# Is the physics mathematically consistent

## How many dimensions does the world have?

### Research Report 2007 - Max Planck Institute for Gravitational Physics

In its now more than 30-year history, string theory has given rise to a multitude of highly interesting cross-connections to other areas of theoretical physics and also to sub-disciplines of pure mathematics. In a number of cases - the examples range from knot theory to theoretical solid state physics - it has provided valuable suggestions for other branches of research. The key question of whether the long-sought, large, unified theory of all elementary particles and their interactions has now been found with string theory is still open. Ultimately, this question can only be answered experimentally, and that has proven extremely difficult.

### The problem

A central finding of modern physics is that the variety of matter that we perceive in everyday life is made up of a comparatively small number of basic building blocks, the elementary particles. Four basic forces work between them: electromagnetism, which binds electrons to the atomic nucleus, for example, the strong nuclear force which binds quarks to form protons and neutrons and these to form atomic nuclei, the weak nuclear force, which is responsible for certain radioactive decays, and universal gravity . Every interaction comes about through the exchange of elementary particles, so-called gauge bosons: of photons, gluons, W and Z bosons and gravitons. The forces are thus reduced to fundamental particles.

With the exception of gravitation, the particles of matter and the forces acting between them can be described in accordance with the laws of quantum theory and special relativity. The result is the so-called standard model of elementary particle physics, the prime example of a quantum field theory.

Although impressively confirmed by many experiments, the standard model is not a completely satisfactory description of the fundamentals of our world. On the one hand, it contains around two dozen free parameters, the values of which do not follow from the theory itself, but have to be determined experimentally. These include, for example, the mass ratios of the particles and the strength of their interactions. On the other hand, one can hope for a complete theory that it has no free parameters. The second drawback is that there is no gravity whatsoever in the Standard Model world. Admittedly: For all earthly experiments, for example at particle accelerators, this force is negligibly weak compared to the other three forces, so that this shortcoming is of very little importance in experimental particle physics. In the very early universe, on the other hand, according to conventional cosmological models, there should have been phases in which all four forces were of comparable strength. Although these ratios can never be achieved in the laboratory, we need an extension of the Standard Model in order to better understand the origin of our universe, which must be in accordance with the laws of quantum theory and general relativity. Because the energies typical for such a quantum gravity theory are extremely high, experimental verification will be very difficult. But even the theoretical description brings with it problems: Attempts to fall back on the quantum field theory, which has proven so extraordinarily well in the standard model of elementary particle physics, for the formulation of quantum gravity, are shipwrecked. The result of such an exercise is a theory devoid of any predictive power.

### The solution

A starting point of the Standard Model is the idea of idealized point particles, as they are known from classical physics. There is some evidence that this fact could be responsible for the failure of the direct generalization of the Standard Model to gravity. This leads to the core idea of string theory, which does not start from point-like particles, but whose basic building blocks are one-dimensional, tiny vibrating strings. The same string can vibrate in different ways; Put simply, the variety of "overtones" corresponds to the variety of elementary particles. The strings are so extremely short that they appear point-like in the experiments, but depending on the vibration state as point-like particles with different properties. There are open and closed strings that can transform into each other. Among the vibrational states of the closed string there is always one that has the right properties to play the role of the exchange particle of gravitation, the graviton. However you formulate a consistent string theory, that particular vibrational state will always be included. Gravitation is therefore an inevitable consequence of a string theoretical description of the world. In string theory, the idea of unifying all elementary particles and their interactions is realized in a very economic way.

### From ten to four dimensions?

However, this standardization has its price. It turns out that strings cannot vibrate in any space-time. A particularly striking limitation resulting from the laws of quantum theory is that the cosmos must have not only the usual three, but nine or even ten spatial dimensions. In a world with one time and ten space dimensions, string theory would lead to a complete model with no free parameters, known as M-theory, but still largely not understood. 'M' stands for matrix, membrane, magic or mystery, depending on which aspect you want to emphasize.

There are ways in which a higher-dimensional space can appear three-dimensional to us. One of them is that the six extra dimensions are "tightly rolled up": just as we can treat a wire as an effectively one-dimensional object when viewed from a distance, a universe with six of its ten dimensions rolled up into tiny bits would be for its inhabitants effectively four-dimensional (three spatial directions and the time direction). There is a huge number of possibilities for how exactly one gets from nine to three spatial dimensions by rolling up dimensions, which are compatible with all consistency conditions of string theory. The permissible three-dimensional worlds differ in several respects, for example by the size of the rolled-up dimensions and, closely related to this, by the spectrum of the elementary particles present, whose masses and other properties depend on how the strings are defined by the properties of the extra dimensions, for example Conditions can oscillate.

However, gravity is always one of the existing interactions. The question of whether the number of these possibilities, which are called “string vacuums”, is finite or infinite is still open. In any case, it is enormous - the number 10 is often read in this context^{500}, so a 10 followed by 500 zeros!

The string theorists have already found admissible worlds whose properties come very close to those of our own world - this is by no means a matter of course in view of the confusing huge number of possibilities. The search for the exact standard model within string theory continues.

Whether there is a principle of selection that distinguishes this particular model - and thus the world in which we live - from the other possibilities, or whether string theory offers other possibilities to explain why our world is the way it is is an open one and currently hotly debated issue.

### holography

Extra dimensions aren't the only surprise string theory has to offer when it comes to dimensions. A closer look shows that the concept of dimension is not as clearly defined as is commonly assumed.

The first indications of this came from the theory of black holes. Black holes are created by the collapse of heavy stars. Huge black holes are in the centers of galaxies - including ours. The interface between a black hole and the outside world is called the event horizon. Everything that crosses this horizon necessarily falls into the black hole; escape and even communication with the outside world are impossible. Even light is captured, hence the name black hole.

It has been known since the 1970s that the area of the event horizon is a measure of the amount of information that the black hole has already swallowed - in physical terms: its entropy. One of the successes of string theory is that it can explain, for at least some types of black holes, how information is stored on the horizon.

It is noteworthy that the information content of the three-dimensional volume is given by the content of a two-dimensional surface, the horizon surface. The idea of a holographic world goes one step further by asking the question: is our world a hologram? Is four-dimensional space-time based on a three-dimensional reality? Are there two equivalent, in the terminology of physicists, dual, descriptions of one and the same reality: the hologram and the reconstructed higher-dimensional image?

For ten years we have known a very concrete realization of this holographic principle outside the physics of black holes - an explicit four-dimensional description of a five-dimensional world. The four-dimensional hologram is a so-called Yang-Mills theory, a quantum field theory that is closely related to that which describes the strong interaction within the framework of the standard model. This four-dimensional quantum field theory is equivalent to a string theory in which five of the nine space dimensions are rolled up in a certain way, while the remaining five-dimensional space-time is curved in a certain way. This equivalence is known as the AdS / CFT correspondence, after the abbreviations for the relevant quantum field theory (“Conformal Field Theory”) and the five-dimensional spacetime in which the strings exist (“anti-de-sitter spacetime”). It is often called the “Maldacena Conjecture” after Juan Maldacena, who first put forward this thesis.

While rigorous evidence of this correspondence is pending, it has already led to an impressive variety of developments that have contributed to a deeper understanding of both string theory and Yang-Mills theories. The AdS / CFT correspondence and a large number of its generalizations have established a highly topical research area.

One of the most exciting questions is whether there are also dual descriptions for the models of conventional elementary particle physics, such as a higher-dimensional string theory. If this were the case, it would have unexpected consequences, because the holographic principle can be exploited in two directions. Understanding the higher dimensional string theory provides interesting information about the hologram. Conversely, the hologram allows conclusions to be drawn about the higher-dimensional theory. Certain calculations in the context of one theory can turn out to be much easier than their counterparts in the other model. If it could provide a higher-dimensional image of the theory of strong nuclear forces, for example, then string theory could become a useful tool in conventional elementary particle physics. There are a number of indications that developments could go in precisely this direction.

### outlook

Since string theory forces us to think about the dimensionality of our world, we can ask ourselves whether the additional dimensions can be measured not only indirectly, as indicated above via the elementary particle spectrum, but also directly. Indeed, these possibilities have been investigated and models have been constructed that lead to measurable consequences. At the particle accelerator LHC (the Large Hadron Collider of CERN), which will be put into operation in 2008, one would see additional dimensions as apparent violations of energy conservation: some of the energy disappears in the fourth, fifth or even higher dimensions.

Mind you, that would not be proof of the correctness of string theory. In any case, such proof will be very difficult. Nevertheless, string theory has meanwhile provided so much impetus for developments both in elementary particle physics and in mathematics that there are no longer any doubts about its internal consistency. Whether it is the “Theory of Everything” we are looking for remains an open question. An alternative role would be as a theoretical substructure which, similar to quantum field theory, exists in many facets and can be used to deal with different questions without being an all-encompassing superstructure. Perhaps it is just a theoretical substructure similar to quantum field theory - a framework that can be filled with a wide variety of concrete content and thus used to deal with very different questions without the unified description of all elementary particles and basic forces being one of the applications.

We don't yet know the answer to this question. But in any case, at least for the already large and still growing string community, we do not know of a more elegant and multifaceted theory so far, and that we can hope that string theory will come up with many surprises in the years to come.

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