Can you apply product rule to arg of a bra-ket?

Question

I found the following expression in a paper:

$$ \frac{d}{dt}\arg\langle\phi_+|\dot{\phi_-}\rangle $$

where the $\arg$ term is the argument of the complex number given by inner product between two eigenstates $\phi_+$ and $\phi_-$, and the dot indicates a time-derivative.

Does the product rule apply in the usual sense in this case? i.e., does the following hold? If not, why?

$$ \frac{d}{dt}\arg\langle\phi_+|\dot{\phi_-}\rangle = \arg\langle\dot\phi_+|\dot{\phi_-}\rangle + \arg\langle\phi_+|\ddot{\phi_-}\rangle $$ I could not find much about $\arg$ terms in quantum mechanics and bra-ket literature, and so would appreciate any help you might have to offer.

Answer 1

No it definitely doesn't work that way. Take $z\equiv \langle \phi_+ |\dot{\phi}_-\rangle$. It is true you can use the product rule on derivatives of $z$. But $Arg$ is a nontrivial function and you need to use a chain rule.

$$z=|z|e^{iArg (z)}$$ $$\log z = \log |z|+iArg(z)$$ where $Arg$ is only defined up to an integer times $2\pi$ but it won't matter since we're taking a derivative, $$\frac{d}{dt}Arg(z)=\frac{d}{dt}\text{Im}\log z=\text{Im}\frac{\dot{z}}{z}$$ So the product rule appears in $\dot{z}$ but otherwise it looks nothing like it.