Tetrad formalism and fields with half integer spin

Question

The main aim of the tetrad formalism is to apply action principle in general theory of relativity. But why to incorporate general relativity with field theory of particle with half integer spin?

Answer 1

1. On one hand, matter fields with integer spin can be coupled to gravity via the metric tensor (and the Levi-Civita connection) without using a vielbein (aka. a vierbein or a tetrad in 4D).

2. On the other hand, matter fields with half integer spin needs a vielbein in order to interact with gravity in geometrically covariant manner. Geometrically, a vielbein provides us with an pseudo-orthonormal frame in each spacetime point.

   The problem is that a field with half spin is a spinor, and hence carries a spinor index. To write down a covariant kinetic term for a spinor, we need a covariant derivative of a spinor, and hence a spin connection. We also need a representation of the Clifford algebra, i.e. gamma matrices. Furthermore, to convert between curved and flat indices on the gamma matrices, we need a vielbein. See e.g. eq. (1) in my Phys.SE answer here for explicit formulas.

   There is a similar story for fields with higher half integer spin, such as, e.g., a Rarita-Schwinger spin $\frac{3}{2}$ field.

Answer 2

The main aim of the tetrad formalism is to apply action principle in general theory of relativity.

Well, the action principle in General Relativity doesn't need the tetrad formalism. This is a physical and mathematical byproduct of the principle of local covariance, i.e. in the neighbourhood of any point of the curved spacetime the laws of special relativity apply. In other words, for small curvature, we can sometimes consider the metric tensor as a perturbation around the Minkowski spacetime. This entails the possibility to use spinors locally as explained below.

But why to incorporate general relativity with field theory of particle with half integer spin?

In order to learn the theory of https://en.wikipedia.org/wiki/Supergravity, one needs two steps. The first is to "deform" GR into a semiclassical theory, i.e. allow the possibility that the matter in GR be also fermionic (via the regular "classical QFT"). For this, one needs to employ the principle of local covariance to allow a passage from curved spacetime to flat neighbourhoods. In this way "classical" spinors (Dirac or Weyl) can enter the picture and one can study their dynamics on curved spacetimes. The second step is to allow supersymmetry, i.e. the possibility that the local Poincaré/Lorentz symmetry allowed by the local covariance be extended to a generic supersymmetry algebra.