Compactification of dimensions: why we only perceive four dimensions

Compactification of dimensions: why we only perceive four dimensions
Compactification of dimensions: why we only perceive four dimensions

One of the main candidates for a theory of everything is string theory, or its more generalized version, M-theory. But it makes one prediction that we’ll hardly ever be able to verify - hidden, compactified dimensions.


String theory tries not only to combine quantum mechanics with General Relativity, but also to explain the spectrum of particles and forces observed in nature. In the most recent formulation of the theory - matrix theory - there are 11 dimensions. Its proponents are faced with one of the biggest problems of string theories - explaining exactly how extra dimensions are "compactified", which makes them impossible to observe in our four-dimensional world. Compactification also clarifies the most interesting properties of the theory.

String theory states that the world is made up of incredibly small vibrating strings in ten-dimensional space-time. In 1995, during the second superstring revolution, Edward Witten proposed M-theory that combined all five different types of string theory. This is an 11-dimensional theory that includes supergravity. There is no single answer among scientists as to what the "M" means in the name, but many theorists agree that this letter means "membranes", since the theory contains vibrating surfaces of several different dimensions. M-theory lacks exact equations of motion, but in 1996 Tom Banks of Rutgers University and his colleagues proposed a description of it as a "matrix theory" whose main variables are matrices.

Compactifying this 11-dimensional theory to four changes was by no means easy. To compactify literally means to "roll up" the extra dimensions of a theory to a very small size. For example, to fold two dimensions, take a donut - or a torus (it's a two-dimensional surface) - and squeeze it to a circle or loop with a small cross section, and then squeeze that loop to a point. Without a sufficiently sensitive probe that could register "squeezed" measurements, this loop looks one-dimensional, while the point is zero-dimensional. In M-theory it is assumed that we are talking about dimensions of the order of 10-33 centimeters, which, in turn, can in no way be recorded by modern equipment. It turns out that after compactification of seven dimensions, the world around us looks four-dimensional.


But what is a dimension in itself? Intuitively, it might seem that each dimension is an independent direction in which we (or any object) can move. So it turns out that we live in three spatial dimensions - "forward-backward", "left-right" and "up-down" - and one time - "past-future". In general, these are four dimensions. But our perception of dimensions is tightly tied to scales.

Imagine that you are watching a ship sailing from a distance to the port. At first, it looks like a zero-point on the horizon. After a while, you realize that it has a mast pointing to the sky: now it looks like a one-dimensional line. Then you notice its sails - and the object looks already two-dimensional. As the ship gets closer to the dock, you finally notice that it has a long deck - the third dimension.

There is nothing strange in this, as well as in the fact that a donut, reduced to an incredible size, appears to be a zero-dimensional point.The fact is that we are not able to determine measurements from long distances. This logically leads to what was described above: there may be other dimensions, but they are so small that we do not perceive them.

Let's return to compactification of measurements. Imagine that you are a squirrel living on an infinitely long tree trunk. In one way or another, a tree trunk is a cylinder. You can move in two independent directions - "along" and "around". Once you get bored, you move to a tree with a thinner trunk, the circumference of which is much smaller. Now your 'around' dimension is much smaller than before. You only need two steps to completely bypass the barrel. You jump to an even thinner tree. Now, in one step, you wrap the barrel a hundred times! The “around” dimension has become too small for you to notice. The thinner the tree trunks become, the more the dimensions of your world are reduced to one.


This is exactly what happens in string theory with six (seven for M-theory) extra dimensions. Each time you move your hand through space, you turn around the hidden dimensions an incredible number of times.

As mentioned above, the dimensions of the compactified measurements are of the order of 10-33 centimeters, which is comparable to the Planck length (1.6 × 10-33 centimeters). It should be noted that it is unlikely that in the near future we will have the opportunity to directly register them experimentally. Nevertheless, scientists hope for some tests, the results of which, however, largely depend on a successful combination of circumstances.

The shape and size of the strings is extremely important for simulating their vibrations and interactions. You need to understand how they twist around the six curled up dimensions. The precise structure of the surface formed by compactification changes the physics driven by the strings.

There are several ways in which the extra dimensions can fold into such a small space. However, it is not yet known which of these methods ultimately leads to traditional physics.

There have been many previous attempts to compactify matrix theory using a six-dimensional toroid, but nothing came of it. No one thought that the supposedly more difficult compactification problem with Calabi-Yau manifolds would provide workable solutions for a working theory. Compactification of dimensions with Calabi-Yau manifolds avoids some of the complications of matrix theory.

Current research in string theory is more about Calabi-Yau manifolds. This is undoubtedly a promising group of compactifications, but there is still no clear answer, and the number of manifolds discovered has already increased to 10500as string theorist Brian Green recently pointed out in a podcast with Sean Carroll.


String theorists are still far from a clear and unambiguous understanding of whether M-theory actually describes the world on the smallest scales. However, as Edward Witten noted: "It's amazing how you can build a theory that includes gravity, but which was originally based only on gauge theory."

String theory is a complex mathematical apparatus. As Clifford Johnson and Brian Greene pointed out in our magazine interviews, it's hard to say that this theory actually describes reality. But even if it turns out that it has nothing to do with reality, then it will definitely be an important step towards something bigger - towards a theory that describes the Universe more accurately and more elegantly than anything we knew before.

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