# What is quantum field theory

## Quantum field theory - what is it?

Quantum field theory is a synthesis of special relativity and quantum theory - the two great theories that emerged at the beginning of the 20th century.

The special theory of relativity describes space and time, while the quantum theory describes the behavior of subatomic particles. When physicists tried to unite the two theories, solutions with negative energies inevitably appeared. It took them fifty years to interpret all of the consequences consistently and to resolve all apparent contradictions and paradoxes.

Even without going into the mathematical details, one can understand why the problem of negative energies arises. Probably the best-known equation in physics is Einstein's relationship between the energy and mass of a particle: energy is equal to mass times the square of the speed of light. But this simple relationship only applies as long as the particle is at rest. If the particle moves, further terms are added. The problem here is that the energy now appears as a square. Because the square of a negative number is also positive and for this reason the equation always has two solutions - one with positive energy and one with negative energy. In the special theory of relativity one can simply demand that only the positive solutions are valid. In quantum mechanics, however, this does not work: Here both solutions are equal and every attempt to ban one variety leads to insoluble contradictions.

### Particles and Antiparticles

However, the existence of physical particles with negative energies would lead to nonsensical consequences. For example, by emitting energy, for example in the form of electromagnetic radiation, a particle could achieve any desired low energy values ​​(in other words: arbitrarily large negative energy values). In principle, an unlimited amount of energy can be radiated in this way. However, since we do not observe such a thing in nature, this naive interpretation of the negative solutions cannot be correct. The key to correct interpretation is provided by the observation that a particle with negative energy can be interpreted as an antiparticle with positive energy. Such antiparticles are, so to speak, mirror images of the “normal” particles: all their physical properties such as charge, sense of rotation (“spin”) and the more abstract quantum numbers are exactly opposite, only the mass and thus the energy are identical. In the case of a few particles - such as photons - the properties of particles and antiparticles are completely identical. Here it is said that these particles are their own antiparticles.

From the fact that one always finds one with negative energy for every solution with positive energy, the following statement simply follows: for every particle there is an antiparticle. Indeed, physicists have succeeded in finding a corresponding antiparticle for every existing particle. The conservation laws for charge and quantum numbers mean that in all experiments only particle-antiparticle pairs can arise and disappear.

The symmetry between particles and antiparticles is not only an experimental fact, but also extremely aesthetically pleasing. However, this symmetry in combination with Heisenberg's uncertainty principle leads immediately to the next fundamental problem. The uncertainty relation allows, among other things, a violation of the law of conservation of energy, provided that the duration of this violation is only sufficiently short. But it also follows from this that a particle and an antiparticle can arise out of nowhere at any time, provided that both destroy each other again within the time given by the uncertainty principle.

### Virtual particles

We can illustrate the creation of virtual particle-antiparticle pairs with the following comparison. Think of a nightly fireworks display with rockets of different sizes, with each rocket that explodes, the shorter the light the brighter it is. All of these exploding rockets jointly illuminate the night sky, with each rocket size contributing equally to the average brightness. If there were no upper limit for the size of the rockets, the night sky would become infinitely bright. Mathematically, one would then speak of a divergence of the surface brightness.

If there is no upper limit for the energy of the virtual particle-antiparticle pairs, one finds within the framework of quantum field theory, analogous to this example, that in almost all calculations for observable quantities only the meaningless result "infinite" comes out. In the case of previously unexplored, very high energies, there must therefore be unknown physical effects that prevent these divergences. Now, in principle, we could speculate arbitrarily about what the laws of physics look like in areas of energy about which we have neither direct nor indirect information. But obviously a theory would be pointless, in which every prediction of effects in the observable energy range depends on the infinitely many, non-verifiable additional assumptions that can be made. The only solution to obtain a self-contained and meaningful theory is therefore the extremely restrictive requirement: Only those theories are permissible in which the physics at low energies is independent of the way in which the contributions of arbitrarily high energies are suppressed .

In the vivid picture of our fireworks, this would mean that the frequency of extremely bright rockets is suppressed in some way, but that we cannot possibly find out how - for example because we have no measuring instruments to determine their extremely high luminosity and their extremely short burn time. So we find that the night sky only has a finite average brightness, but we cannot say exactly how it comes about.

The requirement formulated above can only be fulfilled because the de facto task of physics is to explain the relationships between the results of different experiments. In order to determine the elementary charge, for example, one can analyze how an electron of low energy is deflected in a magnetic field. On the other hand, we can measure the probability with which certain particles are generated if an electron and a positron annihilate at very high energies. (That was the job of the LEP accelerator at the CERN research center.) The electrical charge of the electron can now be determined from both processes. At school you learn that this elementary charge is a natural constant.

### Gauge theories

In the context of quantum field theory, however, its effective value depends on the quantum fluctuations described above. The value of the elementary charge therefore depends on the measurement process used. It is enough if we are able to calculate this difference. The three researchers David Gross, David Politzer and Frank Wilczek received the Nobel Prize in 2004 for successfully predicting how the value of the so-called “strong coupling constant” depends on the measurement process. In very few of the infinitely many conceivable theories, the differences between the effective values ​​of the elementary charge in all conceivable experiments are finite. Only such theories provide a mathematically consistent description of the world. The experiments CMS and ATLAS

In fact, it has been shown that only the so-called gauge theories can meet this requirement. The simplest form of these gauge theories, which can describe all known empirical facts, is the so-called standard model. For many years now - so far without success - all major accelerator centers around the world have been looking for a case in which the standard model clearly fails. Because such an experiment would allow a first look at the as yet unknown physics beyond the Standard Model. At the moment hope rests primarily on the Large Hadron Collider at CERN, which is due to start operations in 2007.

Finally, the most well-known example of the strength of the mathematical consistency requirements in quantum field theory should be mentioned: In the standard model, all divergences only disappear when the fundamental particles of matter, the so-called fermions, form complete “families”, each made up of a lepton (such as the Electron), a neutrino (such as the electron neutrino) and two quarks (such as the up and down quarks), whereby the latter must appear in three variants (“colors”). In fact, all of the fermions found so far form exactly three such complete families, and there is strong evidence that there are no other fermions.

The fact that quantum field theory allows to derive such concrete requirements from purely mathematical consistency conditions, which are then actually realized in the world, is what makes this theory so fascinating. Driven to extremes, one can hope that there is ultimately only one single fundamental theory, which should then also contain a quantum theory of gravity. This “Theory of Everything” would have to go beyond pure quantum field theory, but would contain the standard model as a limiting case for low energies.