A Local phase invariance of complex scalar field in curved spacetime

Tertius
Messages
57
Reaction score
10
TL;DR Summary
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=
 
Physics news on Phys.org
What is your goal? Do you want to "gauge" the Klein-Gordon field? Then you have to introduce a gauge field and you'll end up with "scalar electrodynamics" (in a curved background spacetime).
 
My main goal is a deeper understanding. Particularly about what gauge fields are, and how a curved background may or may not change their characteristics.
If the Klein Gordon equation is a general relativistic field theory, why would the gauge field be electrodynamics? Is that just because it would be a U(1) gauge symmetry?
I suppose the KG equation is of particular interest to me because every type of particle can be a solution to it.
I am also curious how/if a curved space time affects the resulting gauge field of a lagrangian.
 
I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...
Thread 'Relativity of simultaneity in actuality'
I’m attaching two figures from the book, Basic concepts in relativity and QT, by Resnick and Halliday. They are describing the relativity of simultaneity from a theoretical pov, which I understand. Basically, the lightning strikes at AA’ and BB’ can be deemed simultaneous either in frame S, in which case they will not be simultaneous in frame S’, and vice versa. Only in one of the frames are the two events simultaneous, but not in both, and this claim of simultaneity can be done by either of...
Back
Top