quantum gravity (2004)
The first program will be referred to as "string gas cosmology" (a modern term) while the second program will be referred to as "triangulation".

The problem

Most people can observe one dimension of time and three spatial dimensions. They seem to obey the laws of a smooth, quasi-Euclidean (or Minkowski) geometry, at least in some approximation. In particular, the three spatial dimensions seem to be nearly flat and smooth dimensions following the rules of the Euclidean geometry.

A theory that explains the origin of space and time must be compatible with this very basic observational fact - that 3+1 dimensions exist and behave as smooth Euclidean dimensions in a wide range of phenomena. More ambitiously, a complete theory of spacetime might even provide us with an explanation for this fact. It is not guaranteed that a 3+1-dimensional spacetime is the only one in which physicists may be asking similar questions but such a possibility is not ruled out, either.

There may exist semi-explanations why one dimension has the opposite signature than others: why there is one time. If there were no time, the Universe would be too boring and it would admit no evolution. Because we apparently need some time to ask a question, physicists couldn't ask such questions in a timeless Universe. ;-)

If there were at least two macroscopic temporal dimensions, including t1 and t2, there would exist closed time-like curves - for example, circles in the t1-t2 plane - which would allow you to return to the "past" along a smooth trajectory and to kill your grandmother before she touched your grandfather for the first time. The Universe would be inconsistent. Also, the Hilbert space would contain physical polarizations of spinning particles such as photons whose norm would be negative: probabilities wouldn't be positively definite. One gauge invariance is not enough to remove two sets of unphysical, ghostly time-like oscillators.

Fine. So imagine that the question why there is one time is settled. But the universes with a different number of spatial coordinates seem much more conceivable and realistic. Is there an explanation why we see three macroscopic spatial dimensions? There exist various anecdotes why the life would be rather hard outside 3+1 dimensions. For example, in 2+1 dimensions, a dog would be cut into two pieces by a bone it ate (assuming that the mouth differs from the opposite hole). In 4+1 dimensions (or higher), the electrostatic potential would increase as quickly as 1/r^2 at short distances (or faster). In such a strong field, the uncertainty principle wouldn't be enough to keep the electron at a finite distance and the electron would collapse to the nucleus as tightly as allowed, to reduce its energy. The non-relativistic chemistry as we know it wouldn't exist, to say the least.

But these are kind of "anthropic" arguments from the realm of entertainment. We may prefer a physical explanation that doesn't depend on the existence of life. The two basic programs mentioned at the beginning propose vastly different approaches how to find such an explanation.

String gas cosmology

The string gas cosmology postulates that there exists a new important era of early cosmology in which important processes decide about the number of dimensions that are allowed to expand to astronomical size.

The particular idea of Brandenberger and Vafa was to assume that at the beginning, strings were wound around cycles of our space and this winding generally prevented the dimensions from expanding because the wound strings would be getting very long and massive. The only way how this "confinement" can be overcome is to annihilate the strings with the oppositely wound strings: the resulting strings born in this "merger" have a vanishing winding number and they can be unwrapped continuously.

But if the space has too many dimensions, it is unlikely for a pair of strings to collide and annihilate. The maximum number of spatial dimensions in which the strings tend to collide at some moment is three: it's because each string in the pair has 1 dimension and one more spatial dimension is probed as time goes and the strings move.

More concretely, the first wound string may be assumed to be stretched in the x-direction, in z=0 plane. The second string is stretched in the y-direction, in z=u plane which is different. They don't intersect. But generically, the transverse distance u in between them changes with time so chances are at least 50% that after some time, they inevitably intersect and they can annihilate. If the number of spatial dimensions that macroscopically expanded exceeded three, the strings would be increasingly unlikely to meet, as the dimensions grow in size. The unwinding couldn't proceed, and the expansion would eventually stop.

So that is their rough speculative reason why three dimensions but not more dimensions were allowed to expand into macroscopic size. This explanation links the dimension of space with the dimension of strings: our observed dimension of space is one plus twice the dimension of a string. Incidentally, Lisa Randall suggested an opposite explanation (or mechanism) based on annihilating braneworlds in which the observed spacetime dimension (four) is one-half of the full spacetime dimensions (ten) minus one. ;-)


In the triangulation approach, the physicists construct a discrete model, simulate it with a computer, and define a number that could correspond to the Hausdorff dimension of the network of their discrete building blocks. They are excited that their particular number is 3.10.

Let me now describe the basic points about the nature and status of these two approaches. String gas cosmology
  • is based on degrees of freedom and objects that are known to be important in quantum gravity, at least in some limit, namely strings (or branes)
  • assumes the existence of special phenomena in the very early cosmology in which these fundamental objects were important - which is almost certainly the case, too
  • calculates the dimension of space as a function of some other numbers - the dimension of strings in this case - whose value can be justified by independent physical arguments
  • seems to be ruled out in its simplest incarnation; however, it leads to somewhat well-defined rules whose predictions can be more or less checked in more detailed stringy compactifications that include branes of other kinds and dimensionalities (brane gas cosmology)
On the other hand, the triangulation attempt
  • is based on a toy model that can be shown to be non-local, non-unitary, non-smooth, incompatible with basic qualitative features of physics, i.e. generally inconsistent and unrelated to physics as we know it
  • identifies the dimensionality of space with a number that seems to have no relationship with a smooth geometry
  • calculates the Hausdorff dimension as a function of other parameters and assumptions that are arbitrary and have no other independent justification (garbage in, garbage out)
  • leads to a wrong result, even if you ignore the problems above; the non-integral figure they obtain strongly indicates that smooth space doesn't emerge from the discrete model; moreover, this troubling fractional dimensionality seems to be a universal prediction of this whole class of models
Fine. Let me now discuss these four facts about the status of the two approaches one by one, for both approaches simultaneously.

Underlying theory

Our goal is to calculate the dimensionality of the space around us. Clearly, this can only be calculated from the right theory that actually describes the space around us. The further your theory is separated from the phenomena we actually observe, the less relevant your results will be for the reality.

Because the dimensionality of space is something that general relativity cares about, the theory you build upon should be particularly accurate as a description of the geometry of space. But in principle, you should also incorporate a correct description of all other particles and forces - or to show that their existence doesn't change your result.

Brandenberger and Vafa actually build on objects - strings - that have been shown to be the fundamental degrees of freedom of quantum gravity, at least in the weakly coupled "stringy" regimes of quantum gravity (regimes that incidentally include a significant portion of realistic compactifications). We know that certain massless closed string modes inevitably behave as gravitons, obeying the rules of general relativity at long distances. Strings can be shown to exist in all other consistent quantum gravity formalisms we know as of today, including the AdS/CFT and matrix (string) theory.

On the other hand, the simplices in the triangulation approach have not been shown relevant for physics (i.e. for gravity, other forces, or other known elementary particles) in any way - except for a wishful thinking. These theories share a lot of other universal pathological features. For example, they either violate locality and causality - hugely, macroscopically - or they violate unitarity and transitivity of time evolution - by equally unacceptable amounts.

Why? Because if you want to restore locality, at least approximately, you must prevent the simplices from interacting with very distant simplices. Typically, the non-local and/or acausal histories which lead to very non-trivial spacetime topologies (with a lot of wormholes between any pairs of points etc.) are removed by hand. But such a procedure leads to a violation of causality and unitarity by itself.

In Feynman's path integral approach to quantum mechanics, it is absolutely crucial that one sums over all histories, not just an ad hoc selection of them. Why? Because if you calculate the evolution amplitude from slice A to slice C, you get the result by summing over all histories between A and C. Why? Because you may divide the interval to A-B and B-C evolutions. The A-C evolution is the sum over all intermediate configurations B and over all histories in the A-B and B-C intervals. It is easy to see that any "global" or "topological" restriction on the histories in the A-C interval will destroy this transitivity property unless it can be formulated as a local constraint on some variables. (Requiring that the magnetic fluxes over cycles are integer-valued is about the most non-trivial "global" constraint you should ever impose and it is not really new: it automatically follows from a description in terms of a potential.)

For a simpler example, just imagine that you use Feynman's path integral to calculate the result of a complex double-slit experiment (with two double slits) but you remove the histories in which the particle goes through the left slits twice. Obviously, the amplitudes you calculate in this way won't be equivalent to the results of any Schrödinger's equation. The conservation of probability will fail, too. If you rescale the wave function ad hoc, to keep the sum of probabilities at one, you will violate locality because the amplitude of one outcome will depend even on the amplitudes of other outcomes that won't be realized. In fact, not only locality will be broken: it will no longer be true that the probability of "A or B" is the sum of "P(A)" and "P(B)" if A,B are mutually exclusive. Logic will break down, too.

All these things are inconsistent in quantum mechanics. The only problem is that many (or most) people don't understand the subtle character of quantum mechanics and the robustness of its postulates so they play with tons of models that are demonstrably wrong.

Where does the dimensionality emerge?

The approaches dramatically differ in the "location in spacetime" where the relevant phenomena decide about the spacetime dimensionality, too. In string gas cosmology, one has to study some details of cosmology that tell us which universes are "fit enough" to ever become "mature", assuming that they were born from a Big-Bang-like event. The "Big Bang" is where these decisions take place.

The triangulation paradigm seems to argue that space in quantum gravity cannot possibly have a different dimensionality than the number that they calculate from their randomly picked discrete model: the decision occurs everywhere in the Cosmos. This is, of course, known to be wrong. You may believe that none of the vacua of string theory describes our Universe perfectly accurately. But you can't really deny that they're consistent superselection sectors of a theory of quantum gravity. Many of them have dimensionalities that differ from 3+1, including the well-known AdS5 x S5 compactification with a known holographic description. The extra dimensions may be compactified but they may be, in principle, decompactified, too.

It is simply not true that other "pieces of space" with dimensionalities different from 3+1 cannot exist.

Even if you decided to pretend that these counterexamples don't exist and if you only focused on discrete models, it is simply not true that all of them lead to the same Hausdorff dimension. Depending on the shape of the simplices, the types of interactions in the action, and the additional criteria to a posteriori "censor" the path integral, you will obtain very different results.

In the triangulation approach, the space dimensionality is extracted as a Hausdorff dimension of a set. And they obtain something like 3.10. Does it mean that they have found evidence that there are three dimensions of space in such a discrete model? Well, not at all.

This fractal, the Sierpinski carpet, has Hausdorff dimension of log(8)/log(3) = 1.8928 or so. Does it mean that it is a two-dimensional Euclidean plane? No. The physics inside this carpet simply doesn't behave as physics in a two-dimensional Euclidean plane. This is a very serious problem.

How serious is it? Well it is equally serious as the problem with an alternative Christian theory of causal dynamical triangulation. A priest can calculate the dimensionality of space as the number of entities in the Holy Trinity. There are the Father, the Son, and the Holy Spirit, giving you the x,y,z dimensions of space.

I am not really exaggerating here. The three "objects" in the causal dynamical triangulation are equally disconnected from the three "objects" we want to obtain - coordinates in a theory that obeys the laws of three-dimensional geometry - as is the Holy Trinity. It's just a number. The priest at least got the right answer.

Wasn't the output inserted as input?

Even if you neglect the problem that one of the two approaches uses an inconsistent theory, invalid assumptions about the diversity of a priori allowed dimensionalities, and an incorrect identification of the spacetime dimensions with some numbers in the theory, there are other problems.

One of them is that the calculated dimensionality depends on certain numbers that you inserted. With different rules for the simplices, their interactions, and the censorship of their histories, you would obtain different results.

This problem actually exists in the string approach, too, although it is less serious here. Why? Because the dimensionality calculated by the simple argument due to Brandenberger and Vafa depends on the dimension of a string, one. This dimension looked like a canonical number in string theory. However, once branes were discovered in the mid 1990s to be new important ingredients of string theory, other possible dimensions of objects that may prevent the space from expanding have emerged, too. So the original calculation 1+1+1=3 seems to be just a kind of naive perturbative approximation.

I say that this problem is less serious in string theory because there still exist arguments that the strings continue to be more important than other objects in large regions of the parameter space. And even if they are not, they exist more refined calculations that lead to rather accurate results. These results depend on the particular model we choose (and its spectrum of branes).

In this sense, the string/brane gas cosmology approach still offers a link between the details of observable (!) physics on one side and the dimensionality of a well-behaved and desirable smooth space (!!) on the other side.

On the other hand, the calculated link in the triangulation approach is between unobservable (!) assumptions about the elementary simplices on one side and the Hausdorff dimensionality of an unphysical (!!) fractal on the other side.

Is the result correct when calculated more carefully and can it be fixed?

The result 3.10 for the dimension of the space in the triangulation approach is wrong. It is also true that when the Brandenberger-Vafa calculation is computed more accurately, with all the stringy and relativistic effects and with the additional branes in realistic stringy vacua, the final conclusion is typically altered, too. 3+1 is no longer the unique correct answer.

Such a falsification is surely bad news for those who would like to believe that the original Brandenberger-Vafa paper was the final word. On the other hand, falsification is the way how science makes progress.

The important thing is that there exist transparent criteria for what it means to make all these calculations more reliable. It means to pick a stringy vacuum that has more realistic physics than the previous ones and to incorporate all kinds of high-temperature, short-distance, strong-coupling effects that may be relevant but that were previously neglected. Also, more complete papers could calculate more than just the number of the dimensions - maybe something about the topology of the compact manifold, too. And finally, it could be connected with more diverse observational data, for example with some subtle properties of the cosmic microwave background (or other data and facts from cosmology).

We don't know for sure whether this approach will ever be completely well-established and accepted - which is why it remains speculative - but we know that there exists a meaningful program whose success may be judged by scientific criteria. On the other hand, such a program doesn't exist in the case of triangulations because we don't know of any "refinable" links between the observable physics and the detailed assumptions (or outcomes) of the triangulated calculations which is why these calculations will almost certainly remain balderdash in all of their future reincarnations.

And that's the memo.

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