Physical motivation for the definition of Spin structure

I'm pretty confused about the motivations behind defining a spin structure on a manifold. Let me explain.

In quantum mechanics, particles are represented by irreducible unitary projective representations of the Poincaré group (i.e. the states that characterize a particle are vectors belonging to a representation of $$SO^+(1,3)$$). Knowing that there is a correspondence between the projective representations of a group and the "true" representations of its universal covering group, we are more interested in the "true" representations of the universal covering group (since they are easier to find) of $$SO^+(1,3)$$ i.e. $$SL(2, \mathbb{C})$$.

Now let's turn to QFT and consider that our spacetime is characterized by 4-dimensional Lorentzian manifold $$(M,g)$$. Particles are therefore characterized by fields $$\phi$$ defined over $$M$$ and we naturally expect that for any $$x \in M$$
$$\phi(x) \in V$$ where $$V$$ is a representation space of $$SL(2, \mathbb{C})$$. So, if we want to speak in the language of fiber bundle, we expect the fields representing the particles to be sections of a vector bundle $$E$$ associated to an $$SL(2,\mathbb{C})$$ principal bundle.

So here's my question:

Why do we define a spin structure as:

• a $$SL(2, \mathbb{C})$$-principal bundle $$P$$

• AND a double covering $$\Lambda : P \longrightarrow FM$$ where $$FM$$ is the frame bundle of $$M$$ i.e. a $$SO^+(1,3)$$-principal bundle such

$$\Lambda(pq) = \Lambda(p) \lambda(q)$$ for $$q \in SL(2,\mathbb{C})$$ and $$\lambda: SL(2,\mathbb{C}) \longrightarrow SO^+(1,3)$$ the universal covering map.

In other words, why is the second point important? Why can't we just consider any $$SL(2,\mathbb{C})$$-principal bundle and describe our particles with associated vector bundle?

• Excellent question, apart from the minor error I edited. Have you read the famous chapter 13 of Wald? Nov 18, 2023 at 18:45
• Thank you for your correction. I did not know this reference but I thank you for having given it to me since it seems to answers my question : « We define spinor by […] specifying their behavior under $SL(2,\mathbb{C})$ transformations associated with changes of orthonormal basis » (Wald - General Relativity : p -364/365 ). In practice, we want to see how the fields transform under a change of orthonormal bases. Since we use $SL(2,\mathbb{C})$ to make representation theory easier, we need to link our $SL(2,\mathbb{C})$-principal bundle to the orthonormal frame bundle FM.
– eomp
Nov 19, 2023 at 16:58

You need the second condition because this $$\mathrm{SL}(2,\mathbb{C})$$ isn't just a random internal symmetry group (like the gauge groups of Yang-Mills theories, where you indeed just have a $$G$$-principal bundle without further conditions), it's supposed to be the spacetime symmetry group - the universal cover of the local $$\mathrm{SO}(1,3)$$-symmetry induced by the pseudo-Riemannian structure of spacetime.
As you correctly stated, we get to wanting representations of $$\mathrm{SL}(2,\mathbb{C})$$ by considering projective representations of $$\mathrm{SO}(1,3)$$, and that $$\mathrm{SO}(1,3)$$ is precisely the $$\mathrm{SO}(1,3)$$ from the frame bundle. So the manifold already comes with a $$\mathrm{SO}(1,3)$$-bundle, and of course our $$\mathrm{SL}(2,\mathbb{C})$$-bundle needs to be compatible with this and the precise meaning of compatibility is exactly that we have the covering map $$P\to FM$$.
The way an ordinary tensor field transforms under the spacetime symmetry that the $$\mathrm{SL}(2,\mathbb{C})$$ represents is already fixed by it being the tensor field on this manifold, and one way to state that transformation behavior is by considering both $$TM$$ and $$T^\ast M$$ (and their tensor powers) as associated bundles to the frame bundle $$FM$$. Through the projection $$P\to FM$$, all those bundles also become associated bundles to $$P$$ - the action $$P$$ on them is just projecting down to $$FM$$ and then using the action of $$FM$$.