# Local phase invariance of complex scalar field in curved spacetime

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• Tertius
In summary, the goal of the author is to understand what gauge fields are and how they might be affected by a curved background spacetime. The author is stuck trying to derive the gauge field in a 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. However, if there is additional insight (or correction) here I would gladly read an article or book chapter if there are any suggestions.
Tertius
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=

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.

vanhees71

## 1. What is local phase invariance of a complex scalar field in curved spacetime?

The local phase invariance of a complex scalar field in curved spacetime refers to the property of this field to remain unchanged when the phase of the field is varied at each point in spacetime. This means that the magnitude and direction of the field at a given point is not affected by the local phase change.

## 2. Why is local phase invariance important in physics?

Local phase invariance is important in physics because it is a fundamental symmetry of nature. It is related to the conservation of energy and momentum, and plays a crucial role in the formulation of quantum field theories, such as the Standard Model of particle physics.

## 3. How does local phase invariance affect the behavior of a complex scalar field in curved spacetime?

Local phase invariance affects the behavior of a complex scalar field in curved spacetime by imposing constraints on the allowed interactions and transformations of the field. It also leads to the existence of conserved quantities, such as the electric charge, which are associated with the local phase symmetry.

## 4. Can local phase invariance be violated in certain situations?

Yes, local phase invariance can be violated in certain situations, such as in the presence of external fields or in the presence of certain types of interactions. This can lead to the breaking of the symmetry and the emergence of new physical phenomena, such as the Higgs mechanism in particle physics.

## 5. How is local phase invariance related to general relativity?

Local phase invariance is related to general relativity through the concept of gauge symmetry. In general relativity, the local phase symmetry is equivalent to the diffeomorphism invariance, which is a fundamental principle of the theory. This connection allows for the incorporation of quantum field theories into the framework of general relativity.

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