The basic geometric and topological stucture of the space $\mathcal{S}$

In Chapter 1 the bundle $\mathcal{S}$ of Lorentz frames has been introduced as a useful auxiliary tool, while the spacetime $\mathcal{M}$ was considered as the basic geometric concept of physics. In the following, a theory based on a geometry of this kind is called a normal theory or, more exactly, a theory with a normal geometry. However, there are several arguments that suggest a change of perspective, namely that $\mathcal{S}$ has a more direct physical interpretation, while $\mathcal{M}$ should be considered as a mathematical construction, possibly justified only under some assumptions, which might have an approximate character.

In the presentation of these ideas we shall follow essentially refs. [3,4,5]. Similar points of views, with various motivations, have been presented by various authors [49,50,51,52,53,54]. Some of them are discussed with some detail in the follwing Sections.

A first simple argument in favour of the new point of view is that many physical observables have a vector or tensor nature and take a definite value only when a local frame, namely a point of $\mathcal{S}$ is given. A point $x \in \mathcal{M}$ does not convey enough information. Other arguments for choosing $\mathcal{S}$ as the geometric arena of physical theories will be given in the Sections 2.2 and 2.4.

We propose to drop the assumption that $\mathcal{S}$ is a principal fiber bundle with a connection and to endow it with a simpler structure suggested by the treatment given in Section 1.7 and justified by the operational discussion of Section 2.2. Further assumptions will be added when necessary.

More precisely, we describe the geometric background of a physical theory (including the gauge fields) by means of the $(10 + n)$-dimensional differentiable manifold $\mathcal{S}_n$ and the $10 + n$ differentiable vector fields $A_{\alpha}$ linearly independent at all the points $s \in \mathcal{S}_n$. The physical meaning of these fields given in Section 1.7 is only a useful suggestion. In Section 6.1 a slightly modified interpretation is suggested.

As we have remarked in Section 1.7, the vector fields $A_{\alpha}$ permit us to identify all the tangent spaces $T_s \mathcal{S}_n$ with a single $(10 + n)$-dimensional vector space $\mathcal{T}_n$, namely they define on $\mathcal{S}_n$ a structure called absolute parallelism or teleparallelism. It can be described as a trivialization of the tangent bundle, namely a diffeomorphism between $T \mathcal{S}_n$ and $\mathcal{S}_n \times \mathcal{T}_n$

Adopting a more clear point of view, we define $\mathcal{T}_n$ as the vector space composed of all the vector fields of the form $A = b^{\alpha} A_{\alpha}$, where the coefficients $b^{\alpha}$ are constant. Note that all the vector fields on $\mathcal{S}_n$ can be written in the form $b^{\alpha}(s) A_{\alpha}(s)$ with variable $b^{\alpha}(s)$ and that the elements of $\mathcal{T}_n$ are vector fields of a particular kind, that we call fundamental vector fields. If $A \in \mathcal{T}_n$ is a fundamental vector field we have $A(s) \in T_s \mathcal{S}_n$ and in this way one establishes the isomorphism between $\mathcal{T}_n$ and $T_s \mathcal{S}_n$. The physical properties that charcterize the fundamental vector fields are discussed in Section 2.2.

We define the structure coefficients $F_{\alpha \beta}^{\gamma}$ by means of eq. (1.52) and the differential one-forms $\omega^{\beta}$ by means of eq. (1.53). They satisfy the equations (1.54) and (1.55), which have been discussed, in a particular context, in Section 1.7. Also the dynamical quantities which form the components of the $(10 + n)$-momentum (see Section 1.9), when they have an approximate meaning in the general formalism, are indicated by the compact notation $p_{\alpha}$.

All the other properties of $\mathcal{S}_n$, and in particular its structure of principal fiber bundle with a connection, if it maintains an approximate validity, are shifted from the realm of geometry to the realm of dynamics. For instance, all the structure coefficients $F_{\alpha \beta}^{\gamma}$ should be considered as dynamical fields, namely their values should be determined by the field equations and the action principle. In the normal theories a dynamical role is recognized only to the coefficients $F_{ik}^{\gamma}$.

Also the equations (1.14) and (1.25), that determine the transformation properties of the fields with respect to the Lorentz and internal symmetry groups, should be considered as dynamical field equations, to be derived, as the other field equations which contain the spacetime derivatives, from the action principle.

If we consider an arbitrary constant positive definite $(10 + n) \times (10 + n)$ matrix $G_{\alpha \beta}$ and we consider its elements as the components of a metric tensor in the basis defined by the vector fields $A_{\alpha}$, the manifold $\mathcal{S}_n$ acquires a structure of Riemannian manifold. One can easily see that the topology and the uniform structure defined by this metric do not depend on the choice of the matrix $G$. In particular, we can introduce without any ambiguity the concepts of completeness of the space $\mathcal{S}_n$ and of boundedness of a vector field. A vector field $b^{\alpha} A_{\alpha}$ is bounded if all its components $b^{\alpha}$ are bounded functions.

These concepts are very useful, because a theorem proven by Palais [55] assures that, if $\mathcal{S}_n$ is a complete Riemannian manifold, every bounded differentiable vector field $A$ generates a one-parameter group of diffeomorphisms of $\mathcal{S}_n$ onto itself, that we indicate by $\exp(\tau A)$, where $\tau$ is a real parameter. More in general, a Lie algebra of bounded vector fields generates a right action of the corresponding simply connected Lie group on $\mathcal{S}_n$. The physical meaning of the completeness property is discussed in Section 2.5.