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regions which general relativity can only represent as singularities—we desperately need a quantum theory of gravity. Only then can we obtain a reliable description of high-density states such as the big bang or black holes. A complete theory, alas, is not available; rather, there are several very different approaches, all having distinct strengths and weaknesses. Even in their basic principles they strongly deviate from one another. The reason for this diversity is that the mathematical foundations of general relativity and quantum theory, as well as the understanding of their concepts, can be combined in many ways. At present, we don’t know which of the proposed combinations is the correct one—or whether we will have to look for yet another route to reach the goal.

APPROACHES TO QUANTUM GRAVITY:

STRENGTH IN DIVERSITY

For a wide, experimentally untested field such as quantum gravity it is not surprising that vastly different routes to a potential completion are being taken. A count of researchers and publications indicates that the most-traveled roads are string theory and loop quantum gravity, which are indeed, as we will see, very different from each other. In addition to these two, intriguing quantum-gravity-related proposals such as twistor theory, noncommutative geometry, causal sets, and causal dynamical triangulations are being developed by smaller numbers of researchers, where progress is correspondingly slower. Many of the key ideas in quantum gravity can already be illustrated by string theory and loop quantum gravity, and so we will focus on those.

STRING THEORY: THE RICHEST SYMPHONY

String theory has attracted the greatest interest. Having started from older developments in particle physics, its strengths rest in particular on a quantum theoretical description of excitations such as gravitational waves on a fixed space-time stage. In string theory, there are particle-like objects, gravitons, which transport the gravitational force in packages just as photons carry light. But this is almost a side effect, for the basic concept of string theory is in fact much more radical, as it proposes to leave behind the fundamental picture of particles as pointlike or as hard solid balls with tiny extensions. Even the indistinct but still somewhat localized wave function of quantum mechanics is surpassed by the string concept.

Instead, particles such as gravitons as well as the constituents of matter—the electron, the quarks forming protons and neutrons, and additional elementary particles generated at high energies of accelerator experiments—are understood as excitations of one single elementary object: the string. As the string of a musical instrument can create diverse sounds by differently excited vibrations, a fundamental string can oscillate in many ways. Just as sounds are distinguished from one another by different frequencies, the vibrations of strings have varying energies or masses. In principle, they could explain the observed masses of elementary particles, if the calculated numbers agree with the precisely measured values known from accelerator experiments.

By reducing all phenomena of particle physics to a single object, string theory promises to unify all the known fundamental forces—gravity, electromagnetism, and the strong and weak interactions—in a single force formula. There would no longer be different concepts such as space-time as the carrier of gravity and the electromagnetic field as the messenger of the electric force, but just a single object from whose vibrations would emerge all forces and the matter particles on which they act. This object in its elementary form is the eponymous string.

Unifications of different theories and forces have played major roles in the development of physics, for instance when Maxwell combined the initially unconnected electrical and magnetic phenomena in one theory of electromagnetism. From such combined theories spring, almost inescapably, predictions of new phenomena that might be exploited technologically, or that can be used for independent tests