Can the transformation properties of spinor fields on a manifold be *derived* similar to how vector field transformations can be derived?

Question

I'm asking about transformation properties of vector and spinor fields. I'm trying to better understand statements like "vectors/spinors transform in a certain way under way under [symmetry operations]" where the symmetry operation might be a rotation or Lorentz transformation.

I want to work in the mathematical context of differential manifolds.

We have a manifold and on that manifold we have (local) coordinates. Within the context of differential geometry I can define the tangent space at each point on the manifold. The tangent space is related to derivatives along coordinate direction on the manifold. So in this way, the tangent space is closely related to, and derived from, coordinates on the manifold. We can undergo a procedure where for each point on the manifold we select a single vector from its tangent bundle. In this way we define a vector field on the manifold. But, importantly, because of the relationship between elements of the tangent space and coordinates I described above, it is possible to derive how vectors transform when we apply coordinate transformations to the manifold.

My question is: Can we do something similar for spinors? That is, can we define a "spinor space" at each point on the manifold whose transformation properties can be derived from some relationship between the spinor space and the coordinates on the manifold? And then can we select a spinor from the spinor space at each point on the manifold to build up a spinor space? OR INSTEAD, do we simply define an abstract spinor space at each point on the manifold and somehow just declare how elements of that space transform when we transform the coordinates on the manifold? That is, can we define spinors in such a way that their transformation properties under coordinate transforms are derived from their definition in relation to a manifold?

Answer 1

Briefly, there is an important difference. Let us consider an arbitrary curved $n$-dimensional differential manifold. The structure group is a priori the general linear group $GL(n,\mathbb{R})$.

1. On one hand, a vector field $X\in\Gamma(TM)$ can be defined directly on $M$.

2. On the other hand, a Lorentz transformation and a spinor field require additional geometric data and conditions so that (among other things) the structure group of the manifold $M$ is reduced to the Lorentz group $O(n\!-\!1,1;\mathbb{R})$. For more details, see e.g. spinor bundle on Wikipedia.

   An important example is when $M$ is endowed with a choice of vielbein/solder form $e$. Recall that for each spacetime point $p\in M$ a vielbein/solder form $$(T_pM,g_p)\stackrel{e_p}{\longrightarrow}(V,\eta)$$ is essentially a choice of a frame for each tangent space $T_pM$. Here $\eta$ denotes the $n$-dimensional Minkowski metric. We can then use the corresponding real Clifford algebra $Cl(V,\eta)$ and Dirac/gamma matrices to define spinor representations.