Where is the Error in My Derivation of the Magnetic Field Equation?

[Gaussian97]

As I said my goal is to derive the equation $\tilde{B}^k(\vec{q})=-i\varepsilon^{ijk}q^i\tilde{A}^j_{cl}(\vec{q})$

As far as I know, the magnetic field is defined using the potential as $\vec{B}=\vec{\nabla}\times\vec{A}$

Then in equation 6.6 they define $A^\mu(x)=\int \frac{1}{(2\pi)^3}\tilde{A}^\mu(\vec{k})e^{-ikx}\text{d}^3 k$ and an equivalent equation for $\vec{B}$. So, using the definition
$$B^k=\varepsilon^{kij}\partial_i A^j=\int \frac{\varepsilon^{kij}}{(2\pi)^3}\tilde{A}^j(\vec{q})\partial_ie^{-iqx}\text{d}^3 q=\int \frac{1}{(2\pi)^3}\left[\varepsilon^{kij}\tilde{A}^j(\vec{q})(-iq_i)\right]e^{-iqx}\text{d}^3 q\Longrightarrow \tilde{B}^k=-i\varepsilon^{ijk}q_i\tilde{A}^j(\vec{q})$$

But this is not the equation the book gives, because we are using the metric $(+---)$ and then $q^i=-q_i$ so the equation I get is $$\tilde{B}^k=i\varepsilon^{ijk}q^i\tilde{A}^j(\vec{q})$$

Someone can tell me where is my error?

Thank you very much :)

 

[vanhees71]

Let's do it in the 3D formalism first. The integrand is
$\vec{A}_{\vec{q}}(x)=\vec{\tilde{A}}(\vec{q}) \exp(-\mathrm{i} q x)=\vec{\tilde{A}}(\vec{q}) \exp(-\mathrm{i} k t + \mathrm{i} \vec{k} \cdot \vec{x}),$
from which
$$\vec{\nabla} \times \vec{A}_{\vec{q}}(x)=-\vec{\tilde{A}}(\vec{q}) \times \exp(-\mathrm{i} q^0 t + \mathrm{i} \vec{q} \cdot \vec{x})=-\vec{\tilde{A}}(\vec{q}) \times \mathrm{i} \vec{q} \exp(-\mathrm{i} q^0 t + \mathrm{i} \vec{q} \cdot \vec{x})=\vec{\tilde{B}}_{\vec{q}} \exp(-\mathrm{i} q^0 t + \mathrm{i} \vec{q} \cdot \vec{x}).$$
In the Ricci calculus this reads
$$\tilde{B}^k=-\mathrm{i} \epsilon^{ijk} \tilde{A}_{\vec{q}}^i q^j=+\mathrm{i} \epsilon^{ijk} q^i \tilde{A}_{\vec{q}}^j.$$
So you are right.

Note that Peskin and Schroeder is full of typos. You always have to check any formula yourself ;-)).

 

[Gaussian97]

Ok, thank you very much