# Quantum Field Theory

The next contradiction that physicists faced was between quantum mechanics (which had been developed over the thirty years following Planck's seminal insight) and the special theory of relativity. Most of the work in quantum mechanics was in the Galilean (or non-relativistic) approximation.
To be sure, Dirac had developed a relativistic wave equation for the electron, which was an important advance, but there was still a basic contradiction that needed to be resolved. The new feature that is required in a successful union of quantum mechanics and special relativity is the possibility of the creation and annihilation of quanta (or `particles'). The non-relativistic theory does not have this feature.
The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. It is based on three basic principles: two of them, of course, are quantum mechanics and special relativity. The third one, which I wish to emphasize, is the postulate that elementary particles are point-like objects of zero intrinsic size. In practice, they are smeared over a region of space due to quantum effects, but their descripton in the basic equations is as mathematical points.
Now the general principles on which quantum field theory are based actually allow for many different consistent theories to be constructed. (The consistency has not been established with mathematical rigour, but this is not a concern for most physicists.)
Among these various possible theories there is a class of theories, called `gauge theories' or `Yang-Mills theories' that turn out to be especially interesting and important. These are characterized by a symmetry structure (called a Lie group) and the assignment of various matter particles to particular symmetry patterns (called group representations). There is an infinite set of possibilities for the choice of the symmetry group, and for each group there are many possible choices of group representations for the matter particles.
One of this infinite array of theories has been experimentally singled out. It is called the ``standard model''. It is based on a Lie group called SU(3) X SU(2) X U(1). The matter particles consist of three families of quarks and leptons. (I will not describe the representations that they are assigned to here.) There are also addition matter particles called ``Higgs particles'', which are required to account for the fact that part of the symmetry is spontaneously broken.
The standard model contains some 20 adjustable parameters, whose values are determined experimentally. Still, there are many more things that can be measured than that, and the standard model is amazingly successful in accounting for a wide range of experiments to very high precision. Indeed, at the time this is written, there is only one clear-cut piece of experimental evidence that the standard model is not an exactly correct theory. This evidence is the fact that the standard model does not contain gravity!

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