Understanding Quantum Field Theory

https://www.youtube.com/watch?v=Bd06l3UOuec
Understanding 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 rigor, 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!

The results described above constitute quite an achievement for one century, but it leaves us with one fundamental contradiction that still needs to be resolved. General relativity and quantum field theory are incompatible. Many theorical physicists are convinced that superstring theory will provide the answer. There have been major advances in our understanding of this subject, which I consider to constitute the "second superstring revolution,"during the past few years.
After presenting some more background, I will describe the recent developments and their implications.
There are various problems that arise when one attempts to combine general relativity and quantum field theory. The field theorist would point to the breakdown of the usual procedure for eliminating infinities from calculations of physical quantities. This procedure is called "renormalization", and when it fails the theory is said to be "non-renormalizable.".
In such theories the short-distance behavior of interactions is so singular that it is not possible to carry out meaningful calculations. By replacing point-like particles with one-dimensional extended strings, as the fundamental objects, superstring theory overcomes the problem of non-renormalizability.
An expert in general relativity might point to a different set of problems such as the issue of how to understand the causal structure of space-time when the geometry has quantum-mechanical excitations. There are also a host of problems associated to black holes such as the fundamental origin of their thermodynamic properties and their apparent incompatibility with quantum mechanics. The latter, if true, would mean that a modification in the basic structure of quantum mechanics is required.
In fact, superstring theory does not modify quantum mechanics; rather, it modifies general relativity. The relativist's set of issues can not be addressed properly in the usual approach to quantum field theory (perturbation theory), but the recent discoveries are leading to non-perturbative understandings that should help in addressing them.
Most string theorists expect that the theory will provide satisfying resolutions of these problems without any revision in the basic structure of quantum mechanics. Indeed, there are indications that someday quantum mechanics will be viewed as an implication (or at least a necessary ingredient) of superstring theory.
When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made.
In my opinion, success in such enterprises requires a better understanding of the theory than has been achieved as yet. It is very difficult to assess whether this level of understanding is just around the corner or whether it will take many decades and several more revolutions.
In the absence of this kind of confirmation, we can point to three qualitative "predictions" of superstring theory. The first is the existence of gravitation, approximated at low energies by general relativity. No other quantum theory can claim to have this property (and I suspect that no other ever will).
The second is the fact that superstring solutions generally include Yang-- Mills gauge theories like those that make up the "standard model" of elementary particles.
The third general prediction is the existence of supersymmetry at low energies (the electroweak scale). Since supersymmetry is the major qualitiative prediction of superstring theory not already known to be true before the prediction, let us look at it a little more closely. (One could imagine that in some other civilization, the sequence of discoveries is different.).

Supersymmetry is a theoretically attractive possibility for several reasons. Most important from my viewpoint, is the fact that it is required by superstring theory. Beyond that is the remarkable fact that it is the unique possibility for a non-trivial extension of the known symmetries of space and time (which are described in special relativity by the Poincare group).
Mathematically, it can be described in terms of extra dimensions that are rather peculiar. Whereas ordinary space and time dimensions are described by ordinary numbers, which have the property that they commute: X · Y= Y · X, the supersymmetry directions are described by numbers that anti-commute: X · Y = - Y · X.