- #1

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- Homework Statement
- Given the Dirac-Hamiltonian

$$\hat{ \vec H} = -i \gamma^0 \gamma^j \partial_{j} + m \gamma^0$$

And the Angular momentum operator

$$\hat{ \vec L}_T = -i \vec r \times \vec \nabla + \frac{1}{2} \vec \sigma$$

Show that

$$[\hat{ \vec H}, \hat{ \vec L}_T]=0$$

- Relevant Equations
- $$\vec \sigma := (\sigma^{23}, \sigma^{31}, \sigma^{12}) := (\sigma^1, \sigma^2, \sigma^3)$$

$$\sigma^{\mu \nu} = \frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}]$$

$$\{\gamma^{\mu}, \gamma^{\nu} \} = 2 \eta^{\mu \nu}$$

We want to show that ##[\hat{ \vec H}, \hat{ \vec L}_T]=0##. I made a guess: we know that ##[\hat{ \vec H}, \hat{ \vec L}_T]=[\hat{ \vec H}, \hat{ \vec L}] + \frac 1 2 [\hat{ \vec H}, \vec \sigma]=0## must hold.

I have already shown that

$$[\hat{ \vec H}, -i \vec r \times \vec \nabla]= - \gamma^{0} \sum_{i=1}^3 \varepsilon_{ijk}\gamma^{j}\partial^{k}$$ Hence we expect to get

\begin{equation*}

[\hat{ \vec H}, \vec \sigma] = 2 \gamma^{0} \sum_{i=1}^3 \varepsilon_{ijk}\gamma^{j}\partial^{k}

\end{equation*}

Let's look at ##[\hat{ \vec H}, \vec \sigma]## carefully. We first note that ##[\hat{ \vec H}, \vec \sigma] = [\hat{ \vec H}, \sigma^1] + [\hat{ \vec H}, \sigma^2] + [\hat{ \vec H}, \sigma^3]##. Let's evaluate ##[\hat{ \vec H}, \sigma^1]## explicitly

$$[\hat{ \vec H}, \sigma^1] = -i[ \gamma^0 \gamma^j \partial_{j} + m \gamma^0, i\gamma^2 \gamma^3] $$

$$= [ \gamma^0 \gamma^j \partial_{j}, \gamma^2\gamma^3] $$

$$= [ \gamma^0 \gamma^1 \partial_{1}, \gamma^2\gamma^3] + [ \gamma^0 \gamma^2 \partial_{2}, \gamma^2\gamma^3] + [ \gamma^0 \gamma^3 \partial_{3}, \gamma^2\gamma^3] $$

$$= \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^2\gamma^3\gamma^0 \gamma^1\partial_{1} + \gamma^0 \gamma^2 \gamma^2\gamma^3\partial_{2} - \gamma^2\gamma^3\gamma^0 \gamma^2\partial_{2} $$

$$+ \gamma^0 \gamma^3 \gamma^2\gamma^3\partial_{3} - \gamma^2\gamma^3\gamma^0 \gamma^3\partial_{3} $$

$$= \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^0 \gamma^3\partial_{2}- \gamma^0 \gamma^3\partial_{2} $$

$$+ \gamma^0 \gamma^2\partial_{3} + \gamma^0 \gamma^2\partial_{3} $$

$$= 2\gamma^0 (\gamma^2 \partial_3-\gamma^3 \partial_2) $$

$$= -2\gamma^0 (\gamma_2 \partial_3-\gamma_3 \partial_2)$$

Where I've used ##\sigma^1 := \sigma^{23}, \ \sigma^{\mu \nu} = \frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}], \ \{\gamma^{\mu}, \gamma^{\nu} \}=2 \eta^{\mu \nu}## and ##\gamma_{\mu} = \eta_{\mu \nu} \gamma^{\nu}##, with the convention for the Minkowski metric ##(+,-,-,-)##

The funny thing is that I do not match the answer due to a sign (i.e. the right answer is ##[\hat{ \vec H}, \sigma^1]=2\gamma^0 (\gamma_2 \partial_3-\gamma_3 \partial_2)##).

Could you please help me out spotting my mistake?

Thank you!

I have already shown that

$$[\hat{ \vec H}, -i \vec r \times \vec \nabla]= - \gamma^{0} \sum_{i=1}^3 \varepsilon_{ijk}\gamma^{j}\partial^{k}$$ Hence we expect to get

\begin{equation*}

[\hat{ \vec H}, \vec \sigma] = 2 \gamma^{0} \sum_{i=1}^3 \varepsilon_{ijk}\gamma^{j}\partial^{k}

\end{equation*}

Let's look at ##[\hat{ \vec H}, \vec \sigma]## carefully. We first note that ##[\hat{ \vec H}, \vec \sigma] = [\hat{ \vec H}, \sigma^1] + [\hat{ \vec H}, \sigma^2] + [\hat{ \vec H}, \sigma^3]##. Let's evaluate ##[\hat{ \vec H}, \sigma^1]## explicitly

$$[\hat{ \vec H}, \sigma^1] = -i[ \gamma^0 \gamma^j \partial_{j} + m \gamma^0, i\gamma^2 \gamma^3] $$

$$= [ \gamma^0 \gamma^j \partial_{j}, \gamma^2\gamma^3] $$

$$= [ \gamma^0 \gamma^1 \partial_{1}, \gamma^2\gamma^3] + [ \gamma^0 \gamma^2 \partial_{2}, \gamma^2\gamma^3] + [ \gamma^0 \gamma^3 \partial_{3}, \gamma^2\gamma^3] $$

$$= \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^2\gamma^3\gamma^0 \gamma^1\partial_{1} + \gamma^0 \gamma^2 \gamma^2\gamma^3\partial_{2} - \gamma^2\gamma^3\gamma^0 \gamma^2\partial_{2} $$

$$+ \gamma^0 \gamma^3 \gamma^2\gamma^3\partial_{3} - \gamma^2\gamma^3\gamma^0 \gamma^3\partial_{3} $$

$$= \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^0 \gamma^1 \gamma^2\gamma^3\partial_{1} - \gamma^0 \gamma^3\partial_{2}- \gamma^0 \gamma^3\partial_{2} $$

$$+ \gamma^0 \gamma^2\partial_{3} + \gamma^0 \gamma^2\partial_{3} $$

$$= 2\gamma^0 (\gamma^2 \partial_3-\gamma^3 \partial_2) $$

$$= -2\gamma^0 (\gamma_2 \partial_3-\gamma_3 \partial_2)$$

Where I've used ##\sigma^1 := \sigma^{23}, \ \sigma^{\mu \nu} = \frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}], \ \{\gamma^{\mu}, \gamma^{\nu} \}=2 \eta^{\mu \nu}## and ##\gamma_{\mu} = \eta_{\mu \nu} \gamma^{\nu}##, with the convention for the Minkowski metric ##(+,-,-,-)##

The funny thing is that I do not match the answer due to a sign (i.e. the right answer is ##[\hat{ \vec H}, \sigma^1]=2\gamma^0 (\gamma_2 \partial_3-\gamma_3 \partial_2)##).

Could you please help me out spotting my mistake?

Thank you!