# Commutation of alpha dirac matrix

I want to calculate the commutation of $$[\hat{x},\vec{\alpha}\;\vec{p}]$$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\left[\hat{x},\hat{\alpha_y}\right]\hat{p_y} +\left[\hat{x},\hat{\alpha_z}\right]\hat{p_z}$$

My problem is calculating $$[\hat{x},\hat{\alpha_x}]$$, $$[\hat{x},\hat{\alpha_y}]$$ and $$[\hat{x},\hat{\alpha_z}]$$. Are they just zero? If so why?

Also is there a way I can represent $$\hat{x}$$ explicitly so that I can multiply it with $$\hat{\alpha}$$?

$$\hat{\pmb{\alpha}} = \begin{bmatrix} 0 & \pmb{\sigma} \\ \pmb{\sigma} & 0 \\ \end{bmatrix}$$

Where $$\pmb{\sigma}$$ the Pauli matrices.

Edit(Answer): Since the matrices are arrays of numbers they do commute with $$\hat{x}$$ hence $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}$$

• Dirac matrices, Pauli matrices, etc... are arrays of numbers, not operators, so they all commute with Hilbert space operators. – Cosmas Zachos May 23 at 14:22
• Oh, so that would actually make them zero. Yeah that makes sense. Just a follow up question: If we consider $\frac{\partial{<\hat{x}>}}{\partial{t}} = -\frac{ic}{\hbar}<[\bar{x},\vec{\alpha}\;\vec{p}]>$ what physical interpretation does $\alpha$ take? – fielder May 23 at 14:39
• possible duplicates: physics.stackexchange.com/q/476458/84967, physics.stackexchange.com/q/104241/84967 – AccidentalFourierTransform May 23 at 15:00
• Velocity? Is this part of your homework? – Cosmas Zachos May 23 at 15:08

Since the matrices are arrays of numbers they do commute with $$\hat{x}$$. Furthermore we have $$\frac{\partial{<\hat{x}>}}{\partial{t}} = -\frac{ic}{\hbar}<\left[\hat{x},\vec{\alpha}\;\vec{p}\right]>$$
Now $$\left[ \hat{x},\vec{\alpha}\;\vec{p}\right] = \vec{\alpha}[\hat{x},\vec{p}] + [\hat{x},\vec{\alpha}]\vec{p} = i\hbar\hat{\alpha_x}$$ hence $$\frac{\partial{<\hat{x}>}}{\partial{t}} = -\frac{ic}{\hbar} = c\hat{\alpha_x}$$ This means we can classically interpret $$\pmb{\vec{\alpha}}$$ as the velocity.