Introduction

These notes contain a revised and updated account of a series of articles published by the author and collaborators in the last three decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], concerning the physical applications of the space of inertial local frames.

In the standard approach, summarized in Chapter 1, one starts from a $4$-dimensional pseudo-Riemannian spacetime $\mathcal{M}$ and builds, by means of a standard mathematical procedure [25,26,27], the $10$-dimensional principal fiber bundle $\mathcal{S}$ of the Lorentz frames, which is a very useful, but not necessary, instrument for the treatment of the known relativistic theories of gravitation. Also Maxwell and Yang-Mills theories can be included in this scheme by considering an ``extended'' principal fibre bundle $\mathcal{S}_n$ with a higher dimension.

There is no need for a new detailed exposition of this argument, which has been treated by many authors. We only present in Chapter 1 the basic ideas, in order to introduce some definitions and notations and to write some important formulas to be used later.

Our main purpose is to present a substantially different point of view, namely we start from the manifold $\mathcal{S}$ with a direct physical interpretation, and, if some conditions to be physically verified are satisfied, we build the spacetime manifold $\mathcal{M}$ by means of a suitable mathematical procedure. If these conditions are not satisfied, we have a nonlocal theory. In this way, we can introduce in a classical geometry a fundamental length $\ell$, which is suggested by quantum gravity.

The development of this alternative point of view gives the opportunity for a discussion of some general aspects of physics concerning, for instance, the relativity principle, symmetry transformations and conserved quantities. We shall try to give some indications of this kind whenever we think that it may be useful.

We restrict our attention to the classical aspects of geometry, with some applications to quantum theories of matter in a classical geometric background. Since we are not interested in nonrelativistic mechanics, by ``classical'' we always mean ``nonquantum''. We indicate, whenever it is necessary, the points in which a classical geometry is not consistent with quantum theory. We hope that a new approach to classical geometry can provide new starting points and new ideas for the construction of a quantum theory of gravitation, namely a quantum geometry.

In Chapter 2 we treat the geometry of the space $\mathcal{S}$, defined in terms of transformations, namely mappings of $\mathcal{S}$ into itself, interpreted as physical procedures that have the purpose of building a new frame starting from a pre-existent one. The transformations have a strict operational interpretation, while the frames, namely the single points of $\mathcal{S}$, are not operationally defined. This remark provides a foundation for a general formulation of the relativity principle, that states that all the frames are, a priori, physically equivalent.

We dedicate some attention to the problems raised by the very concept of inertial local frame, which has to be defined in terms of some material objects which inavoidably interact with the objects under investigation and with the measuring instruments. This remark is connected with the difficulties encountered in the construction of a quantum theory of gravitation.

In Chapter 3 we introduce in the tangent spaces of $\mathcal{S}$ a cone that characterizes the infinitesimal ``feasible'' transformations. The symmetry group of this cone is $GL(4, \mathbf{R})$ and we think that it is not an accident that $4$ also is the number of components of the Dirac fields that describe matter in the Standard Model of elementary particles. We also discuss some mathematical properties of this group, of some of its subgroups and of some of their representations. $GL(4, \mathbf{R})$ has a subgroup isomorphic to $SL(2, \mathbf{C})$ the universal covering of the proper orthochronous Lorentz group. We advance the idea that a field theory may have a spontaneouly broken symmetry with respect to a larger subgroup of $GL(4, \mathbf{R})$.

In Chapter 4 we deal with a Lagrangian approach to the classical field theories defined on $\mathcal{S}$ and with the connection between symmetries and conservation laws (Noether's theorem). In Chapter 5 we apply this formalism to several classical field theories usually defined on the spacetime manifold. In Chapter 6 we treat a scalar-tensor theory, giving a geometrical interpretation to the scalar field that replaces the gravitational constant.

In Chapter 7 we look for Lagrangian field theories with a symmetry group larger than the Lorentz group. This is a rather difficult problem and only preliminary, not completely satisfactory results are presented. We hope to give more complete results in a future version of these notes.

In Chapter 8 we describe the motion of a test particles by associating to it a set of frames that form a submanifold of $\mathcal{S}$. If we consider the particle as a small region in which the fields are particularly strong, the particle dynamics is provided by the balance equations of the underlying field theory. Alternatively, one can introduce an independent Lagrangian or Hamiltonian particle dynamics. The treatment of test particles is relevant for the physical interpretation of field theories.

The Chapters 9, 10 and 11 are not yet complete and contain only some references to the original papers. In the following versions they will present some other aspects of the formalism based on local inertial frames.

The present notes contain mainly already published material, though many ideas and calculations have been clarified and improved. We hope that a consistent description of the state of the art will be useful for a further progress. We devote a special attention to the best choice of the notations and conventions, modifying in some cases the choices adopted in the original articles. We cite the most relevant contributions of other authors, also if they do not agree completely with our point of view, but we do not claim that the list of references is complete. It will be improved in the following versions of these notes.

We do not try to give a set of references to the wide literature on quantum gravity. Even giving some references to review articles and books would imply a relative evaluation of the different approaches, which is neither necessary nor useful for the purposes of the present notes. In any case, an extended analysis of the classical theory is an important step for the construction of a quantum theory.