How to define spinors in general relativity?

Question

In Schwartz's QFT and the Standard Model, a vector is defined an object that transforms like a vector. For instance, $V$ is a 4-vector if its components transform like this under Lorentz boost,

\begin{equation} V_\mu \rightarrow \Lambda_{\mu\nu}V_\nu \end{equation}

Where $\Lambda_{\mu\nu}$ are elements of Lorentz matrix. The book then talks about the commutator of Lorentz group, and how other representations of Lorentz group can be found by constructing objects whose transformation rule obey the same commutation relation. In particular, a spin 1/2 fermion can be represented by a Dirac spinor. In later part of that book, they check if an operator is vector by testing its transformation rules under Lorentz transformation. It is a vector only if it follows the aforementioned transformation rule.

After finishing that book, I moved on to read Schutz's textbook on general relativity and am surprised to learn that they defined vector in an alternative way. They define it as

"a vector is a tangent to some curve, and is the function that gives $d\phi/ds$ when it takes $\tilde{d}\phi$ as an argument".

Or if I understand correctly, vector $\vec{V}$ is an object that when supplied as an input to a one-form with components $(d\phi/dx^i)$, produces an invariant number $d\phi/ds$ where $s$ is a parametrization of a curve with tangent $\vec{V}$ and $\phi$ is any scalar field.

This definition actually makes more sense to me as it doesn't rely on the apparently tautological definition of "transform like a vector" (I know it's not tautology as the rules of transformation are defined in advance, but still). Furthermore, this definition immediately extends to vectors in Riemannian spaces. I have no doubt that this definition leads to transformation rules that obeys the Lorentz commutator in a flat Minkowski space as both definitions lead to the same 4-vector.

But now I wonder, it is possible to define spinors in similar manner as well? Can spinors be defined in terms of one-forms and derivatives of $\phi$? The first idea that comes to my mind is to define $\phi$ as a Dirac spinor field rather than a scalar field, but that may ends up with 4 independent scalar rather than a Dirac spinor field. I guess I have to somehow uses gauge invariance properties? It seems to get technical and tedious soon.

At the moment, I am not interested (yet) in quantizing fields in curved spaces. I just want to know if there is an alternative definition of spinors in the context of differential geometry.

Answer 1

Defining tensors in curved spacetime is really easy and straightforward. You can use Schutz's definition of vector and work out the algebraic properties of tensor products, dual spaces, and etc to get to all other tensors.

Spinors are ridiculously more complicated. I will just mention the very basics and in no way will I attempt at giving a precise construction in this answer.

Tensors are a constructions that is available for any vector space. Spinors, however, are a different beast. They are defined in flat spacetime as particular (projective) representations of the Lorentz group, which is just a fancy way of saying "a spinor is something that transforms like a spinor". Here is where the difficulties begin: how do you bring the Lorentz group to curved spacetimes? Lorentz symmetry is explicitly broken in more general spacetimes.

You use something we call a vielbein (or tetrad in four-dimensions). This is a smooth attribution of an ordered orthonormal base of the tangent space for each point on some subset of the manifold (there might not be a global vielbein). Hence, a vielbein can be thought of as a map $e \colon U \to (T_pM)^4$ that takes $p \mapsto \{e_{\mu}^p\}$, where $p$ is a point in the subset $U$ and $\{e_{\mu}^p\} = \{e_{0}^p,e_{1}^p,e_{2}^p,e_{3}^p\}$ is an orthonormal basis at $p$. By orthonormal I mean $g(e_{\mu},e_{\nu}) = \eta_{\mu\nu}$, where $g$ is the metric and $\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)$.

Notice that the vielbein gives us a way of talking about the Lorentz group in curved spacetime: different orthonormal bases at a point $p$ are related by Lorentz transformations (think of the analogy with rotations), so we get a Lorentz symmetry in general relativity.

Using this notion of Lorentz symmetry, you can describe spinors at a point by using the appropriate representation. The problem is to move from points to fields. I don't know of any simple way to do this: every time I read about it the author used the language of fiber bundles. Fiber bundles can be thought of as manifolds that locally look like a product of manifolds, but might not do so globally. For example, the Möbius strip locally looks like the product of the circle $\mathbb{S}^1$ with the real line $\mathbb{R}$, but when you look at it globally you see it is not globally this product.

Fiber bundles are interesting because they allow you to write manifolds that locally look like $M \times V$, where $V$ is the vector space of the representation you're working with. Globally, however, the structure may be much more complicated. Tensor fields are sections of fiber bundles (which is a specific sort of function that essentially assigns a tensor to each spacetime point), although most general relativity books do not introduce them in that language (which is indeed far more complicated). Spinors too are sections of fiber bundles, but there is no short cut (that I know of) to defining them. You need to use the language of fiber bundles to be able to define a smooth spinor field, which is, of course, a smooth mapping of a spinor to each spacetime point. This is not a map $\psi \colon M \to V$, and the reason deep down for this is that the spinor space at each spacetime point is not trivially related to the spinor spaces at neighboring points, which forbids me from writing them all as $V$. Notice how no book in general relativity defines tensor fields as maps $T \colon M \to V$ for a suitable vector space $V$: the reason is the same, the vector spaces at each point are not trivially related to each other.

Wald's General Relativity has a chapter on spinors (Chap. 13). He does all of this construction in there, but I find it a bit rushed and difficult to follow if it is your first time dealing with fiber bundles (but the section on spinors in Minkowski spacetime, Sec. 13.1, is beautiful and extremely instructive, in my opinion). Currently, my favorite pedagogical reference on spinors in curved spacetime is Hamilton's Mathematical Gauge Theory. As the name suggests, the book is actually on gauge theory, but Chap. 6 discusses the classical theory of spinors in detail. I should mention that fiber bundles are also a natural geometric language to discussing gauge theory in curved spacetimes.

Oh, as a final remark, why no one ever talks about fiber bundles in quantum field theory in flat spacetime? There is a nice result about fiber bundles that says that if a space is contractible (meaning it can be continuously deformed to a point), then all fiber bundles defined on it are trivial, meaning they all are indeed just products. $\mathbb{R}^4$ is contractible, so indeed all fiber bundles are trivial. Hence, in flat spacetime, we actually can just say that a spinor is a map $\psi \colon M \to V$ for a suitable vector space $V$.