A compact formalism

In the present Section we introduce some notations that permit us to write the formulas of the preceding Sections in a very compact form. In this way we simplify some calculations, but, at this stage, we do not introduce any new mathematical or physical idea. We also write all the relevant formulas using the concepts of differential geometry that do not refer to a particular local coordinate system or to a local section of the fiber bundle. In the following Chapter 2 we shall give a different, more general, interpretation of this compact formalism and use it to introduce new physical ideas.

We have seen that the (possibly extended) bundle of Lorent frames is a manifold $\mathcal{S}_n$ with dimension $10 + n$, where $n$ is the dimension of an internal gauge group $\mathcal{G}$. We have also seen that its most important geometric properties are described by the vector fields $A_{i}$, $A_{[ik]}$ and, if $n > 0,$ $A_a$. We use for all these fields a unified notation $A_{\alpha}$, where $\alpha$ takes the values $0,\ldots, 9 + n$.

More precisely, the fields $A_0,\ldots, A_3$ generate parallel displacements of the tetrads along the directions of the tetrad vectors, $A_4 = A_{[32]}$, $A_5 = A_{[13]}$, $A_6 = A_{[21]}$ generate rotations around the spatial vectors of the tetrad, $A_7 = A_{[01]}$, $A_8 = A_{[02]}$, $A_9 = A_{[03]}$ generate Lorentz boosts along the same spatial vectors and $A_{10},\ldots, A_{9 + n}$ generate the infinitesimal transformation of the internal gauge group. If $n = 1$, in order to avoid a two digits index, we write $A_{\bullet}$ instead of $A_{10}$.

The vectors $A_{\alpha}(s)$, $\alpha = 0,\ldots, 9 + n$ are linearly independent and they provide a basis in every tangent space $T_s \mathcal{S}_n$, $s \in \mathcal{S}_n$. By means of this basis, one can identify in a natural way all the tangent spaces $T_s \mathcal{S}_n$ with a single $(10 + n)$-dimensional vector space $\mathcal{T}_n$.

The subspace of $T_s \mathcal{S}_n$ generated by the vectors $A_0(s),\ldots, A_3(s)$ is called the horizontal subpace, while the subspace generated by the vectors $A_4(s),\ldots,$ $A_{9 + n}(s)$ is called the vertical subpace. The vertical subspaces are tangent to the fibers, while the horizontal subspaces define a connection in the principal bundle $\mathcal{S}_n$. We also consider these subspaces as subspaces of the vector space $\mathcal{T}_n$ and we indicate them, respectively, by $\mathcal{T}_H$ and $\mathcal{T}_V$. The last subspace can be identified with the Lie algebra of the structural group. If $n > 0$, the vertical subspace $\mathcal{T}_V$ is the direct sum of the subspace $\mathcal{T}_L$ generated by $A_{[ik]}$ and the subspace $\mathcal{T}_I$ generated by $A_a$.

The vector fields $A_{\alpha}$ can be considered as first order differential operators and their commutators (Lie brackets) can be written in the form

$$[A_{\alpha}, A_{\beta}]= F_{\alpha \beta}^{\gamma} A_{\gamma}.$$ (1.52)

The quantities $F_{\alpha \beta}^{\gamma} = - F_{\beta \alpha}^{\gamma}$, are called structure coefficients.

We also introduce in the space $\mathcal{S}_n$ the differential 1-forms $\omega^{\beta}$, dual to the vector fields $A_{\alpha}$, defined by

$$i(A_{\alpha}) \omega^{\beta} = \omega^{\beta}(A_{\alpha}) = \delta_{\alpha}^{\beta},$$ (1.53)

where $i(X)$ is the interior product operator acting on the differential forms. Their exterior derivatives are given by

$$d \omega^{\gamma} = - 2^{-1} F_{\alpha \beta}^{\gamma} \, \omega^{\alpha} \wedge \omega^{\beta}.$$ (1.54)

The exterior products of these 1-forms provide a basis in the space of differential forms of higher degree. We say that a term containing the product of $d_H$ forms of the kind $\omega^i$ has horizontal degree $d_H$, a term containing the product of $d_L$ forms of the kind $\omega^{[ik]}$ has Lorentz vertical degree $d_L$ and a term containing the product of $d_I$ forms of the kind $\omega^a$ has internal degree $d^I$. We use the notation $(d_H, d_L, d_I)$ to describe the partial degrees of a term. The total degree is the sum of the partial degrees. These concepts are very useful in the calculations.

From the Jacobi identity satisfied by the commutators (1.52) or considering the vanishing exterior derivation of eq. (1.54), we find the generalized Jacobi identity

$$A_{\alpha} F_{\beta \gamma}^{\delta} + A_{\beta} F_{\gamma ... ...{\eta \beta}^{\delta} = J_{\alpha \beta \gamma}^{\delta} = 0.$$ (1.55)

We see from eqs. (1.11), (1.19), (1.33), (1.34) and (1.35) that the structure coefficients coincide with the structure constants $\hat F_{\alpha \beta}^{\gamma}$ of the Lie algebra of the extended Poincaré group $\mathcal{P} \times \mathcal{G}$, with the exception of the coefficients $F_{ik}^{\gamma}$, which give the anholonomic components of the torsion, curvature and gauge field strength tensors defined by eqs. (1.37), (1.38) and (1.39).

The generalized Jacobi identity (1.55) represents, in a very compact form, a large number of physically relevant formulas. In particular:

• $J_{ijk}^{a} = 0$ is the homogeneous set of the Maxwell or Yang-Mills equations;
• $J_{ijk}^{[mn]} = 0$ is the Bianchi identity for the curvature;
• $J_{ijk}^{l} = 0$ is the Bianchi identity for the torsion or a symmetry property of the curvature if the torsion vanishes;
• $J_{aik}^{b} = 0$ represents the gauge transformation property of the Maxwell or Yang-Mills field strength;
• $J_{aik}^{[mn]} = 0$ represents the gauge invariance of the curvature;
• $J_{aik}^{l} = 0$ represents the gauge invariance of the torsion.
• $J_{[ij]lk}^{a} = 0$ represents the tensor nature of the Maxwell or Yang-Mills field strength;
• $J_{[ij]lk}^{[mn]} = 0$ represents the tensor nature of the curvature;
• $J_{[ij]lk}^{l} = 0$ represents the tensor nature of the torsion.

For other values of the indices we obtain the Jacobi identity for the structure constants of the extended Poincaré algebra.