The fifth gamma matrix and fermion fields I am aware of the various relations with Dirac spinors and chirality but how does the fifth gamma matrix $\gamma^5$ behave with fermion fields, $\psi$?
Does the fifth gamma matrix have any particular commutation relation with $\psi$ that can be manipulated?
My issue has arisen out of the evaluation of the commutator, $[Q^5_a, Q^5_b]$ where, $$Q_a \equiv \int d^3x\ \psi^{\dagger}_r(x)(T_a)_{rs}\gamma^5\psi_s(x)$$
I have determined the algebra for $[Q_a, Q_b]$ (as above without $\gamma^5$ in) but I am unsure how to proceed with the inclusion of $\gamma^5$.
 A: In spinor space, the fermionic fields $\psi$ are vectors, for example in the 4x4 representation of the $\gamma$-matrices:
$\psi^\dagger = \begin{pmatrix}
         \psi_1^* &  \psi_2^* & \psi_3^* & \psi_4^*
        \end{pmatrix}^{T}$
and
$\psi=\begin{pmatrix}
         \psi_1 \\
         \psi_2 \\
         \psi_3\\
         \psi_4
        \end{pmatrix}$
where $\psi_1$ to $\psi_4$ denote the spinor components. So the expression you gave above is of the form: row vector times matrix times column vector. Which overall is just a number. This also makes clear why the matrices can possibly commute with the field, the matrix multiplication would not make sense anymore.
To proceed in your question (The following are advices since I do not want to solve the problem for you and rather general. If you'd like more specific advice I'm happy to help). Usually one has scattering between particles of certain momentum. Therefore one would first want to use a Fourier transform (plane wave expansion) of the field to transform to momentum space.
After that a useful trick is to note that:
$\psi^\dagger \Gamma \phi = trace\left( \Gamma \phi \psi^\dagger \right)$
for any two spinors $\psi$, $\phi$ where $\Gamma$ is any combination of Dirac matrices. You can convince yourself of this identity by writing it out in components as I did above.
Then one can use completeness identities in Fourier space to get the required results in terms of momentum etc. Also note that usually the matrix element squared is computed, which is your expression squared (or something with similar structure). This simplifies calculations if the matrices are arranged in a convenient way (note that you can transpose Dirac numbers since they have no Dirac structure.).
