Question

Q1: Consider complex scalar free field given by the following Lagrangian density $mathcal{L} = frac{1}{2} partial_{mu} phi^{} partial^{mu} phi - frac{1}{2} m^{2} |phi|^{2}$, 1) Find all equation of motions associated with the field. Justify your answer. 2) Assume the following transformation of the field $phi ightarrow phi^{'} = e^{ialpha} phi$, $(1 + ialpha)phi = phi + ialphaphi$ where $alpha$ is constant, infinitesimal, and real. Prove that the Lagrangian is invariant under the above transformation. (Hint: it is sufficient to show that $delta mathcal{L} = 0$). 3) Prove that the conserved current $j^{mu}(x)$ associated with this transformation may be written as $j^{mu} = i [(partial^{mu} phi^{}) phi - phi^{*} (partial^{mu} phi)]$ 4) Prove that the current is conserved (you need to show it obeys the continuity relation $partial_{mu} j^{mu} = 0$).

          Q1: Consider complex scalar free field given by the following Lagrangian density
$mathcal{L} = frac{1}{2} partial_{mu} phi^{*} partial^{mu} phi - frac{1}{2} m^{2} |phi|^{2}$,
1) Find all equation of motions associated with the field. Justify your answer.
2) Assume the following transformation of the field
$phi 
ightarrow phi^{'} = e^{ialpha} phi$, $(1 + ialpha)phi = phi + ialphaphi$
where $alpha$ is constant, infinitesimal, and real. Prove that the Lagrangian is invariant under the above
transformation. (Hint: it is sufficient to show that $delta mathcal{L} = 0$).
3) Prove that the conserved current $j^{mu}(x)$ associated with this transformation may be written as
$j^{mu} = i [(partial^{mu} phi^{*}) phi - phi^{*} (partial^{mu} phi)]$
4) Prove that the current is conserved (you need to show it obeys the continuity relation $partial_{mu} j^{mu} = 0$).

$Q1: Consider complex scalar free field given by the following Lagrangian density mathcalL = frac12 partialmu phi^* partial^mu phi - frac12 m^2 |phi|^2, 1) Find all equation of motions associated with the field. Justify your answer. 2) Assume the following transformation of the field phi ightarrow phi^' = e^ialpha phi, (1 + ialpha)phi = phi + ialphaphi where alpha is constant, infinitesimal, and real. Prove that the Lagrangian is invariant under the above transformation. (Hint: it is sufficient to show that delta mathcalL = 0). 3) Prove that the conserved current j^mu(x) associated with this transformation may be written as j^mu = i [(partial^mu phi^*) phi - phi^* (partial^mu phi)] 4) Prove that the current is conserved (you need to show it obeys the continuity relation partialmu j^mu = 0).$

Added by Kathryn L.

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