String Theory & Fermions: Exploring Differences & Why They Matter

[moriheru]

This is a stupid question (good that I ask). String theory treats bosons and fermions in different ways e.g fermion potential are differ from boson potentials,actions differ and so on...
My question is simply : why?
Yes I know fermions and bosons are different groups of particles (integer and non-integer spin) and have many differences, but how are these incoperated in the action or potential.
Specifically why does one need anticomuting dynamical variables for fermions?
Thanks for any clarifications.

 

[fzero]

Using anticommuting objects to represent fermions is a characteristic of quantum field theory rather than string theory. As you say, bosons and fermions have certain different properties and the important one here is the observation that no two identical fermions are ever seen in the same quantum state. For example, in the ground state of helium, the electrons always have opposite spin, If we found a helium atom with both electrons with the same spin, it would always be in an excited state, where the electrons were in different orbitals.

In quantum mechanics, we account for this by using an antisymmetric wavefunction for the state describing two identical fermions. Say we are in some position basis and we label the fermions as 1,2, then we can write:

$$\Psi(x_1,x_2;1,2) = \psi(x_1;1)\psi(x_2;2) - \psi(x_2;1)\psi(x_1;2).$$

If we consider an operation where we now place fermion 2 at the point $x_1$ and fermion 1 at the point $x_2$, our new wavefunction is

$$\Psi(x_2,x_1;1,2) = - \Psi(x_1,x_2;1,2).$$

This is also called the exchange symmetry for fermions. In this case, we find that it is impossible to find the two fermions at the same point in space: $\Psi(x,x;1,2)=0$ because of the antisymmetry of the wavefunction.

In quantum field theory, quantum states are replaced by quantum fields $\hat{\psi}$. These are operators that act on the vacuum state to produce one particle states:

$$\hat{\psi}_1(x_1) |0\rangle = |x_1,1\rangle.$$

If we demand that the quantum fields describing fermions anticommute, $\hat{\psi}_1(x_1)\hat{\psi}_2(x_2)=-\hat{\psi}_2(x_2)\hat{\psi}_1(x_1)$, then we guarantee that the multiparticle states that the fields create will satisfy the correct exchange symmetry for fermions. We no longer have to force it by hand since it is a fundamental property of the theory.

When you see anticommuting variables being used for fermions in the description of the string worldsheet, that is because the tools of two-dimensional quantum field theory are used. The treatment of fermions at that level is not something unique to string theory.