II.3.1 Super Symmetry and Super Space

Super symmetry is one of the preferred methods for extending the Standard Model by adding another class of symmetries that exchange fermionic particles with bosonic ones. This predicts the existence of super symmetric particles, abbreviated as sparticles, which include the sleptons, squarks, neutralinos and charginos. Each particle in the Standard Model would have a super partner whose spin differs by  from the ordinary particle. If the super symmetry is perfectly unbroken, each pair of super partners would share the same mass and internal quantum numbers besides spin. However, if that was the case, we would expect to find these super partners using present-day equipment, but the fact is that all efforts in trying to detect them have been disappointing so far. Therefore, it is currently believed that if super symmetry exists then it consists of a spontaneously broken symmetry allowing super partners to differ in mass. Due to this breaking of super symmetry, the sparticles could be much heavier than their ordinary counterparts; they are so heavy that existing particle colliders may not be powerful enough to produce them. The reason why super symmetry is still supported by physicists is that the current theories are known to be incomplete and their limitations are well established. The first runs of the LHC found no previously-unknown particles other than the Higgs boson.

Super symmetry, often abbreviated as SUSY, provides an elegant solution to most of the major problems mentioned above. For example, the hierarchy problem in gauge theory will cease because the quadratic divergences of all orders will cancel out in perturbation theory. In the Standard Model, the electroweak scale receives enormous Planck-scale quantum corrections. The observed hierarchy between the electroweak scale and the Planck scale must be achieved with extraordinary fine tuning. In a super symmetric theory, Planck-scale quantum corrections cancel between partners and super partners. Additionally, in the current model, the weak, strong and electromagnetic couplings fail to unify at high energy, which is preventing the expected Grand Unification between these fundamental interactions, whereas super symmetry introduces some modifications on the running of the gauge couplings that lead to precise high-energy unification of the gauge couplings. The modified running also provides a natural mechanism for radiative electroweak symmetry breaking. The first realistic super symmetric version of the Standard Model, which is known as the Minimal super symmetric Standard Model, was proposed in 1977 by Pierre Fayet.

Super symmetric Quantum Field Theory is often much easier to analyze, as many more problems become mathematically tractable. When super symmetry is imposed as a local symmetry, Einstein s theory of General Relativity is included automatically, and the result is said to be a theory of super gravity.

One reason that physicists explored super symmetry is because it offers an extension to the more familiar symmetries of Quantum Field Theory. These symmetries are grouped into the Poincar group, since internal symmetries and the Coleman Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincar group with a compact internal symmetry group, or if there is no mass gap, the conformal group with a compact internal symmetry group. In 1971 Golfand and Likhtman were the first to show that the Poincar algebra can be extended through introduction of four anti-commuting spinor generators (in four dimensions), which later became known as supercharges. in 1975 the Haag-Lopuszanski-Sohnius theorem analyzed all possible super-algebras in the general form, including those with an extended number of the super-generators and central charges. This extended super Poincar algebra paved the way for obtaining a very large and important class of super symmetric field theories.

Traditional symmetries of physics are generated by objects that transform by the tensor representations of the Poincar group and internal symmetries. super symmetries, however, are generated by objects that transform by the spin representations. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anti-commute. Combining the two kinds of fields into a single algebra requires the introduction of a -grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a Lie super-algebra.

Therefore, it is possible to have more than one kind of super symmetry transformation. Theories with more than one super symmetry transformation are known as extended super symmetric theories. The more super symmetry a theory has, the more constrained are the field content and interactions. Typically the number of copies of a super symmetry is a power of 2, i.e. 1, 2, 4, 8. In four dimensions, a spinor has four degrees of freedom and thus the minimal number of super symmetry generators is four, in four dimensions, and having eight copies of super symmetry means that there are 32 super symmetry generators, which is the maximal number of super symmetry generators. This corresponds to an  super symmetry theory. Theories with 32 super symmetries automatically have a graviton, or massless fields with spin greater than 2. It is also possible to have super symmetry in dimensions other than four.