A resent discussion about complex conjugates in some other threads reminded me of one strange thing with the Dirac's field. I'll first show why the Lagrangian density
<br />
\mathcal{L}(\psi) = \overline{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi<br />
with the action principle implies the Dirac equation. Let \xi(x) be some arbitrary variation, that vanishes at the end points of some time interval [t_1,t_2]. We then demand
<br />
0 = D_{\alpha} \int d^4x\; \mathcal{L}(\psi + \alpha\xi)\Big|_{\alpha=0}<br />
\;=\; D_{\alpha} \int d^4x\;\Big((\overline{\psi} \;+\; \alpha\overline{\xi})\big(i\gamma^{\mu}\partial_{\mu} \;-\; m\big)(\psi \;+\; \alpha\xi)\Big)\Big|_{\alpha=0}<br />
<br />
= \int d^4x\;\Big( \overline{\xi}(i\gamma^{\mu}\partial_{\mu} \;-\; m)\psi \;+\; \overline{\psi}(i\gamma^{\mu}\partial_{\mu} \;-\; m)\xi\Big)<br />
= \int d^4x\;\Big(i\gamma^{\mu}(\overline{\xi}\partial_{\mu}\psi \;+\; \overline{\psi}\partial_{\mu}\xi) \;-\; 2m\textrm{Re}(\overline{\xi}\psi)\Big)<br />
When we perform integration by parts to move the second derivative like this \overline{\psi}\partial_{\mu}\xi\to -(\partial_{\mu}\overline{\psi})\xi, we obtain
<br />
= 2\int d^4x\;\textrm{Re}\Big(\overline{\xi}\big(i\gamma^{\mu}\partial_{\mu}\psi - m\psi\big)\Big)<br />
and we get the Dirac's equation
<br />
(i\gamma^{\mu}\partial_{\mu} - m)\psi = 0.<br />
However, this is not the way physicists derive the Dirac's equation. Physicists way goes like this. We first derive the Euler-Lagrange equation
<br />
\partial_{\mu}\frac{\partial\mathcal{L}(\phi)}{\partial(\partial_{\mu}\phi)} - \frac{\partial\mathcal{L}(\phi)}{\partial\phi} = 0<br />
for real valued fields, and then assume, that we can use it for complex fields, by assuming that the complex conjugate \phi^* is constant with respect to \phi. This assumption enables one to use derivation formulas like D|z|^2 = D (z^* z) = z^* which seem to be so ridiculously wrong, that one might think it is impossible to ever get correct results with them. However, this turns out to be a false assumption. See how IT WORKS!
We have
<br />
\frac{\partial\mathcal{L}}{\partial\psi} = -\overline{\psi}m,<br />
\quad\quad\quad<br />
\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\psi)} = \overline{\psi}i\gamma^{\mu}<br />
so the Euler-Lagrange equation is
<br />
\partial_{\mu}\overline{\psi}i\gamma^{\mu} + m\overline{\psi} = 0<br />
By taking the complex conjugate of this, we get
<br />
-i(\gamma^{\mu})^{\dagger}\gamma^0 \partial_{\mu}\psi + m\gamma^0\psi = 0<br />
Here the knowledge (\gamma^0)^{\dagger} = \gamma^0 was used. By multiplying with -\gamma^0 from left, and using identities \gamma^0 \gamma^0 = 1 and \gamma^0(\gamma^{\mu})^{\dagger}\gamma^0 = \gamma^{\mu} we get
<br />
(i\gamma^{\mu}\partial_{\mu} - m)\psi = 0<br />
which is the Dirac's equation again. Okey, so it works...
Why do physicists want to do it this way? Often the answer is that they want to do things the easier way. I'm not fully convinced the physicists way is easier in this case, it's a matter of opinion. For example you need to know some properties of gamma matrices, which you don't need to know when you do the variations manually.
It is somewhat puzzling how the assumption that complex conjugates can be assumed to be constants works like this. According to my intuition it shouldn't work for anything, because it is plain wrong. But surprisingly, I find it comforting that actually this assumption doesn't usually work! Suppose we used the Lagrangian
<br />
\mathcal{L} \;=\; \frac{i}{2}\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi \;-\; \frac{i}{2}(\partial_{\mu}\overline{\psi})\gamma^{\mu}\psi \;-\; m\overline{\psi}\psi\quad \Big(= \textrm{Re}\big(i\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi \;-\; m\overline{\psi}\psi\big)\Big)<br />
The action principle gives precisely the same equation of motion from this Lagrangian, but the assumption of \overline{\psi} being constant and the Euler-Lagrange equations do not, because you get the factor 1/2 wrong.
(ARGH! EDIT EDIT: I just noted I made a mistake here. It is still working! Noooooo...
)
So in oder to get the correct equation of motion the physicists way, you first need to "choose" the "correct form" of the Lagrangian!
Peskin & Schroeder describe the derivation of Dirac's equation like this
I disagree with the statement a little bit. The Euler-Lagrange equation for \overline{\psi} doesn't immediately imply the correct result because you have
<br />
\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\overline{\psi})} = 0.<br />
You need to first, with integration by parts, to move the derivative from \psi onto the \overline{\psi}, and then apply the Euler-Lagrange equations. If you give a physicists an exercise where he is supposed to derive the Dirac's equation like P&S instruct, he will first do the integration by parts, and then use the Euler-Lagrange equations, and then get the correct results. If I ask "why do you do the integration by parts first?", the answer is "because otherwise you don't get the correct result".
You cannot derive the equation of motion the physicists way unless you already know the result before!
BUT WHY?
Wouldn't it be so much easier for everyone if we just wrote 0=D_{\alpha} \int dt\; L(\psi+\alpha\xi)\Big|_{\alpha=0}, and understood what we are doing? Why do physicists want to start making this simple thing into so much more complicated?
It seems that the authors of the physics books think, that the readers are so intelligent, that it doesn't matter what they write into the books, because the readers are not going to develop major misunderstanding in anyway. This has been a fatal mistake IMO. I once tried to ask about this from one physicist, and in the end he was explaining to me, that similarly as \textrm{Re}(\phi) and \textrm{Im}(\phi) are independent variables, also \phi and \phi^* are independent variables. In other words, he was explaining, that if x and y can be chosen independently, then components of (x,y) do not fix the components of (x,-y). Despite this, he also understood, that components of \phi^* actually do fix the components of \phi, but still on the other hand, they did not fix each others components, because they were describing "physical dynamics". They are always these "physical arguments", where I get lost.
