- #1
Tertius
- 57
- 10
- TL;DR
- Trying to derive the gauge field for the complex scalar field in curved spacetime.
I am stuck deriving the gauge field produced in curved spacetime for a complex scalar field. If the underlying spacetime changes, I would assume it would change the normal Lagrangian and the gauge field in the same way, so at first guess I would say the gauge field remains unchanged. If there is additional insight (or correction) here I would gladly read an article or book chapter if there are any suggestions.
Ok, here's where I am getting stuck. Starting with the complex scalar field Lagrangian (where covariant derivatives have been replaced with partials because it is a scalar field): $$ L = (g^{\mu \nu}d_\mu \phi d_\nu \phi^* -V(\phi, \phi^*)) \sqrt{-g}$$ We can then make the substitutions $$ \phi \rightarrow \phi e^{i\theta(x^\mu)} $$ and $$ \phi^* \rightarrow \phi^* e^{-i\theta(x^\mu)} $$ And the Lagrangian becomes $$ L = (g^{\mu \nu} (d_\mu \phi d^{i\theta} + i d_\mu \theta e^{i\theta} \phi)(d_\nu \phi^* e^{-i\theta} - i d_\nu \theta e^{-i\theta} \phi^*) - V(\phi, \phi^*)) \sqrt{-g} $$ After expanding, which I'm not sure is the best idea, we get $$ L = ( g^{\mu \nu}(d_\mu \phi d_\nu \phi^* - i d_\nu \theta d_\mu \phi~\phi^* + i d_\mu \theta d_\nu \phi^* ~ \phi + d_\mu \theta d_\nu \theta~ \phi \phi^*) - V(\phi, \phi^*)) \sqrt{-g} $$
At this point, I'm not sure how to make progress to distill this into a single field that takes all of those extra terms. Maybe there is a better route to determine the gauge field?
L=
Ok, here's where I am getting stuck. Starting with the complex scalar field Lagrangian (where covariant derivatives have been replaced with partials because it is a scalar field): $$ L = (g^{\mu \nu}d_\mu \phi d_\nu \phi^* -V(\phi, \phi^*)) \sqrt{-g}$$ We can then make the substitutions $$ \phi \rightarrow \phi e^{i\theta(x^\mu)} $$ and $$ \phi^* \rightarrow \phi^* e^{-i\theta(x^\mu)} $$ And the Lagrangian becomes $$ L = (g^{\mu \nu} (d_\mu \phi d^{i\theta} + i d_\mu \theta e^{i\theta} \phi)(d_\nu \phi^* e^{-i\theta} - i d_\nu \theta e^{-i\theta} \phi^*) - V(\phi, \phi^*)) \sqrt{-g} $$ After expanding, which I'm not sure is the best idea, we get $$ L = ( g^{\mu \nu}(d_\mu \phi d_\nu \phi^* - i d_\nu \theta d_\mu \phi~\phi^* + i d_\mu \theta d_\nu \phi^* ~ \phi + d_\mu \theta d_\nu \theta~ \phi \phi^*) - V(\phi, \phi^*)) \sqrt{-g} $$
At this point, I'm not sure how to make progress to distill this into a single field that takes all of those extra terms. Maybe there is a better route to determine the gauge field?
L=