$SL(4, \mathbf{R})$ spinors and $SO(3,3)$ tensors

The symmetry group $\mathcal{F}^G$ introduced in Section 2.5 leaves the geometry of $\mathcal{T}$ and in particular the cone $\mathcal{T}^+$ invariant and therefore it must be a subgroup of the 16-dimensional group $GL(4, \mathbf{R})$. In the present Section we study some tensor representations of this group and of its subgroup $SL(4, \mathbf{R})$. In order to avoid confusion with the tensors of the Lorentz group and of other pseudo-orthogonal groups, we use the word spinors.

We have already met the symmetric contravariant spinor $b^{AB}$, which determines an element of the vector space $\mathcal{T}$. In a similar way one can describe the 10-momentum introduced in Section 1.9 by means of a symmetric covariant spinor $p_{AB}$ that transforms according to a different inequivalent representation of $SL(4, \mathbf{R})$. It is given by

$$p = p_{\alpha} \breve\Gamma^{\alpha}, \qquad p_{\alpha} = 2^{-2} \mathrm{Tr}\,(\Gamma_{\alpha} p)$$ (3.47)

and we have

$$b^{\alpha} p_{\alpha} = b^i p_i + 2^{-2} b^{[ik]} p_{[ik]} = 2^{-2} \mathrm{Tr}\,(b p).$$ (3.48)

The determinant of the matrix $p$ which is invariant under $SL(4, \mathbf{R})$, can be written in the form

$$\det p = \left(\ell^2 (p_0)^2 - \ell^2 \Vert\mathbf{p}\Vert^2 - \Ver... ...thbf{p}''\Vert^2 \right)^2 - 4 \ell^2 \Vert\mathbf{p} \times \mathbf{p}'\Vert^2$$

$$- 4 \Vert\mathbf{p}' \times \mathbf{p}''\Vert^2 - 4 \ell^2 \Vert\mat... ...f{p}\Vert^2 - 8 \ell^2 p_0 \, \mathbf{p} \cdot \mathbf{p}' \times \mathbf{p}''&$$ (3.49)

where we have introduced the 3-dimensional vector notation (1.74)

In the following, we shall also meet antisymmetric spinors. A covariant antisymmetric spinor $f_{AB}$ can always be written in the form

$$f = f_u \breve\Theta^u, \qquad f_u = - 2^{-2} \mathrm{Tr}\,(\Theta_u f),$$ (3.50)

where $u = 0,\ldots, 5$. We have introduced the antisymmetric matrices

$$\Theta_i = \gamma_i \gamma_5 C^{-1}, \qquad \Theta_4 = - C^{-1}, \qquad \Theta_5 = - \gamma_5 C^{-1} = - G^{-1}.$$ (3.51)

$$\breve\Theta^u = - (\Theta_u)^{-1}, \qquad \breve\Theta^i = ... ...ad \breve\Theta^4 = C, \qquad \breve\Theta^5 = C \gamma^5 = G.$$ (3.52)

The matrix $\gamma_5$ is defined in Section 0.3. In the Majorana representation it is real and has the property

$$\gamma_5^T = C \gamma_5 C^{-1}.$$ (3.53)

We have

$$- 2^{-2} \mathrm{Tr}\,(\breve\Theta^u \Theta_v) = \delta^u_v.$$ (3.54)

In a similar way, a contravariant antisymmetric spinor $h^{AB}$ can always be written in the form

$$h = h^u \Theta_u, \qquad h^u = - 2^{-2} \mathrm{Tr}\,(\breve \Theta^u h).$$ (3.55)

The group $GL(4, \mathbf{R})$ has two connected components with different signs of $\det a$. In the following we treat with some detail only the connected component of the identity $GL(4, \mathbf{R})_0$. The 15-dimensional subgroup $SL(4, \mathbf{R}) \subset GL(4, \mathbf{R})_0$ is defined by the condition $\det a = 1$ and is connected. The totally antisymmetric spinors $\epsilon^{ABCD}$ and $\epsilon_{ABCD}$ are invariant under this subgroup. They define an invariant (not positive) quadratic form (namely the Pfaffian) in the spaces of the covariant and contravariant antisymmetric spinors and $SL(4, \mathbf{R})$ acts on these spaces by means of pseudo-orthogonal transformations.

By means of eq. (3.18), we obtain

$$2^{-3} \epsilon^{ABCD} \breve\Theta^u_{AB} \breve\Theta^v_{C... ...d 2^{-3} \epsilon_{ABCD} \Theta_u^{AB} \Theta_v^{CD} = g_{uv},$$ (3.56)

$$2^{-3} \epsilon^{ABCD} f_{AB} f_{CD} = g^{uv} f_u f_v, \qquad 2^{-3} \epsilon_{ABCD} h^{AB} h^{CD} = g_{uv} h^u h^v,$$ (3.57)

where $g^{uv} = g_{uv}$ is the diagonal 6-dimensional metric tensor with $g^{11} = g^{22} = g^{33} = 1$, $g^{00} = g^{44} = g^{55} = - 1$. The representation of $SL(4, \mathbf{R})$ in the space of the antisymmetric spinors defines a homomorphism $SL(4, \mathbf{R}) \to O(3,3)$. Since both these Lie groups have dimension 15, this is a local isomorphism and the Lie algebras $sl(4, \mathbf{R})$ and $o(3,3)$ are isomorphic. The group $SL(4, \mathbf{R})$ is mapped onto the connected component of the identity $SO(3,3)_0 \subset SO(3,3)$ and the kernel of the homomorphism is composed of the two matrices $\pm 1$.

We can write the infinitesimal transformations of $SL(4, \mathbf{R})$ in the form

$$a \sim 1 + 2^{-1} \zeta^{[uv]} \Sigma_{[uv]}, \qquad u, v = 0,\ldots, 5,$$ (3.58)

where the $4 \times 4$ real matrices $\Sigma_{[ik]}$ are given by eq. (1.17) and

$$\Sigma_{[i4]} = - \Sigma_{[4i]} = 2^{-1} \gamma_i \gamma_5, \qquad \Sigma_{[i5]} = - \Sigma_{[5i]} = 2^{-1} \gamma_i,$$

$$\Sigma_{[45]} = - \Sigma_{[54]} = 2^{-1} \gamma_5.$$ (3.59)

They provide a basis for the Lie algebra $sl(4, \mathbf{R}) = so(3, 3)$ and satisfy the commutation relations

$$[\Sigma_{[uv]}, \Sigma_{[xy]}] = g_{vx} \Sigma_{[uy]} - g_{ux} \Sigma_{[vy]} - g_{vy} \Sigma_{[ux]} + g_{uy} \Sigma_{[vx]}.$$ (3.60)

Note that the elements of the Lie algebra, on which the adjoint representation of the group operates, can be represented by traceless mixed rank two spinors or by antisymmetric rank two tensors.

The infinitesimal transformations of Dirac spinors and of covariant and contravariant antisymmetric spinors are given by

$$\delta \Psi = 2^{-1} \zeta^{[uv]} \Sigma_{[uv]} \Psi,$$ (3.61)

$$\delta f = - 2^{-1} \zeta^{[uv]} (\Sigma^T_{[uv]} f + f \Sig... ...h = 2^{-1} \zeta^{[uv]} (\Sigma_{[uv]} h + h \Sigma^T_{[uv]}).$$ (3.62)

One can easily check the formulas

$$\Sigma_{[uv]} \Theta_w + \Theta_w \Sigma^T_{[uv]} = g_{vw} \Theta_u - g_{uw} \Theta_v,$$

$$\Sigma^T_{[uv]} \breve\Theta^w + \breve\Theta^w \Sigma_{[uv]} = \delta_u^w \breve\Theta_v - \delta_v^w \breve\Theta_u$$ (3.63)

and from eqs. (3.50) and (3.55) one obtains the expected 6-vector infinitesimal transformations

$$\delta f_u = \zeta_u{}^v f_v, \qquad \delta h^u = \zeta^u{}_v h^v.$$ (3.64)

The infinitesimal transformations of covariant and contravariant symmetric spinors are

$$\delta p = - 2^{-1} \zeta^{[uv]} (\Sigma^T_{[uv]} p + p \Sig... ...b = 2^{-1} \zeta^{[uv]} (\Sigma_{[uv]} b + b \Sigma^T_{[uv]}).$$ (3.65)

In order to introduce the corresponding $SO(3,3)$ tensors, we define the quantities

$$p_{[uvw]} = 2^{-2} \mathrm{Tr}\,(\Theta_u\ \breve \Theta_v \... ...2} \mathrm{Tr}\,(\breve \Theta^u\ \Theta^v \breve \Theta^w b).$$ (3.66)

It is easy to show that they transform as tensors of rank 3, namely

$$\delta p_{[uvw]} = \zeta_u{}^x f_{[xvw]} + \zeta_v{}^x f_{[uxw]} + \zeta_w{}^x f_{[uvx]},$$

$$\delta b^{[uvw]} = \zeta^u{}_x b^{[xvw]} + \zeta^v{}_x b^{[uxw]} + \zeta^w{}_x b^{[uvx]}.$$ (3.67)

By computing the traces, we find that, as it is suggested by the notation, these quantities are antisymmetric with respect to their three indices and are given by

$$p_{[i45]} = - \ell p_i \qquad p_{[ijk]} = - \ell \epsilon_{ijk}{}^l p_l,$$

$$p_{[ik4]} = p_{[ik]}, \qquad p_{[ik5]} = 2^{-1} \epsilon_{ik}{}^{jl} p_{[jl]},$$ (3.68)

$$b^{[i45]} = - \ell^{-1} b^i \qquad b^{[ijk]} = \ell^{-1} \epsilon^{ijk}{}_l b^l,$$

$$b^{[ik4]} = b^{[ik]}, \qquad b^{[ik5]} = - 2^{-1} \epsilon^{ik}{}_{jl} b^{[jl]}.$$ (3.69)

We enclose an antisymmetric set of three indices into square brackets to indicate that they replace a greek index that labels a basis in the space $\mathcal{T}^*$ or $\mathcal{T}$. In a similar way we define the vector fields $A_{[uvw]}$ and the 1-forms $\omega^{[uvw]}$.

The general 6-dimensional antisymmetric tensors of rank 3 have 20 independent components and, in order to describe the vectors of a 10-dimensional space, they must satisfy some constraint. In fact, from the formulas given above, we have

$$6^{-1}\epsilon_{uvw}{}^{xyz} p_{[xyz]} = p_{[uvw]}, \qquad 6^{-1}\epsilon^{uvw}{}_{xyz} b^{[xyz]} = - b^{[uvw]}.$$ (3.70)

This means that these tensors are respectively self-dual and anti-self-dual.

In terms of the usual Lorentz components, the transformation formulas (3.67) take the form

$$\delta p_i = \zeta_i{}^j p_j + \ell^{-1} \zeta^{[k5]} p_{[ik]} - 2^{-1} \ell^{-1} \zeta^{[j4]} \epsilon_{ij}{}^{kl} p_{[kl]},$$

$$\delta p_{[ik]} = \zeta_i{}^j p_{[jk]} + \zeta_k{}^j p_{[ij]} + \ell (\zeta_{[i5]} p_k - \zeta_{[k5]} p_i)$$

$$- \ell \zeta^{[j4]} \epsilon_{ikj}{}^{l} p_l - 2^{-1} \zeta^{[45]} \epsilon_{ik}{}^{jl} p_{[jl]}$$ (3.71)

$$\delta b^i = \zeta^i{}_j b^j + \ell \zeta_{[k5]} b^{[ik]} - 2^{-1} \ell \zeta^{[j4]} \epsilon^{i}{}_{jkl} b^{[kl]},$$

$$\delta b^{[ik]} = \zeta^i{}_j b^{[jk]} + \zeta^k{}_j b^{[ij]} + \ell^{-1} (\zeta^{[i5]} b^k - \zeta^{[k5]} b^i)$$

$$- \ell^{-1} \zeta^{[j4]} \epsilon^{ik}{}_{jl} b^l + 2^{-1} \zeta^{[45]} \epsilon^{ik}{}_{jl} b^{[jl]}.$$ (3.72)

These formulas can be interpreted as the transformations of the components corresponding to the infinitesimal transformations of the basis vectors $A_{\alpha}$ and $\omega^{\alpha}$ in the vector spaces $\mathcal{T}$ and $\mathcal{T}^*$. It follows that $A_{\alpha}$ and $\omega^{\alpha}$ transform, respectively, in the same way as the components $p_{\alpha}$ and $b^{\alpha}$.