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I've been trying to learn about how to express the Dirac equation in curved spacetime, and in looking through various resources, I've found that the concepts of the spinor bundle and spin connection keep popping up. I keep getting confused behind the formalism that attempts to describe spin. I'm familiar with the concepts of vector bundles and connections on vector bundles, and I would like to understand how those concepts are related to spin. To make this question a bit less open-ended, I'll explain my intuition behind vector bundles and their connection, explain why I'm confused about the spin connection and the spinor bundle, and ask more pointed questions at the end.

Vector Bundles: Given a manifold $M$ we would like to attach a k-dimensional vector space to each point in such a way that locally on the manifold, the vector spaces look like the product space $M \times \mathbb{R}^n$. A priori, these vector spaces don't really talk to each other in the sense that there isn't a well-defined way to add and subtract vectors that lie in different vector spaces, i.e. the vector spaces attached to different points on the manifold. So, we introduce a connection on the bundle to be able to subtract points near each other on the manifold. This allows us to take derivatives, and defines the covariant derivative $D_\mu = \partial_\mu + A_\mu$, where the matrices $A_\mu$ are the Christoffel symbols (or gauge fields, etc.) in a certain coordinate system on the manifold and the bundle's frame. In physics literature, the fields $A_\mu$ are often introduced as a fudge factor meant to impose the condition of gauge covariance on $D_\mu$. Covariance means the transformation $\psi \rightarrow U \psi$ and the corresponding $D_\mu \psi \rightarrow U D_\mu \psi$, which in the bundle language corresponds to changing coordinates on the bundle.

Spin connection: As I understand it (from Wiki), the spin connection describes generalized tensors that are constructed in terms the local orthonormal frames. First, you construct the local frame fields, or vierbeins, $e^a _\mu$. Then you treat the latin $a,b,c,...$ indices as a new kind of tensor object while you treat the greek indices $\mu,\nu,...$ as the standard tangent vectors or co-vectors. There are different connection coefficients ${{\omega_\mu}^b}_c$ for the latin indices than for the Christoffel coefficients ${\Gamma}^\lambda_{\mu \nu}$. The $\omega$ are computed by stipulating that the covariant derivative kills the vierbeins, i.e. that $D_\rho e^a _\mu = \partial_\rho e^a _\mu + {{\omega_\mu}^a}_b e^a _\mu - {\Gamma}^\lambda_{\mu \rho} e^a _\lambda = 0$ and that $D_\rho \eta_{a b} = 0$.

Spinor Bundle: This is the concept that's been confusing me the most. As I understand, the spin group a double cover of $SO(n)$ corresponding to spinors, since a $360^o$ rotation should equal -1 like for spinors, and a $720^o$ rotation should be the identity. The algebra that goes into this construction is very confusing to me, especially since I don't have the intuition/motivation to see where it comes from. As such, I also don't understand how a bundle picture comes from this

So finally, my questions are:

  1. It seems that the spin connection is entirely dependent on the metric tensor $g_{\mu \nu}$, since the vierbein frames are constructed from $g_{\mu \nu}$ and the defining equations of the tensor rely only on the vierbeins and the Christoffel connections. Does the spin connection add any additional structure to the metric, or is it entirely metric-dependent?
  2. What are these generalized spin-tensor, latin-indexed objects such as $e^a_\mu$ and why should they be significant for spinors?
  3. What are the fibers associated to the spinor bundle, and how does the spin connection correspond geometrically to 'connecting' them?

Overall, I'd ideally like to see how all these different concepts are all 'connected' (bad pun I'm sorry) in as intuitive a sense as possible. Any relevant resources or additional comments are greatly appreciated!

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  • $\begingroup$ Hi Joe, if you reckon your answers will be long in explaining​ it to you, (sorry I don't know the material very well, so you tell me), could I suggest you split your questions into three separate ones. I say this because it looks from your text above the questions that you might need a longish answer. Anyway,.just a thought , best of luck with it. $\endgroup$ – user154420 Jun 13 '17 at 23:52
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    $\begingroup$ Hello, thanks for the suggestion! But, the reason I'm putting these all as one question is because they're all separate angles of asking "what is spin/spinor bundles/spin connection?". The text above was just put there to clarify where I'm coming from, and give some background vocabulary and ideas that might be helpful in constructing replies. $\endgroup$ – Joe Jun 14 '17 at 0:02
  • $\begingroup$ I think this article arxiv.org/abs/1010.5822 will help you a lot $\endgroup$ – jak Jun 14 '17 at 15:15
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Some comments:

First, let me describe the construction of the spinor bundle in a couple sentences. Given the smooth oriented Lorentzian manifold $(M,g)$, we have the tangent bundle $TM\to M$. We also have the $\mathrm{SO}(n,1)$ principal frame bundle, $FM\to M$. Now, $TM$ is the bundle associated to $FM$ via the natural representation of orthonormal frames as elements of $\mathrm{SO}(n,1)$. Since $\mathrm{SO}(n,1)$ has the double cover $\mathrm{Spin}(n,1)$, we suppose the existence of a double cover bundle of $FM$, $\mathrm{Spin}(n,1)\to P\to M$. We then take a vector space $\Delta$ on which there is a representation of $\mathrm{Spin}(n,1)$, this is where the Clifford algebra comes in. The spinor bundle is finally $S=P\times_{\mathrm{Spin}(n,1)}\Delta$. So the spinor bundle is a vector bundle associated to the double cover of the frame bundle.

  1. The spin connection is indeed entirely dependent on the metric. The additional structure you added is the spin bundle (which need not be unique). The spin connection is in a sense a lift of the Levi-Civita connection from the tangent bundle.

  2. The vielbeins $e^a{}_\mu$ are in a sense mixed frame-vector field objects. They can be thought of as representing the $TM\leftrightarrow FM$ duality from above.

  3. The fibers of the spinor bundle are elements in the $\Delta$ from above, roughly speaking. They are spinors in the representation theory sense. The connection connects them in the same sense as for any Ehresmann connection.

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