Supersymmetry

[mimefan599]

Can someone with knowledge on what supersymmetry is please enlighten me. Hawkins literature on the subject has left me scratching my head.

So basically: What is supersymmetry? What does it affect?

 

Thanks

[Severian]

Supersymmetry is an extention of space-time to include fermionic dimensions.

 

We have known for a long time that particles come in 2 types: bosons and fermions. For example, an electron is a fermion and a photon is a boson. The distinction is a bit technical, but essential it comes down to what happens to the quantum mechanical wavefunction when you exchange two identical particles. Exchange two identical bosons and nothing much happens, but exchange two fermions and you have to multiply the wavefunction by -1. (At first sight this doesn't change anything since probabilities are the modulus squared of the wavefunction, but it does affect the interference between different contributions.)

 

Now, for a long time it was thought that the Poincarre algebra was the maximal symmetry of space-time (the Coleman-Mandulla theorem). By that I mean that the laws of physics are invariant under Lorentz boosts and translations. However, in the 80s it was realised that this isn't true.

 

Our 4 space-time dimesnions with which we are familiar as bosonic. If I take 2 steps forward and 2 steps to the right I would end up in the same place as if I went 2 steps to the right and two steps forward. The operations commute, and the dimensions are bosonic. But what if this were not the case - what if I could go in some other directions and pick up minus signs by doing things in a different order? What if there were fermionic dimensions?

 

So space-time can be extended by adding in fermionic dimensions, and then a rotation between bosonic and fermionic dimensions would extend the symmetry. The laws of physics would be invariant under this new enlarged symmetry. This new symmetry is called supersymmetry and it is now thought that it is mathematically the largest symmetry group any space-time can have.

 

But surely this is rubbish - wouldn't we see the extra fermionic dimesnions (new route to the pub anyone?). Actually no, because something weird happens when you work out how this would manifest in the real world.

 

All your fields become functions of not only your bosonic coordinate [math]x^\mu[/math] ([math]\mu=0...3[/math]) but also the fermionic coordinates [math]\theta[/math] and [math] \bar \theta[/math] (I won't go into why there are two). Each of these fermionic coordinates is actually a 2-d vector, so for example [math]\theta = \left( \begin{array}{c} \theta_1 \\\theta_2 \end{array} \right) [/math], so altogether there are four fermionic coordinates just like there were four bosonic ones (although one can imagine including more if you like).

 

But since these coordinates are anticommute, [math]\theta_1 \theta_2 = -\theta_2 \theta_1[/math] and more significantly [math]\theta_1 \theta_1 = -\theta_1 \theta_1 = 0[/math]. So whenever we make a Taylor exansion of the 'superfield' in these new fermionic coordinates, the series will be cut off as soon as we have two identical fermionic components in one term.

 

For example [math]f(x,\theta) = a(x)+b(x)\theta_1 +c(x) \theta_2+d©\theta_1 \theta_2[/math] with no extra terms. The objects a(x) and d(x) look like bosons, while b(x) and c(x) look like a fermions. So suppersymmetry predicts that for even bosonic particle there must be a fermionic particle and vice versa. Fermions are really just projections of this superfield onto the fermionic direction, while bosons are projections onto the bosonic (ie usual) directions.

[ajb]

I agree with everything that Severian said. (He talked about superspace formulation of supersymmetry, it is not the only way, but in some sence it is the best).

 

Even more basically, supersymmetry is a non-trivial symmetry that a theory with both bosonic and fermionic degrees of freedom can possess. Wroughly it means that we can interchange the fermions with the bosons and vice versa leaving the theory unchanged.

 

There is a lot of interesting mathematics that has been developed to deal with such theories. It goes under the general heading of supermathematics, I work on an area of this known as supermanifold theory.