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imprecise position, the picture of a water wave illustrates the long reach of quantum effects: Distant regions, which classically would play no role whatsoever, influence the wave position. An exact description requires knowledge of the whole wave, not only of its maximum. Also, the spread widens in time, and it influences the mean position. Aside from position and spread, many other parameters are necessary for an exact description of the waveform. No finite number of parameters will suffice to describe the complete wave precisely, for this would require knowledge of the height of the wave at all its infinitely many positions. All these parameters are, as a result of their mutual influence on one another, subject to complicated interplay in time. In general, a correct mathematical description can be obtained only by means of the Schrödinger equation and its solutions. Luckily, we are usually interested not in the whole wave but only in a few parameters such as the peak position and the spread. For not too large a spread, for instance in semiclassical states, the set of infinitely many numbers can be reduced to a handful. Behind this reduction lies the mathematical procedure of effective equations.

Such approximation techniques pervade all of physics. They can be found in particle physics, in which complicated interactions of elementary particles can, at moderate energies, be described by effective equations. In high-energy physics, instead of the spread of a wave function, one commonly speaks of the creation of a particle-antiparticle pair—a process based on another consequence of the uncertainty relation: Energy, too, is imprecise and can, for brief periods of time, be used to create a pair consisting of a particle and its antiparticle—on the condition that the pair must annihilate itself shortly to return the borrowed energy. Here, the wave function is being spread out when suddenly two new particles appear.

Just as the propagation and spreading of waves on a lake are influenced by the lakebed, the process of pair creation depends on the ground—here, the form of space-time. Astonishing implications will be seen later in cosmology and in black hole physics. Sometimes pair creation can give rise to drastic phenomena, comparable to the buildup of tsunamis after seaquakes, when rapidly moving ground generates high waves on the surface. When space-time expands, especially in an accelerated manner, it can generate particle waves taking the form of new matter. In contrast to the seabed during a quake, the universe does not vibrate periodically, and consequences are thus not too dramatic. But what is important is the acceleration of the ground. As we will see in chapter 5 on observational cosmology, accelerated expansion, called cosmic inflation, is indeed assumed to have taken place in the very early universe. If this idea is correct, the accelerated expansion active at those times just sufficed to create all the matter now present, like a wave piling up out of a vacuum.

So far, we have looked only at a single wave in its entire extension. But in the universe, as on a lake, there are usually many waves superposed on one another. In such a superposition the waves influence one another’s motion even without direct contact, in contrast to hard classical particles. If waves should grow far apart, they can still retain a memory of the interaction for a long time—a phenomenon called entanglement, following Erwin Schrödinger. In quantum mechanics, this even goes so far that an event happening to one wave—such as breaking on the shore, or a quantum mechanical measurement process—can strongly influence other waves far out in the ocean. For sea waves, such a strong effect would contradict expectations, but in quantum mechanics it is possible due to the so-called nonlinearity of the measurement process: Small changes in a nonlinear system can have disproportionately strong effects throughout the system.

But why, then, are macroscopic objects such as we find in everyday life not entangled in this way? In such cases, after all, direct