Notations and conventions

For the velocity of light, we always use the convention $c = 1$, while we write the Planck constant $\hbar$, the gravitational constant $G$ and the fundamental length $\ell$ explicitly. We use rationalized units, namely a factor $4 \pi$ appears in the Coulomb law, but not in the source terms of Maxwell's equations.

The indices $i, j, k, l, m, n, p, q$ take the values $0, 1, 2, 3$ and label, for instance, the anholonomic components of the $SO(1, 3)$ tensors. The indices $\lambda, \mu, \nu, \sigma, \tau$ take the same values and label the local coordinates in the spacetime $\mathcal{M}$ and the holonomic tensor components. The indices $r, s, t$ take the values $1, 2, 3$. The indices $u, v, w, x, y, z$ take the values $0, 1, 2, 3, 4$ and appear in the tensor calculus of the anti-de Sitter group $SO(2, 3)$. When explicitly stated, they also take the value $5$ and are used in the tensor calculus of the group $SO(3,3)$.

The indices $\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta$ take the values $0,\ldots, 9 + n$, where $n$ is the dimension of the internal gauge group. If $n > 0$, the indices $a, b, c, d$ take the values $10,\ldots, 9 + n$ and label a basis in the Lie algebra of the internal gauge group. If $n = 1$, the index $10$, which labels the generator of the electromagnetic gauge transformations, is replaced by the symbol $\bullet$ for typographical reasons.

The summation over a pair of repeated, upper and lower, indices is understood unless a different indication is given.

The Minkowskian metric tensor $g_{ik} = g^{ik}$ is diagonal and we put $g_{00} = -1$, $g_{rs} = \delta_{rs}$. This convention, different from the one adopted in the preceding articles, is particularly convenient in connection with Hamiltonian mechanics. If we indicate by $p_i$ the generators of the active spacetime translations, conjugate to the Minkowskian spacetime coordinates $x^i$, the three quantities $p^r = p_r$ are the momentum components and $p^0 = - p_0$ is the generator of the passive time translations, namely the energy. Another advantage is that the Dirac $\gamma$ matrices in the Majorana representation are real.

We indicate by $\epsilon$ the antisymmetric Levi-Civita symbol, with the normalization $\epsilon_{0123} = - \epsilon^{0123} = 1$ in relativistic tensor calculus and $\epsilon_{1234} = \epsilon^{1234} = 1$ in Dirac spinor calculus.

For the sign conventions concerning the Riemann curvature tensor, we follow ref. [28], where the conventions used by other authors are also discussed.

The three-dimensional vectors are indicated by bold-face letters. The scalar product, the vector product and the norm are represented respectively by $\mathbf{u} \cdot \mathbf{v}$, $\mathbf{u} \times \mathbf{v}$ and $\Vert\mathbf{u}\Vert$. If $u = (u^0, \mathbf{u})$ and $v = (v^0, \mathbf{v})$ are four-vectors, we write their scalar product in the form $u \cdot v = g_{ik} u^i v^k = \mathbf{u} \cdot \mathbf{v} - u^0 v^0$.

The indices of Dirac spinors and $\gamma$ matrices are usually understood, since we use a matrix notation. When it is necessary, we use for them the capital letters $A, B, C, D,\ldots$ that take the values $1,\ldots, 4$. In a similar way, the components of the nongeometric fields are represented by a one-column matrix and when necessary they are labelled by the capital letters $U, V, W$.

Modifying a convention used in the preceding articles, we assume that the structural group (for instance the Lorentz group) has a right action on the principal bundle, in agreement with the majority of the textbooks of differential geometry. The elements of the structural group act on the local frames, namely they have a passive interpretation. When they operates on the observables, namely they are considered from the active point of view, they have a left action, as it is usually assumed.

We use italic fonts to indicate important concepts that appear for the first time and can be found in the Index.