Dirac Equation and Notation

Question

I need to ask something regarding the Dirac equation (for a charged particle in an electromagnetic field) with the slash notation, which i fail to understand. We have the Dirac equation with the slash notation:

($\gamma^\mu p_\mu -m$)$\psi$=0

In the presence of an electromagnetic field, we get:

($\gamma^\mu (p_\mu -eA_\mu) -m$)$\psi$=0

I know that we derivative the above expression by multiplying with $\beta$ matrix.

And,correct me if i am wrong $\gamma^\mu$ is a vector matrix, what i mean is that $\gamma^\mu$=($\gamma^0,\vec\gamma)$.

Now, in my script we change the Dirac equation by using $\overline \psi$.

I know that $\overline \psi$= $\psi^\dagger\gamma^0$. Therefore we initially conjugate transpose the equation and then multiply it with $\gamma^0 \gamma^0$:

($\gamma^\mu p_\mu -m$)$\psi$=0.

($\gamma^\mu (p_\mu -eA_\mu) -m$)$\psi$=0

($\gamma^\mu (i\partial_\mu - eA_\mu) -m$)$\psi$=0. I think here there must be a mistake because $p^\mu = i\hbar\partial^\mu$, where the index is up and not down,as shown in the equation. But this is not my main question, thought I'd happily accept an explanation or clarification about this thing.

Now we transpose conjugate:

$\psi^\dagger [(\gamma^\mu)^\dagger (-i\partial_\mu -eA_\mu) - m] $ = 0.

Now in order to transform this into an equation where $\overline \psi$ is present, we initially multiply with $\gamma^0 \gamma^0$ (as I stated above). Then we have:

$\psi^\dagger \gamma^0 \gamma^0 [(\gamma^\mu)^\dagger (-i\partial_\mu -eA_\mu) - m] $ = 0.

$\overline \psi \gamma^0 [(\gamma^\mu)^\dagger (-i\partial_\mu -eA_\mu) - m] $ = 0.

Now these is the part that i don't get at all. As i mentioned above $\gamma^\mu$ is a vector matrix of the form $\gamma^\mu$=($\gamma^0,\vec\gamma)$. Then in the skript is said that the following relation holds true:

$(\gamma^\mu)^\dagger = \gamma^0 \gamma^\mu \gamma^0$

How is this possible?

What we are doing here is this:

$(\gamma^\mu)^\dagger = \gamma^0 (\gamma^0,\vec\gamma) \gamma^0$

How does this even work? We have a matrix multiplying a vector, whose components are matricies? How? Like i don't know the rules on how to solve things like these. Nothing about the necessary algebra was taught to us.

Please tell me if i have a mistake in my equations, or reasoning!

Thanks

Answer 1

The expression $(\gamma^\mu)^\dagger = \gamma^0 \gamma^\mu \gamma^0$ is shorthand for $$(\gamma^0)^\dagger = \gamma^0 \gamma^0 \gamma^0$$ $$(\gamma^1)^\dagger = \gamma^0 \gamma^1 \gamma^0$$ $$(\gamma^2)^\dagger = \gamma^0 \gamma^2 \gamma^0$$ $$(\gamma^3)^\dagger = \gamma^0 \gamma^3 \gamma^0$$

It's sloppy - but standard - to write that $\gamma^\mu = (\gamma^0,\vec \gamma)$. $\gamma^\mu$ is the $\mu^{th}$ gamma matrix, where $\mu=0,1,2,$ or $3$. If anything, we should write something like $\boldsymbol \gamma = (\gamma^0,\vec \gamma)$ to denote that we are collecting the four gamma matrices into a matrix-valued 4-vector.

This is exactly the same as saying that $V^\mu$ is a vector. $\mathbf V=(V^0,V^1,V^2,V^3)$ is a vector, while $V^\mu$ is its $\mu^{th}$ component. It's important not to get mixed up and confuse vectors with their components.

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Explicitly, $\psi^\dagger \gamma^\mu p_\mu$ means $\psi^\dagger(\gamma^0p_0 + \gamma^1 p_1 + \gamma^2 p_2 + \gamma^3 p_3)$. Inserting $\gamma^0\gamma^0 = \mathbb I$ yields $$\psi^\dagger\gamma^0\gamma^0(\gamma^0p_0 + \gamma^1 p_1 + \gamma^2 p_2 + \gamma^3 p_3)$$ $$=\overline{\psi}\gamma^0(\gamma^0p_0 + \gamma^1 p_1 + \gamma^2 p_2 + \gamma^3 p_3)$$

Multiplying by $\gamma^0$ from the right and noting the shorthand I mentioned above, this becomes

$$\overline{\psi}\bigg((\gamma^0)^\dagger p_0 + (\gamma^1)^\dagger p_1 + (\gamma^2)^\dagger p_2 + (\gamma^3)^\dagger p_3\bigg)$$