In the course of QFT i learnt that the gauge field emerges from the need of a gauge invariance in the action, as we use the covariant derivative in minimal coupling. Now i'm studying how spin fields couple on curved space and it seems that is instead the gauge invariance that emerges from this attempt to couple spin field with a curved space, since the tetrad formalism has more degrees of freedom than the metric one, which correspond to the dimension of the proper orthochronous Lorentz group:
$$\text{dim}(e^{\mu}_{a})-\text{dim}(g_{\mu\nu})=d^2-\frac{d(d+1)}{2}=\text{dim}(\tilde{L}^+).$$
Anyway i aspect that this gauge invariance is not the same as the one we require in QFT since QFT is made on flat space. What is wrong and what is true in this reasoning? Is there a physical meaning of the additional degrees of freedom in tetrad formalism?
EDIT: I thought about this question and i believe that i was a bit confused when asked it because the teacher made the exemple of covariant electromagnetism when obtained the covariant derivative of a spin vector (with spin connection). Now, i think that the question were:
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Is spin connection a complete different object than connection due to some symmetry of the theory?
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Could we have both spin connection and gauge connection in the covariant derivative?
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As the connection of (i.e.) $U(1)$ gauge symmetry give rise to electromagnetism, what kind of interaction give rise the coupling with spin connection?
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As the generator of gauge $U(1)$ is the electric charge (so an observable), is the generator of the orthochronous Lorentz group an observable too?
EDIT
As a result of conversation with another user it turns out that the physical meaning of the 6 extra dof is that they describe the coupling with gravity, and they are not present for scalar fields because those don't interact with gravity.
See this: Spin field on curved space: meaning of coupling with spin connection
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