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## Unification

In this intermezzo, we expand on the idea of unification. At this moment, it is one of the most important motivations for taking string theory seriously, so it seems appropriate to exhibit some the strong and weak points of the methodology of unification.

### Sharper formulation

Before we discuss the idea of unification, we could try to formulate it a little more sharply, without getting into an endlessly refined definition. String theory tries to unify the theory of General Relativity and the Standard Model. But since they contradict each other in certain basic assumptions, how can they be unified ? Certainly, by the unification of two physical theories we do not mean that the two physical theories should be an integral part of the unified theory. The important point is that the experimentally verified predictions of the two theories will be taken up in the unified theory.

Many physical theories are based on general principles that guide their formulation. Once formulated, a successful theory describes accurately some phenomena and predicts a host more. The point is that it often predicts phenomena that have never been verified. There resides the power of applying a general principle, but also the danger. Often a theory carries the principles on which it is based over to physical regimes in which those principles may not be valid.

There is no reason to take over such a general principle in a new unified theory that encompasses the first, partially successful theory. Indeed, the new unified theory may be based on entirely different principles. The only requirement it has to satisfy is that it reduces to the old, successful, theory in the regime in which that old theory was tested. That is all.

### Regime of validity

Crucial in the previous exposition is the idea of a physical regime. To every physical phenomenon we can associate certain measurable parameters that determine in which physical regime the phenomenon takes place. For example, when an apple falls from a tree, the large mass of the apple (100 g), the size of the apple (15 cm diameter), and its speed (5 m/s) place it in the physical regime where classical mechanics is to a very good approximation valid. We can therefore use classical mechanics to predict its further fall to the ground. We would be very successful with our prediction on the basis of classical mechanics.

Furthermore, one can define a physical regime in which a particular physical theory should be valid. And to each physical theory we can associate a physical regime in which it is tested, and perhaps another, larger, regime in which we assume the theory will remain valid. A fundamental physical theory would be one for which we would not need to specify in which physical regime it is valid. It would be valid in all regimes. In that sense, it is not clear that we have a single fundamental theory of physics, in stark contrast to what many people seem to think. We may have a few fundamental principles on which such a fundamental theory is based, but that's about it.

### A limit of special relativity

A simple illustration of the idea of regimes of validity is given by Galilean mechanics as a limit of special relativity. We could describe the motion of everyday objects like bikes or cars using special relativity. In that framework we would notice that car clocks tick slower when the car is moving, that the car becomes shorter for an observer on the sidewalk, etcetera. But the main thing we would notice if we did this in detail is that, for instance, the clock only goes an extremely tiny bit slower than the wristwatch of the observer. And that the accuracy of those clocks would not be enough to measure the distinction. And that therefore, for the measurements we want to perform, we might as well not take into account that special effect of special relativity at all. That is what we do in our daily lives.

In the limit of slow speeds (compared to the speed of light), we can neglect the corrections that special relativity makes to Galilean mechanics. (Galilean mechanics is a fancy name for the description of moving bodies that we are most familiar with in our everyday lives, so I won't bother to explain it in detail.) In other words, Galilean mechanics is a valid physical theory when speeds are small, and we neglect corrections of special relativity. That illustrates the idea of a regime of validity, and, at the same time, the emergence of one physical theory out of another when taking a limit (of, say, small speeds).

### Unifying light and space-time

A first, fairly simple example of the application of the idea of unification is the following. At the beginning of the century, there was on the one hand the successful theory of Maxwell, describing most phenomena in his classical theory of electromagnetism, and on the other hand the Galilean description of the movement of bodies through space. It turned out that taken together they predicted that one should be able to measure variations in the speed of light, but those were not observed. Nevertheless, both theories separately seemed very sound, and especially Galilean mechanics looked intuitively very plausible. It turned out to be false at high velocities.

A theory had to be found that would unify Galilean mechanics and Maxwell's equations. The technical tool that made that possible was Special Relativity. It replaced Galilean mechanics. As we saw, it recuperates the predictions of Galilean mechanics at small velocities. Galilean mechanics is recuperated as a limit of Special Relativity. And Special Relativity is not only compatible with Maxwell's equations, but also turns out to provide a much more natural framework for them (that later made the development of the quantum field theory of electromagnetism possible). Special Relativity and the new formulation of Maxwell's equations successfully unified two good theories of which one was only valid (to an excellent approximation) in a special regime -- namely at small speeds.

### The regime of Quantum Gravity

It may be that the unification problem we face today is similar in spirit than the simple unification problem just described. It may be that the principles of quantum mechanics will remain a fundamental principle, and that the theory of General Relativity will need corrections in a unified theory. That the theory of General Relativity needs correcting, is fairly clear to every expert in the field. Perhaps not everybody would agree with the statement that the principles of quantum mechanics should be seen as fundamental. (There is no convincing alternative organizing principle, so we might as well take it along as a reasonable hypothesis.)

In String theory, General Relativity is indeed corrected at small scales, the principles of quantum mechanics are upheld, but the principles of Quantum Field Theory are also modified to a large extent. Nevertheless, it recuperates the predictions of General Relativity at the scales where it is tested, and reinstates the principles of quantum field theory at energies where quantum field theories are accurate physical theories. That means in particular that General Relativity is at latest recuperated at scales of a tenth of a millimeter, and quantum field theory at energies of a few TeV (i.e. at energy scales that are presently just out of reach of our accelerators).

On general grounds, one can argue that the regime of Quantum gravity will at latest kick in at distances of about 10^{-35} meters or energy scales of about 10^{19} GeV. But at present it seems equally likely that corrections indeed become important at the millimeter scale for pure gravity, or 10 TeV for particle accelerator experiments. We just do not know which scale will be the relevant one.

### Remarks

• We do realize we painted a naive science-philosophical picture of what actually happens in the history of science. Science philosophers, science historians, and other experts make it their business to ponder subtler points than the ones we tried to make.
• Varying speeds of light are obtained when light is shone through different media. But light in vacuum should always have the same speed, and all observations up till now confirm that prediction.