Infinitesimal Lorentz transformations

For many purposes, it is useful to consider infinitesimal Lorentz transformations of the kind

$$\Lambda^i{}_k \sim \delta^i_k + \zeta^i{}_k, \qquad \zeta^{ik} = - \zeta^{ki}.$$ (1.7)

Their action on $\mathcal{S}$ is generated by the vector fields (or differential operators) $A_{[ik]}$ defined by

$$f(s \Lambda) \sim f(s) + 2^{-1} \zeta^{[ik]} A_{[ik]} f(s),$$ (1.8)

where $f$ is an arbitrary differentiable function and $\Lambda$ is given by eq. (1.7).

These vector fields are tangent to the fibers, which are diffeomorhic to $\mathcal{L}$. They can be considered as generators of right translations of the group $\mathcal{L}$ and they define a basis of its Lie algebra $o(1, 3)$. Here and in the following, in order to obtain more readable formulas, we always enclose into square brackets the pairs of antisymmetric indices which label the elements of this basis.

The sign of the parameters $\zeta^{[ik]}$ depends on the choice of the metric $g_{ik}$. With our choice (see sect. 0.2), for instance, $\zeta^{[21]} = \zeta^2{}_1$ describes a counter-clockwise rotation around $e_3$ and $\zeta^{[03]} = \zeta^0{}_3$ describes a boost along $e_3$.

If we introduce in an open region of $\mathcal{M}$ a set of local coordinates $x^{\mu}$, we can parametrize locally $\mathcal{S}$ by means of the redundant coordinates $(x^{\mu}, e^{\mu}_i)$ constrained by eq. (1.1). Then we can write the explicit formula

$$A_{[ik]} = (g_{kj} e^{\mu}_i - g_{ij} e^{\mu}_k) \frac{\partial}{\partial e^{\mu}_j}.$$ (1.9)

One has to be careful in dealing with partial derivatives with respect to variables which are not independent. First we define the vector fields (1.9) in the space of the unconstrained variables and then we check that they are tangent to the manifold defined by eq. (1.1) namely that, on the same manifold, we have

$$A_{[ik]} (g_{\mu \nu} e^{\mu}_j e^{\nu}_l) = 0.$$ (1.10)

By computing the commutator we obtain

$$[A_{[ik]}, A_{[jl]}] = 2^{-1} \hat F_{[ik][jl]}^{[mn]} A_{[mn]},$$ (1.11)

where the quantities

$$\hat F_{[ik][jl]}^{[mn]} = \delta_i^m g_{kj} \delta_l^n - \delta_k^m g_{ij} \delta_l^n - \delta_i^m g_{kl} \delta_j^n + \delta_k^m g_{il} \delta_j^n$$

$$- \delta_i^n g_{kj} \delta_l^m + \delta_k^n g_{ij} \delta_l^m + \delta_i^n g_{kl} \delta_j^m - \delta_k^n g_{il} \delta_j^m$$ (1.12)

are the structure constants of the Lorentz Lie algebra.

The behavior of tensor and spinor fields under Lorentz transformations, described in Section 1.2 can be written as a differential equations. For the scalar and vector fields we obtain

$$A_{[ik]} S = 0, \qquad A_{[ik]} V^j = - (\delta^j_i g_{kl} - \delta^j_k g_{il}) V^l$$ (1.13)

and in general, if we write the components of a tensor or spinor in the form of a one-column matrix $\Psi$, we have

$$A_{[ik]} \Psi = - \Sigma_{[ik]} \Psi.$$ (1.14)

The square matrices $\Sigma_{[ik]}$ are defined by

$$\Sigma(\Lambda) \sim 1 + 2^{-1} \zeta^{[ik]} \Sigma_{[ik]},$$ (1.15)

where $\Lambda$ is given by eq. (1.7).

They form a representation of the Lie algebra $o(1, 3)$ of $\mathcal{L}$, namely we have

$$[\Sigma_{[ik]}, \Sigma_{[jl]}] = 2^{-1} \hat F_{[ik][jl]}^{[mn]} \Sigma_{[mn]}.$$ (1.16)

For a Dirac spinor we have

$$\Sigma_{[ik]} = 2^{-2}(\gamma_i \gamma_k - \gamma_k \gamma_i).$$ (1.17)

$$\Sigma_{[ik]} \gamma_j - \gamma_j \Sigma_{[ik]} = g_{kj} \gamma_i - g_{ij} \gamma_k.$$ (1.18)

The properties of the $\gamma$-matrices are summarized in Section 0.3.