My approach to SUSY is slightly unusual. In my earlier post, I discussed the famous Coleman-Mandula theorem, which states that the only symmetries consistent with both quantum mechanics and special relativity are the symmetries of special relativity, of electromagnetism, and of certain well-defined generalisations of electromagnetism (gauge symmetries). The loop-hole is that Coleman and Mandula assumed that symmetry groups are characterised by the commutator,
while some groups can be described instead by the anti-commutator
In both these expressions, f and g are elements of the (symmetry) group.
Let's consider what that means a little more. The simplest possibility for the characteristic function is zero. In the case of the former type of group, this gives us
This is the usual rule for multiplication. Instead, for the latter type of group we have
Quantities which obey these rules are said to anti-commute. Clearly, these cannot be ordinary numbers; but there a lots of mathematical objects which behave like this.
What is more interesting, however, is that there are physical objects that behave like this. I already discussed this in an earlier post, but the fundamental object in quantum mechanics is the wavefunction. Physical measurements depend only on the square of the wavefunction, so flipping its sign has no physical effect. Exchanging identical particles can therefore either leave the wavefunction unchanged, or flip its sign. Consistency demands that for each type of particle, you either always or never need a sign change. The former particles are called fermions, the latter bosons.
This all suggests that our new type of symmetry relates fermions and bosons, and indeed this is the case. Supersymmetry states that for every fermion in your theory, there must also be a boson, and vice versa. Further, the related particles must have identical properties: mass, electric charge and all other quantum numbers except spin (angular momentum). Spin is the exception because it can also be proven that bosons (fermions) all have spin equal to (half an odd) integer in appropriate units.
The big problem with SUSY is that no such pairs of particles are seen. For example, consider perhaps the best-known particle, the electron. SUSY tells us that there should be a bosonic particle, the selectron, with the same electric charge and mass as the electron, but with different spin. No such particle has been found, ruling this out.
The answer is to assume that SUSY is broken in a specific way. The idea is that the interactions of the theory respect SUSY, but the vacuum (lowest energy state) does not. By way of analogy, consider water. At high temperature, water is a gas (steam) and has no structure. The interactions between water molecules do not pick out any preferred point in space. However, as we cool the water it condenses and freezes, forming ice (a lower energy state). In this state, the water molecules have picked out preferred points in space, simply because they don't have enough energy to be anywhere else. This idea, incidentally, is closely related to the Higgs mechanism in the Standard Model.
So where does this leave us? Broken SUSY means that for every particle we have so far discovered, there must also be another particle with the same electric charge and so on, but much heavier. We also have the Higgs and its partners; in fact, we have more than twice as many Higgs particles than in the Standard Model for technical reasons. This doesn't sound very attractive.
But, SUSY does have a number of advantages. In particular, it solves the hierarchy problem, explains dark matter, and predicts gauge unification. It also is a natural consequence of string theory. All of these things are interesting concepts in their own right, but will have to wait for another post. First, though, and hopefully before the end of the week, I'll discuss current searches for Supersymmetry.
1. If Supersymmetry is true, then I am at a disadvantage compared to those who've been working on it since the nineties (or even eighties). So my reasons are selfish, personal and petty.↩