I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused.

Basically, when one tries to do classical field theory (as in, the branch of physics) in a mathematically precise manner, one considers a field $\psi$ to be a section of some fiber bundle $\pi_E:E\rightarrow M$ over the spacetime $M$, and writes up a lagrangian $\hat L\in \Omega^n_H(J^1(E))$, which is a horizontal $n$-form on $J^1(E)$ (sometimes $J^2(E)$) such that the *action* is $$ S[\psi]=\int_M (j^1\psi)^\ast\hat L. $$

If one wishes to involve gravity, one seeks to consider field theories that are diffeomorphism invariant.

If $\pi_E:E\rightarrow M$ is a fiber bundle whose sections are some kind of matter fields, then to have a well-defined concept of diffeomorphism-invariance, for any diffeomorphism $\phi:M\rightarrow M$ one must be able to lift this diffeomorphism into a fiber bundle automorphism $\phi_E\in\text{Aut}(E)$ in a consistent manner.

For the tensor bundles (or indeed any *natural bundle*), there is a functorial lift given by the tangent map (at least in the case of tensor bundles). For example, if $X\in\Gamma(TM)$ is a smooth vector field, and $\phi:M\rightarrow M$ is a diffeo, then $$ \phi_E=T\phi:TM\rightarrow TM $$ is this vector bundle automorphism.

On the other hand, in physics, very important fields are spinor fields, sections of the spinor bundle $\pi_S:S\rightarrow M$. This, however is not a natural bundle (to my knowledge), and I do not know if there is any "canonical" way to lift diffeomorphisms into $\text{Aut}(S)$.

Since general relativity heavily involves diffeomorphism-freedom, it is extremely important to be able to do that. In particular, I have no idea how to define the *stress-energy tensor* of a spinor field without representing diffeomorphisms on $S$ somehow.

This situation is further confusing me, since so far I have been an ardent defender of the viewpoint that in the usual local tensor calculus-based formalism, there is no essential difference between "active diffeos" (point transformations) and "passive diffeos" (coordinate transformations).

However the behaviour of a "traditional" spinor field under a coordinate transformation is simple and clear, a spinor field transforms as a scalar under coordinate transformations, and "as a spinor" under changes of orthonormal frames.

However in the modern, invariant viewpoint, one cannot afford this approach. For example, if at $x\in M$, one is given a spinor $\psi\in S_x$ and a vector $v\in T_xM$, and one considers the tensor product $\psi\otimes v\in S_x\otimes T_xM$, then a diffeo will move $v$ to $T\phi(v)\in T_{\phi(x)}M$, but it will not move $\psi$ at all, so this tensor product under a diffeo would become $\psi\otimes T\phi(v)\in S_x\otimes T_{\phi(x)}M$, a product of fibers taken over *different* base points - clearly undesirable.

Is there any agreed-upon method of dealing with the action of spinors under diffeos? If so, is there a simple way of writing it down/stating it?