How do I obtain the Dirac equation from the Euler-Lagrange equation?

Question

Knowing that the free Dirac Lagrangian is :

$$\tag{1} \mathcal{L}= \bar{\psi} (i \gamma^\mu \partial_\mu -m ) \psi$$

and that the Euler-Lagrange equation is:

$$\tag{2} \frac{\partial \mathcal{L}}{\partial \psi}= \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi)}\right)$$

I am trying to obtain the standard form of the Dirac equation $(3)$ without differentiating by $\bar{\psi}$:

$$\tag{3} (i\gamma^\mu \partial_\mu - m) \psi =0$$

I've been told that the relations $(\gamma^0)^2=1$ and $\gamma^{\dagger \mu}= \gamma^0 \gamma^\mu \gamma^0$ might be useful (I don't see how)

My take on it:

After expanding $(1)$: $$\tag{4} \frac{\partial \mathcal{L}}{\partial \psi}=- \bar{\psi}m$$

and

$$\tag{5} \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi)}=\bar{\psi} i \gamma^\mu$$

so

$$\tag{6} -\bar{\psi}m - \partial_\mu (\bar{\psi}i \gamma^\mu)=0$$

this is where I become lost:

I think I can use the first term on $(6)$ and rewrite it as $ -\bar{\psi}m= m \bar{\psi}$, but how do I proceed from here?

I think I must find a way to place on $\bar{\psi}$ on the RHS of both terms in $(6)$, proceed by multiplying everything by $\psi$ twice, once to cancel the $\bar{\psi}$ and twice to terminate in the standard form shown in $(3)$. How do I do this? Is there another way?

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I have seen other questions and links on the website but these don't quite do it as I intend and tend to differentiate by $\bar{\psi}$ which isn't what I want. Some of the links visited are: Going from the Dirac Lagrangian to the adjoint Dirac equation , Derivation of Dirac equation using the Lagrangian density for Dirac field .

Answer 1

You start with $$\bar{\psi}(i\gamma^\mu\overleftarrow{\partial}_\mu+m) = 0$$ if you take the hermitian conjugate of all

$$\begin{align}\left(\bar{\psi}(i\gamma^\mu\overleftarrow{\partial}_\mu+m)\right)^\dagger &= \left(\psi^\dagger\gamma^0(i\gamma^\mu\overleftarrow{\partial}_\mu+m)\right)^\dagger= (i\gamma^\mu\overleftarrow{\partial}_\mu+m)^\dagger\gamma^{0\dagger}\psi\\ &= (-i\gamma^{\mu\dagger}\partial_\mu+m)\gamma^0\psi = (-i\gamma^0\gamma^\mu\gamma^0\partial_\mu+m)\gamma^0\psi\\ &=(-i\gamma^0\gamma^\mu\gamma^0\gamma^0\partial_\mu+m\gamma^0)\psi =(-i\gamma^0\gamma^\mu\partial_\mu+\gamma^0m)\psi\\ &=\gamma^0(-i\gamma^\mu\partial_\mu+m)\psi= 0 \end{align}$$ Now multiply from the left with a $\gamma^0$ and use the property $\gamma^0\gamma^0=1$ $$(-i\gamma^\mu\partial_\mu+m)\psi=0$$ which is exactily $$(i\gamma^\mu\partial_\mu-m)\psi = 0$$

Answer 2

Your question is based on a false premise. You can't get it except by differentiating with respect to $\bar{ψ}$. If you differentiate with respect to $ψ$, you get the adjoint Dirac equation for $\bar{ψ}$.

If you differentiate with respect to $ψ$, to get the adjoint equation for $\bar{ψ}$, and then take the adjoint of the equation - then all you've really done was just differentiate with respect to $\bar{ψ}$ in the first place. So, no matter what you do, it all comes back to that.

You have to treat $ψ$ and $\bar{ψ}$ as independent variables. So, you're getting two equations, not one. You can see that more clearly in my reply here. In fact, as I point out there, it's actually more correct to write the Lagrangian density symmetrically as: $$\mathcal{L} = \bar{ψ} \left(\frac{iħ}{2} \left(γ^μ \overrightarrow{∂_μ} - \overleftarrow{∂_μ} γ^μ\right) - mc\right) ψ,$$ as some authors do. Then, in the variational: $$Δ\mathcal{L} = Δ\bar{ψ} \left(iħ γ^μ \overrightarrow{∂_μ} - mc\right) ψ - \bar{ψ}\left(iħ \overleftarrow{∂_μ} γ^μ + mc\right) Δψ + ∂_μ \left(\frac{iħ}{2} \bar{ψ} \left(γ^μ \overrightarrow{Δ} - \overleftarrow{Δ} γ^μ\right) ψ\right),$$ you get symmetric contributions from $\overrightarrow{Δ}ψ ≡ Δψ$ and $\bar{ψ}\overleftarrow{Δ} ≡ Δ\bar{ψ}$ to the boundary term.