Superstring theory: do all things exist in 11 dimensions?

2024 Author: Seth Attwood | [email protected]. Last modified: 2023-12-16 15:55

You've probably heard that the most popular scientific theory of our time, string theory, involves many more dimensions than common sense suggests.

The biggest problem for theoretical physicists is how to combine all fundamental interactions (gravitational, electromagnetic, weak and strong) into a single theory. Superstring Theory claims to be the Theory of Everything.

But it turned out that the most convenient number of dimensions required for this theory to work is ten (nine of which are spatial, and one is temporary)! If there are more or less measurements, mathematical equations give irrational results that go to infinity - a singularity.

The next stage in the development of superstring theory - M-theory - has already counted eleven dimensions. And one more version of it - F-theory - all twelve. And this is not a complication at all. F-theory describes 12-dimensional space by simpler equations than M-theory - 11-dimensional.

Of course, it is not for nothing that theoretical physics is called theoretical. All her achievements so far exist only on paper. So, in order to explain why we can only move in three-dimensional space, scientists started talking about how the unfortunate other dimensions had to shrink into compact spheres at the quantum level. To be precise, not into spheres, but into Calabi-Yau spaces. These are such three-dimensional figures, inside which their own world with its own dimension. A two-dimensional projection of such manifolds looks like this:

More than 470 million of such figurines are known. Which of them corresponds to our reality, is currently being calculated. It is not easy to be a theoretical physicist.

Yes, it seems a little far-fetched. But maybe this is precisely what explains why the quantum world is so different from what we perceive.

Let's dive into history a little

In 1968, the young theoretical physicist Gabriele Veneziano pored over the many experimentally observed characteristics of the strong nuclear interaction. Veneziano, who at the time was working at CERN, the European Accelerator Laboratory in Geneva (Switzerland), worked on this problem for several years, until one day he was struck by a brilliant guess. Much to his surprise, he realized that an exotic mathematical formula, invented about two hundred years earlier by the famous Swiss mathematician Leonard Euler for purely mathematical purposes - the so-called Euler beta function - seems to be able to describe in one fell swoop all the numerous properties of particles involved in strong nuclear force. The property noted by Veneziano provided a powerful mathematical description of many features of strong interaction; it sparked a flurry of work in which the beta function and its various generalizations were used to describe the vast amounts of data accumulated in the study of particle collisions around the world. However, in a sense, Veneziano's observation was incomplete. Like a memorized formula used by a student who doesn't understand its meaning or meaning, Euler's beta function worked, but no one understood why. It was a formula that needed an explanation.

Gabriele Veneziano

This changed in 1970 when Yohiro Nambu of the University of Chicago, Holger Nielsen of the Niels Bohr Institute, and Leonard Susskind of Stanford University were able to uncover the physical meaning behind Euler's formula. These physicists showed that when elementary particles are represented by small vibrating one-dimensional strings, the strong interaction of these particles is exactly described using the Euler function. If the string segments are small enough, these researchers reasoned, they will still look like point particles and, therefore, will not contradict the results of experimental observations. Although this theory was simple and intuitively appealing, it was soon shown that the description of strong interactions using strings was flawed. In the early 1970s. high-energy physicists have been able to look deeper into the subatomic world and have shown that some of the predictions of the string model are in direct conflict with observations. At the same time, the development of quantum field theory - quantum chromodynamics - in which the point model of particles was used, was going on in parallel. The successes of this theory in describing the strong interaction led to the abandonment of string theory.

Most particle physicists believed that string theory was forever in the trash bin, but a number of researchers remained true to it. Schwartz, for example, felt that "the mathematical structure of string theory is so beautiful and has so many striking properties that it should undoubtedly point to something deeper."²). One of the problems physicists faced with string theory was that it seemed to offer too many choices, which was confusing.

Some of the vibrating string configurations in this theory had properties that resembled those of gluons, which gave reason to really consider it a theory of strong interactions. However, in addition to this, it contained additional particles-carriers of interaction, which had nothing to do with the experimental manifestations of strong interaction. In 1974, Schwartz and Joel Scherk of the French Graduate School of Technology made a bold assumption that turned this perceived flaw into a virtue. Having studied the strange vibration modes of strings, reminiscent of carrier particles, they realized that these properties coincide surprisingly exactly with the alleged properties of a hypothetical carrier particle of gravitational interaction - the graviton. Although these "tiny particles" of gravitational interaction have not yet been discovered, theorists can confidently predict some of the fundamental properties that these particles should have. Scherk and Schwartz found that these characteristics are exactly realized for some vibration modes. Based on this, they hypothesized that the first advent of string theory ended in failure due to physicists overly narrowing its scope. Sherk and Schwartz announced that string theory is not just a theory of the strong force, it is a quantum theory that includes gravity, among other things).

The physical community reacted to this assumption with a very restrained attitude. In fact, as Schwartz recalled, "our work was ignored by everyone."⁴). The paths of progress have already been thoroughly littered with numerous failed attempts to combine gravity and quantum mechanics. String theory failed in its initial attempt to describe strong interactions, and many felt it was pointless to try to use it to achieve even greater goals. Subsequent, more detailed studies of the late 1970s and early 1980s. showed that between string theory and quantum mechanics, their own, albeit smaller in scale, contradictions arise. The impression was that the gravitational force was again able to resist the attempt to build it into the description of the universe at the microscopic level.

This was the case until 1984. In their landmark paper that summarized more than a decade of intense research that was largely ignored or rejected by most physicists, Green and Schwartz found that the minor contradiction with quantum theory that plagued string theory could be resolved. Moreover, they showed that the resulting theory is broad enough to cover all four types of interactions and all types of matter. News of this result spread throughout the physics community: hundreds of particle physicists stopped working on their projects to take part in what appeared to be the last theoretical battle in a centuries-old assault on the deepest foundations of the universe.

The news of the success of Green and Schwartz eventually reached even the graduate students of their first year of study, and the former discouragement was replaced by an exciting sense of involvement in a turning point in the history of physics. Many of us sat deep after midnight, studying weighty tomes on theoretical physics and abstract mathematics, knowledge of which is necessary to understand string theory.

However, string theory physicists have run into serious obstacles over and over again along the way. In theoretical physics, you often have to deal with equations that are either too complex to understand or difficult to solve. Usually in such a situation physicists do not give up and try to get an approximate solution of these equations. The state of affairs in string theory is much more complicated. Even the derivation of the equations turned out to be so complicated that so far it has been possible to obtain only their approximate form. Thus, physicists working in string theory find themselves in a situation where they have to look for approximate solutions to approximate equations. After several years of astounding progress during the first revolution in superstring theory, physicists were faced with the fact that the approximate equations used were unable to provide the correct answer to a number of important questions, thereby inhibiting the further development of research. Lacking concrete ideas for going beyond these approximate methods, many string physicists experienced growing frustration and returned to their previous research. For those who stayed, the late 1980s and early 1990s. were the testing period.

The beauty and potential power of string theory beckoned researchers like a gold treasure locked securely in a safe, visible only through a tiny peephole, but no one had a key to unleash these dormant forces. A long period of "drought" from time to time was interrupted by important discoveries, but it was clear to everyone that new methods were required that would allow one to go beyond the already known approximate solutions.

The end of the stagnation came with a breathtaking talk given by Edward Witten at the 1995 String Theory Conference at the University of Southern California - a talk that stunned an audience packed with the world's leading physicists. In it, he unveiled the plan for the next phase of research, thus initiating the "second revolution in superstring theory." Now string theorists are energetically working on new methods that promise to overcome the obstacles they meet.

For the widespread popularization of the TS, humanity should erect a monument to Columbia University professor Brian Greene. His 1999 book Elegant Universe. Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory”became a bestseller and received a Pulitzer Prize. The scientist's work formed the basis of a popular science mini-series with the author himself in the role of the host - a fragment of it can be seen at the end of the material (photo by Amy Sussman / Columbia University).

clickable 1700 px

Now let's try to understand the essence of this theory at least a little

Start over. The zero dimension is a point. She has no dimensions. There is nowhere to move, no coordinates are needed to indicate a location in such a dimension.

Let's put the second next to the first point and draw a line through them. Here is the first dimension. A one-dimensional object has a size - a length - but no width or depth. Movement within the framework of one-dimensional space is very limited, because the obstacle that has arisen on the way cannot be avoided. It only takes one coordinate to locate on this line.

Let's put a point next to the segment. To fit both of these objects, we need a two-dimensional space that has length and width, that is, an area, but without depth, that is, volume. The location of any point on this field is determined by two coordinates.

The third dimension arises when we add a third coordinate axis to this system. For us, the inhabitants of the three-dimensional universe, it is very easy to imagine this.

Let's try to imagine how the inhabitants of two-dimensional space see the world. For example, here are these two people:

Each of them will see their friend like this:

But in this situation:

Our heroes will see each other like this:

It is the change of point of view that allows our heroes to judge each other as two-dimensional objects, and not one-dimensional segments.

Now let's imagine that a certain volumetric object moves in the third dimension, which crosses this two-dimensional world. For an outside observer, this movement will be expressed in a change in two-dimensional projections of an object on a plane, like broccoli in an MRI machine:

But for an inhabitant of our Flatland, such a picture is incomprehensible! He is unable to even imagine her. For him, each of the two-dimensional projections will be seen as a one-dimensional segment with a mysteriously variable length, arising in an unpredictable place and also unpredictably disappearing. Attempts to calculate the length and place of origin of such objects using the laws of physics of two-dimensional space are doomed to failure.

We, the inhabitants of the three-dimensional world, see everything as two-dimensional. Only the movement of an object in space allows us to feel its volume. We will also see any multidimensional object as two-dimensional, but it will amazingly change depending on our relationship with it or time.

From this point of view, it is interesting to think about gravity, for example. Everyone has probably seen similar pictures:

It is customary to depict on them how gravity bends space-time. Bends … where? Precisely in none of the dimensions we are familiar with. And what about quantum tunneling, that is, the ability of a particle to disappear in one place and appear in a completely different place, moreover, behind an obstacle through which in our realities it could not penetrate without making a hole in it? What about black holes? But what if all these and other mysteries of modern science are explained by the fact that the geometry of space is not at all the same as we used to perceive it?

The clock is ticking

Time adds another coordinate to our Universe. In order for a party to take place, you need to know not only in which bar it will take place, but also the exact time of this event.

Based on our perception, time is not so much a straight line as a ray. That is, it has a starting point, and the movement is carried out only in one direction - from the past to the future. And only the present is real. Neither the past nor the future exists, just as there is no breakfasts and dinners from the point of view of an office clerk at lunchtime.

But the theory of relativity does not agree with this. From her point of view, time is a full-fledged dimension. All events that existed, exist and will exist, are as real as the sea beach is real, no matter where the dreams of the sound of the surf took us by surprise. Our perception is just something like a searchlight that illuminates some segment on a straight line of time. Humanity in its fourth dimension looks like this:

But we see only a projection, a slice of this dimension at each separate moment in time. Yes, like broccoli on an MRI machine.

Until now, all theories have worked with a large number of spatial dimensions, and temporal has always been the only one. But why does space allow the appearance of multiple dimensions for space, but only one time? Until scientists can answer this question, the hypothesis of two or more time spaces will seem very attractive to all philosophers and science fiction writers. Yes, and physicists, what is really there. For example, the American astrophysicist Yitzhak Bars sees the second time dimension as the root of all troubles with the Theory of Everything. As a mental exercise, let's try to imagine a world with two times.

Each dimension exists separately. This is expressed in the fact that if we change the coordinates of an object in one dimension, the coordinates in others can remain unchanged. So, if you move along one time axis that intersects another at a right angle, then at the point of intersection time around will stop. In practice, it will look something like this:

All Neo had to do was position his one-dimensional time axis perpendicular to the time axis of the bullets. Sheer trifle, agree. In fact, everything is much more complicated.

The exact time in a universe with two time dimensions will be determined by two values. Is it hard to imagine a two-dimensional event? That is, one that extends simultaneously along two time axes? It is likely that such a world will require specialists in time mapping, as cartographers map the two-dimensional surface of the globe.

What else distinguishes two-dimensional space from one-dimensional space? The ability to bypass an obstacle, for example. This is already completely beyond the boundaries of our mind. An inhabitant of a one-dimensional world cannot imagine what it is like to turn a corner. And what is this - a corner in time? In addition, in two-dimensional space, you can travel forward, backward, but at least diagonally. I have no idea what it's like to walk diagonally through time. I'm not even talking about the fact that time is the basis of many physical laws, and it is impossible to imagine how the physics of the Universe will change with the appearance of another temporal dimension. But thinking about it is so exciting!

A very large encyclopedia

Other dimensions are not yet discovered and exist only in mathematical models. But you can try to imagine them like this.

As we found out earlier, we see a three-dimensional projection of the fourth (time) dimension of the Universe. In other words, every moment of the existence of our world is a point (similar to the zero dimension) in the time interval from the Big Bang to the End of the World.

Those of you who have read about time travel know how important the curvature of the space-time continuum plays in them. This is the fifth dimension - it is in it that the four-dimensional space-time is "bent" in order to bring together some two points on this straight line. Without this, the journey between these points would be too long, or even impossible. Roughly speaking, the fifth dimension is similar to the second - it moves the "one-dimensional" line of space-time into the "two-dimensional" plane with all the ensuing opportunities to wrap around a corner.

Our especially philosophical-minded readers a little earlier, probably, thought about the possibility of free will in conditions where the future already exists, but is not yet known. Science answers this question like this: probabilities. The future is not a stick, but a whole broom of possible scenarios. Which one will come true - we'll find out when we get there.

Each of the probabilities exists as a "one-dimensional" segment on the "plane" of the fifth dimension. What is the fastest way to jump from one segment to another? That's right - bend this plane like a sheet of paper. Where to bend? And again it is correct - in the sixth dimension, which gives "volume" to this entire complex structure. And, thus, makes it, like a three-dimensional space, "finished", a new point.

The seventh dimension is a new straight line, which consists of six-dimensional "points". What is any other point on this line? The whole infinite set of options for the development of events in another universe, formed not as a result of the Big Bang, but in different conditions, and acting according to different laws. That is, the seventh dimension is beads from parallel worlds. The eighth dimension collects these "lines" into one "plane". And the ninth can be compared with a book that fits all the "sheets" of the eighth dimension. It is a collection of all the histories of all universes with all the laws of physics and all the initial conditions. Point again.

Here we run into the limit. To imagine the tenth dimension, we need a straight line. And what other point can there be on this line if the ninth dimension already covers everything that can be imagined, and even that which is impossible to imagine? It turns out that the ninth dimension is not another starting point, but the final one - for our imagination, in any case.

String theory states that it is in the tenth dimension that strings vibrate - the basic particles that make up everything. If the tenth dimension contains all universes and all possibilities, then strings exist everywhere and all the time. I mean, every string exists in our universe, and any other. At any given time. Straightaway. Cool, huh?

In September 2013, Brian Green arrived in Moscow at the invitation of the Polytechnic Museum. The famous physicist, string theorist, professor at Columbia University, he is known to the general public primarily as a popularizer of science and the author of the book "Elegant Universe". Lenta.ru spoke with Brian Green about string theory and the recent challenges it has faced, as well as quantum gravity, amplitude and social control.