Covariant derivatives and spin connection

If we consider the anholonomic components $V^j$ of a vector field carrying a charge $Ze$, by means of the useful formula

$$\frac{\partial e_{\rho}^k}{\partial e^{\mu}_j} = - e_{\mu}^k e_{\rho}^j,$$ (1.41)

we obtain

$$A_i V^k = e_i^{\lambda} e^k_{\mu} \left(\frac{\partial V^{\m... ...}^{\mu} V^{\nu} - i Z e a_{\lambda}^{\bullet} V^{\mu} \right),$$ (1.42)

namely the anholonomic components of the covariant derivatives of $V$. This result was expected, if one remembers the meaning of the vector fields $A_i$, and a similar result holds for the anholonomic components of any tensor or spinor field, also with nontrivial transformation properties under the gauge group $\mathcal{G}$. One may consider the differential operator $A_i$ as the covariant derivative in the direction of $e_i$.

For some applications one needs an explicit expression of the kind (1.42) also in the general case. If spinor fields are involved, one has to introduce, besides a local coordinate system in $\mathcal{M}$, a tetrad field [40], namely to assign a tetrad $e_i(x)$ to the points of a region of spacetime. If one also assigns to every point a choice of the gauge, one obtains a local section $x \to s(x)$ of the fiber bundle $\mathcal{S}_n$. The existence of a global section is not assured. Of course, $\pi(s(x)) = x$.

The group element $(\Lambda, g) \in \mathcal{L} \times \mathcal{G}$ represent the element of the structural group that transforms $s(x)$ into $s$ and we have

$$e_i = e_k(x) \Lambda^k{}_i, \qquad \Lambda^k{}_i = e^k_{\mu}(x) e_i^{\mu},$$ (1.43)

$$\Phi(s) = \Sigma(\Lambda^{-1}) \Sigma(g^{-1}) \Phi(x), \qquad \Phi(x) = \Phi(s(x)).$$ (1.44)

The variables $(x^{\lambda}, \Lambda^k{}_i, g)$ provide a new parametrization of the elements $s \in \mathcal{S}$ and, by means of the formulas

$$\left(\frac{\partial}{\partial x^{\lambda}} \right)_e = \le... ...mu}_j} = e_{\mu}^k(x) \frac{\partial}{\partial \Lambda^k{}_j},$$ (1.45)

we obtain from eq. (1.31)

$$A_i = e_i^{\lambda} \left(\left(\frac{\partial}{\partial x^{... ...artial}{\partial \Lambda^j{}_l} + a_{\lambda}^a A^L_a \right).$$ (1.46)

We have introduced the quantities

$$\Gamma_{k \lambda}^j(x) = e^j_{\mu}(x) e_k^{\nu}(x) \Gamma_{... ..._{\mu}(x) \frac{\partial e_k^{\mu}(x)}{\partial x^{\lambda}},$$ (1.47)

namely the connection coefficients in the anholonomic basis, also called the spin connection coefficients. Note that the connection coefficients do not transform as the components of a tensor under a change of the basis in the tangent spaces. It follows from the metricity condition (1.30) that

$$\Gamma_{i \lambda}^k g^{ij} = \Gamma_{\lambda}^{kj} = - \Gamma_{\lambda}^{jk} = \Gamma_{\lambda}^{[kj]}.$$ (1.48)

By means of the formulas

$$\Lambda^k{}_l \frac{\partial \Sigma(\Lambda^{-1})}{\partial ... ...ji]}, \qquad A^L_a \Sigma(g^{-1}) = - \Sigma(g^{-1}) \Sigma_a,$$ (1.49)

we finally obtain

$$A_i \Psi = \Sigma(\Lambda^{-1}) \Sigma(g^{-1}) e_i^{\lambda}... ...Sigma_{[jk]} \Psi(x) - a_{\lambda}^a \Sigma_a \Psi(x) \right).$$ (1.50)

The expression in the parenthesis is the covariant derivative on the section $s(x)$ and the preceding factors transform it into the covariant derivative in the direction of $e_i$ at a generic point $s$.

From the commutation relation (1.33), we see that the covariant derivative has the correct Lorentz transformation property, namely

$$A_{[ik]} A_j \Psi = \hat F_{[ik]j}^l A_l \Psi - \Sigma_{[ik]} A_j \Psi.$$ (1.51)