Tensor and spinor fields

The anholonomic components of tensor fields on $\mathcal{M}$ are uniquely determined when the frame $s \in \mathcal{S}$ is given and have to be considered as scalar fields on $\mathcal{S}$. They behave in a particular way when $s$ moves on a fiber. For instance, a scalar (on $\mathcal{M}$) field $S$ has the property

$$S(s \Lambda) = S(s), \qquad \Lambda \in \mathcal{L},$$ (1.4)

namely it is constant on the fibers. The anholonomic components $V^i$ of a vector field satisfy the condition

$$V^i(s \Lambda) = V \cdot e^i(s \Lambda) = V \cdot ((\Lambda^{-1})^i{}_k e^k(s)) = (\Lambda^{-1})^i{}_k V^k(s).$$ (1.5)

The inverse matrix $\Lambda^{-1}$ appears because $\Lambda$ is interpreted a passive transformation, namely a change of the frame leaving the vector $V$ unchanged. If, instead, we consider an active transformation, that changes the vector $V$ leaving the frame unchanged, the matrix $\Lambda$ appears in the transformation property.

A more general tensor field is characterized by the condition

$$\Psi(s \Lambda) = \Sigma (\Lambda^{-1}) \Psi(s),$$ (1.6)

where the elements of the one-column matrix $\Psi$ are the anholonomic tensor components and the square matrix $\Sigma$ belongs to a linear representation of $\mathcal{L}$.

A similar formula holds for a spinor, but $\Sigma$ is a two-valued representation and the spinor components too are two-valued functions of $s$. A more rigorous approach is to consider a double covering $\tilde \mathcal{S}$ of $\mathcal{S}$, which is a principal fiber bundle with structural group $\tilde\mathcal{L}$, a double covering of $\mathcal{L}$, which contains $SL(2, \mathbf{C})$ and two elements corresponding to the space inversion [39]. Then the components of the spinor fields are one-valued functions on $\tilde \mathcal{S}$ and $\Sigma$ is a one-valued linear representation of $\tilde\mathcal{L}$. The fiber bundle $\tilde \mathcal{S}$ exists only if $\mathcal{M}$ has suitable topological properties and in this case one says that $\mathcal{M}$ admits a spin structure. The use of tetrads to treat spinor fields on a curved spacetime has been introduced by H. Weyl [40].