The spinor representation of the structure coefficients

All the geometric quantities introduced in Section 1.7 can be written in spinor form by means of a change of basis in the spaces $\mathcal{T}$ and $\mathcal{T}^*$. For instance we put

$$A_{(AB)} = \breve\Gamma_{(AB)}^{\alpha} A_{\alpha}, \qquad \omega^{(AB)} = \Gamma^{(AB)}_{\alpha} \omega^{\alpha},$$ (3.84)

$$F_{(AB)(CD)}^{(EF)} = \breve\Gamma_{(AB)}^{\alpha} \Gamma^{(... ...\gamma} \breve\Gamma_{(CD)}^{\beta} F_{\alpha \beta}^{\gamma}.$$ (3.85)

We enclose a symmetric pair of indices into round brackets to indicate that they replace a greek index.

The structure coefficients for a spacetime with constant curvature, given by eqs. (1.12), (1.36) and (1.78) take the form

$$\hat F_{(AB)(CD)}^{(EF)} = 4(3 - \ell^2 \rho)\,\delta_{(A}^{(E}\,C_{B)(C}\, \delta_{D)}^{F)}$$

$$-4(1 + \ell^2 \rho)\,(\gamma_5)^{(E}{}_{(A}\,G_{B)(C}\,\delta_{D)}^{F)}$$

$$-4(1 + \ell^2 \rho)\,\delta_{(A}^{(E}\,G_{B)(C}\,(\gamma_5)^{F)}{}_{D)}$$

$$+4(1 + \ell^2 \rho)\,(\gamma_5)^{(E}{}_{(A}{}\, C_{B)(C}\, (\gamma_5)^{F)}{}_{D)}.$$ (3.86)

where the round brackets indicate the symmetrization of the enclosed indices.

This expression contains the spinors $C$ and $\gamma_5$ and it is not invariant with respect to the whole group $GL(4, \mathbf{R})$ and not even with respect to its symplectic subgroups. It is invariant with respect to a subgroup isomorphic to $SL(2, \mathbf{C})$, namely it is Lorentz invariant, as it is evident from the original definition given by eqs. (1.12), (1.36) and (1.78). Actually, $\gamma_5$ appears an even number of times and the structure constants are also invariant under space reflection. Their sign changes under vertical reflection.

For $\rho = - \ell^{-2}$, $\gamma_5$ disappears from the expression (3.86) and the structure constants are invariant with respect to the vector symplectic subgroup $Sp(4, R)_V$ locally isomorphic to an anti-de Sitter subgroup. These structure constants describe a space-time with a very large constant negative curvature. It has no physical relevance, not even as a highly unstable inflationary vacuum, due to the wrong sign of the curvature $R = 12 \rho$.

The contractions of the expressions (3.86) with respect to one or two pairs of indices are given by the useful formulas

$$\hat F_{(AB)(CF)}^{(EF)} = 6(3 - \ell^2 \rho) \, \delta_{(A... ..._{B)C} - 6 (1 + \ell^2 \rho) \, (\gamma_5)^E{}_{(A} G_{B)C},$$ (3.87)

$$\hat F_{(AE)(FC)}^{(EF)} = 12(4 - \ell^2 \rho) \, C_{AC}.$$ (3.88)

An interestig linear function of the structure coefficients is the antisymmetric spinor

$$48 t_{AB} = F_{(AC)(DB)}^{(CD)},$$ (3.89)

which can be written in matrix form as

$$48 t = \ell C \gamma^i \gamma_m \gamma^j F_{ij}^m + 2^{-1} \ell^2 C \gamma^i \gamma_m \gamma_n \gamma^j F_{ij}^{[mn]}$$

$$- 2^{-1} C (\gamma^i \gamma^k \gamma_m \gamma^j - \gamma^j \gamma_m \gamma^i \gamma^k) F_{[ik]j}^m$$

$$- 2^{-2} \ell C (\gamma^i \gamma^k \gamma_m \gamma_n \gamma^j - \gamma^j \gamma_m \gamma_n \gamma^i \gamma^k) F_{[ik]j}^{[mn]}$$

$$+ 2^{-2} \ell^{-1} C \gamma^i \gamma^k \gamma_m \gamma^j \gamma^l F_... ...3} C \gamma^i \gamma^k \gamma_m \gamma_n \gamma^j \gamma^l F_{[ik][jl]}^{[mn]}.$$ (3.90)

From eq. (3.50), by computing the traces, we can obtain the components of the corresponding 6-dimensional vector $t_u$, which are rather long expressions. In the case of a spacetime with constant curvature we have

$$t_i = 0, \qquad t_4 = 1 - (48)^{-1} \ell^2 R, \qquad t_5 = 0.$$ (3.91)

In the general case, we prefer to use a different method based on the 6 and 5-dimensional tensor calculus. The component $t_5$ is invariant under $SO(2, 3)_A$. The structure coefficients are the components of a 5-dimensional tensor of the kind $F_{[uv5][wx5]}^{[yz5]}$ and, by contraction of the indices, one can obtain only one linear $SO(2, 3)_A$ invariant, namely we have

$$48 t_5 = g^{vw} F_{[xv5][wy5]}^{[xy5]}.$$ (3.92)

The choice of the numerical coefficient will soon be justified.

In a similar way, one can define the following 5-dimensional vectors depending linearly on the structure coefficients:

$$48 f_u = - \epsilon_{yz}{}^{vwx}F_{[uv5][wx5]}^{[yz5]},$$

$$48 f'_u = \epsilon_{uy}{}^{vwx} F_{[vw5][xz5]}^{[yz5]},$$

$$48 f''_u = \epsilon_{uxy}{}^{vw} g^{zz'} F_{[vz5][z'w5]}^{[xy5]}, \$$ (3.93)

where $\epsilon^{uvwxy}$ is the 5-dimensional completely antisymmetric tensor. We have

$$\epsilon_{yzvwx} F_{[uv'5][w'x'5]}^{[yz5]} g^{vv'} g^{ww'} g^{xx'} - 2 \epsilon_{yuvwx} F_{[zv'5][w'x'5]}^{[yz5]} g^{vv'} g^{ww'} g^{xx'}$$

$$- 2 \epsilon_{yzvwu} F_{[xv'5][w'x'5]}^{[yz5]} g^{vv'} g^{ww'} g^{xx'} = 0,$$ (3.94)

because this expression has been antisymmetrized with respect to the 6 indices $u, y, z, v, w, x$, which can take only 5 values. In this way we obtain

$$- f_u + 2f'_u - 2 f''_u = 0$$ (3.95)

and we see that there are only two independent 5-dimensional vectors that are linear functions of the structure coefficients.

The other components $t_u$ with $u = 0,\ldots, 4$ can be determined starting from $t_5$ by means of the transformation property

$$48 \zeta_5{}^u t_u = 48 \delta t_5 = g^{vw} \zeta_5{}^u F_{[xvu][wy5]}^{[xy5]}$$

$$+ g^{vw} \zeta_5{}^u F_{[xv5][wyu]}^{[xy5]} + g^{vw} \zeta^5{}_u F_{[xv5][wy5]}^{[xyu]}.$$ (3.96)

Since the infinitesimal parameters $\zeta_5{}^u$ are arbitrary, we obtain, also using eq. (3.83),

$$t_u = f'_u - 2^{-1} f''_u = 2^{-2} f_u + 2^{-1} f'_u.$$ (3.97)

The quantities defined above can be expressed as 4-dimensional tensors by means of eqs. (3.68) and (3.69) and we obtain

$$48 t_i = \ell \epsilon_{il}{}^{jk} F_{jk}^l + 2 \ell \epsilon_{im}{}... ...km}{}_{jl} F_{[ik]m}^{[jl]} - 2^{-2} \ell \epsilon^{jk}{}_{lm} F_{[jk]i}^{[lm]}$$

$$- 2^{-1} \ell \epsilon_{im}{}^{jl} F_{[jl]k}^{[km]} - 2^{-1} \ell^{-... ...l}{}_{m} F_{[ik][jl]}^m + 2^{-1} \ell^{-1} \epsilon_{i}{}^{jlm} F_{[jl][km]}^k,$$

$$48 t_4 = \ell^2 F_{ik}^{[ik]} + 2 g^{kj} F_{[ik]j}^i + g^{kj} F_{[ik][jl]}^{[il]},$$

$$48 t_5 = - 2^{-1} \ell^2 \epsilon^{ik}{}_{jl} F_{ik}^{[jl]} - \epsilon^{ikj}{}_{l} F_{[ik]j}^l$$

$$- 2^{-1} \epsilon_{mn}{}^{il} g^{kj} F_{[ik][jl]}^{[mn]} - \epsilon_n{}^{kjl} F_{[ik][jl]}^{[in]}.$$ (3.98)

$$48 f_i = - \ell \epsilon^{jk}{}_{lm} F_{[jk]i}^{[lm]} - 2 \ell \epsi... ...{im}{}^{jl} F_{[jl]k}^{[km]} + 2 \ell^{-1} \epsilon_{i}{}^{jlm} F_{[jl][km]}^k,$$

$$48 f_4 = 4 \ell^2 F_{ik}^{[ik]} + 4 g^{kj} F_{[ik]j}^i,$$ (3.99)

In particular, if we consider a bundle of frames, for instance a solution of the Einstein-Cartan equations, we have

$$t_i = (48)^{-1} \ell \epsilon_{il}{}^{jk} F_{jk}^l, \qquad t_4 = 1 - (48)^{-1} \ell^2 R, \qquad t_5 = 0,$$ (3.100)

$$f_i = 0, \qquad f_4 = 1 - (12)^{-1} \ell^2 R.$$ (3.101)

In conclusion, the 5-vectors obtained linearly from the structure coefficients are linear cobinations of $t_u$ and $f_u$. The first 5-vector, together with $t_5$ forms a 6-vector and the second one, which was introduced in ref. [6], is characterized by the property that it does not depend on the torsion.

We have seen that, if the geometric symmetry group $\mathcal{F}^G$ of the field equations is larger than the orthochronous Lorentz group, the vacuum solution has a lower symmetry, namely we have a spontaneous symmetry breaking. In Chapter 7 we shall consider field equations symmetric with respect to the axial symplectic group $Sp(4, \mathbf{R})_A$ and the symmetry breaking can be attributed to a nonvanishing asymptotic value of the 5-vector $f_u$, which plays a role similar to the role played by the Higgs field in the Standard Model of elementary particles [41].