Question

Q5. Consider a Lagrangian for some scalar field φ interacting with electric and magnetic fields in the following way: AL where F μν €μυρσ ρσ/2. Obtain the Maxwell equations in vector form, and show that two of them are same as in regular EM and two are different. This theory is supposed to describe ǎ hitherto undetected particle called the "axion", a possible candidate for dark matter (15 pts).

Answer 1

Data given: - Rightarrow consider a Lagrangian for some scalar field phi interacting with electric and magnetic fields in L = (partial differential phi) + partial differential mu phi - 1/2 m^2 phi^2 - 1/4 F_mu F mu v + g/4 phi F_d^mu v F_uv Where F_d^mu v = elementof mu_v p_d F P_d/2. L S_G ES (D_mu phi) (D_mu phi)^x - n^2 phi phi - 1/4 F_4x F^4x where D_mu = d_mu + i_e A_mu first check partial differential^2 phi: - phi rightarrow elementof^i alpha (x) phi and phi e^-i alpha(x) phi phi^x = e^i alpha(x) phi and phi - e^-i alpha(x) phi phi^x = e^i alpha (x) phi e^-i alpha (x) phi^x phi partial differential^x in value under local gauze transformation. second check (D_mu phi) (D mu phi): - phi rightarrow e^i alpha (x) phi and phi . x'' rightarrow 0^i alpha (x) phi^x D_mu phi = (partial differential_u partial differential_mu + i.e A^x_mu) phi where A_mu rightarrow A_mu - 1/e