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:
- 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?
- What are these generalized spin-tensor, latin-indexed objects such as $e^a_\mu$ and why should they be significant for spinors?
- 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!
