Vector and Axial vector currents

[arestes]

Homework Statement

I have to show that the vector current $$\vec{V}^\mu = \overline{\psi} \gamma^\mu \vec{T} \psi$$ and the vector axial current $$\vec{A}^\mu = \overline{\psi} \gamma^\mu \gamma_5\vec{T} \psi$$ satisfy this

$$\partial_\mu \vec{V}^\mu = i\overline{\psi}[M,\vec{T}] \psi$$$$\partial_\mu \vec{A}^\mu = i\overline{\psi}\{M,\vec{T}\}\gamma_5 \psi$$

The brackets are commutators and the braces are anticommutators

Homework Equations

Dirac's equation for $$\psi$$ and $$\overline{\psi}$$. Also, M is a mass matrix because $$\psi$$ is extended to a vector in flavor space and M is a diagonal matrix (entries are the quark masses). T are the pauli matrices divided by 2 and we are working only with two flavors. The column vector $$\psi$$ contains two entries then, and each of these are a 4-component Dirac spinor.

The Attempt at a Solution

Using Dirac's equations:
$$\overline{\psi}(i\gamma^\mu\stackrel{\leftarrow}{\partial_\mu} + M) &=& 0$$
$$\partial_\mu\overline{\psi} \gamma^\mu &=& i\overline{\psi}M$$

$$(i\gamma^\mu{\partial_\mu} + M)\psi &=& 0$$
$$\gamma^\mu\partial_\mu{\psi} &=& -iM{\psi}$$
when we replace it in the gradient $$\partial_\mu \vec{V}^\mu$$

$$\partial_\mu \vec{V}^\mu = \partial_\mu\overline{\psi} \gamma^\mu \vec{T} \psi + \overline{\psi}\gamma^\mu\vec{T}\partial_\mu \psi$$there's no problem substituting $$\partial_\mu\overline{\psi} \gamma^\mu = i\overline{\psi}M$$ in the first term of this last gradient but for the second term i don't get $$\gamma^\mu\partial_\mu{\psi}$$ but rather $$\gamma^\mu\vec{T}\partial_\mu \psi$$ which differ just by commuting $$\vec{T}$$ and $$\gamma^\mu$$

Same applies to $$\partial_\mu\vec{A}^\mu$$

 

[Dick]

T is a matrix in flavor space. The gammas are matrices in spinor space. I.e. their indices are of different types. You can commute them.