Higgs Mechanism and Lorentz transformation of currents

[CAF123]

Homework Statement

Consider the Higgs mechanism lagrangian, $$\mathcal L = (D_{\mu} \phi)^* (D^{\mu} \phi) -\mu^2 (\phi^* \phi) - \lambda (\phi^* \phi)^2 - \frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ with $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ and $D_{\mu} = \partial_{\mu} + iqA_{\mu}$. One can show that then $\partial_{\mu}F^{\mu \nu} = J^{\nu}$, where $J_{\nu} = iq ((D^{\nu} \phi)^* \phi - \phi^* D^{\nu} \phi)$. Following spontaneous symmetry breaking, the massless gauge field acquires mass and let's suppose it has the following oscillatory behaviour $$A_{\mu} = \cos (M t) \epsilon^{1}_{\mu}$$ where $\epsilon^1_{\mu} = (0,1,0,0)$ What is the four vector current and what is it if I apply a lorentz boost in the z direction?

2. Homework Equations

lorentz boost in z direction is $$t' = \gamma \left( t -\frac{vz}{c^2}\right)$$ and $z' = \gamma(z-vt)$

3. The Attempt at a Solution
I can evaluate the components of $A_{\mu}$ to get $A_{\mu} = (0,\cos (Mt), 0, 0)$ and then I could just evaluate the four current by evaluating each of its four components separately. When I am taking a lorentz transformation, would I just send $t \rightarrow t'$ and use the equation given above in relevant equations? Or would I also need to consider transformation of the differential operator as well?

Thanks!

 

[mark726]

Hello, thank you for your post. It seems like you have a good understanding of the equations and concepts involved in this problem. To answer your question, when taking a Lorentz transformation, you would indeed need to consider the transformation of the differential operator as well. This is because the transformation of a vector field involves both the transformation of its components and the transformation of the derivative operators acting on those components. So in this case, you would need to use the transformation equations for both the vector field $A_{\mu}$ and the derivative operator $D_{\mu}$.

I hope this helps. Let me know if you have any further questions.