Critical remarks

It is evident that there is a serious gap between the operational approach discussed in refs. [1,2] and summarized, with several simplifications, in Section 2.2 and the geometric structure described in Section 2.1. We do not even try to fill this gap, but we think that it is useful to point out some of its aspects.

It looks more and more probable that understanding the very small scale features of the geometric concepts requires a paradigm shift [78] involving several aspects of science. We like to think that the considerations given in the present notes give a small contribution in this direction, but much more work is necessary, expecially concerning the quantum aspects of the problem. The point of view of ref. [2] is more radical, but it is very difficult to formulate in that way a sufficiently rich amount of physical knowledge.

A first simplification present in our approach is to disregard the casual features present in every physical operation, in particular the experimental errors. They, far from being a nuisance, provide an operational justification for the topological concepts of the theoretical models, as it is explained in ref. [57]. The connection between experimental uncertainties and the topological concepts has been pointed out by Poincaré one century ago [79].

By allowing the presence of casual choices in the description of a procedure, one can endowe the varions spaces of procedures with a structure of convex set, similar to the structure of the space of the mixed states in quantum mechanics. We do not use in the following this very useful structure.

A second problem is given by the doubts raised by the interpretation of the manifold $\mathcal{S}$ as the collection of all the possible local inertial frames. The local frames, called more exactly ``situations'' in ref. [2], are defined by material objects that inavoidably interact with each other and with the physical objects under investigation. It is clearly impossible to realize all of them. One could define $\mathcal{S}$ as the collection of all the ``potential'' local frames, but it is difficult to understand what this means.

It may be useful to turn to a fiction, namely assume that the local frames are defined by a very ``thin'' kind of idealized matter that interacts appreciably only with the measuring instruments, in order to transmit to them the geometric informations. If this fiction represents a good approximation, we say that the (classical or quantum) theory has a classical geometry. However, it is inavoidable to take into account:

• the gravitational field generated by the objects that define the frames;
• their quantum properties that do not permit an exact determination of their positions and of their velocities.

The gravitational field can be disregarded if the mass of the object that defines the reference frames is sufficiently small and the quantum effects are not important if this mass is sufficiently large. Only in physical situations in which one can choose a mass that satisfies both these conditions one can adopt a classical geometry. Otherwise, one enters the realm of quantum geometry, namely of quantum gravity. Quantum frames have been discussed in refs. [18,80,81,82].

A third problem concerns the concept of state. In atomic or particle physics a state is defined operationally by a preparation procedures, for instance a suitable instrument produces a beam of atoms with given quantum numbers. Alternatively, one can take a large set of atoms, measure a complete set of observables and choose the atoms that have given the required outcomes. It is clear that these methods do not work when one is investigating a large part of the universe. Other problems arise if one consider with more detail the interpretation of quantum mechanics [83].

Actually, one can formulate a physical theory only in terms of observables (measurement procedures) and relations between them. This is the point of view adopted in refs. [1,2]. Some functions of the concept of state are transferred to the concept of situation. Instead of measuring the observables in various different states, one measures them in various randomly chosen situations and performs a statistical analysis of the outcomes. This is possible because we live in a very large universe and we can choose an arbitrarily large set of situations separated is space and in time. Then, if one likes, one can introduce the states as functionals defined on the space of measurements.

This merging of the concept of state and the concept of local frame in the concept of situation, is, in our opinion, a probable feature of a future theory of physical geometry. The merging of different operationally defined concepts when one enlarges their range of application is a general phenomenon analyzed in ref. [56]. The best known example is the merging of the operational definitions of energy and frequency when one applies them to microscopic physics.

Since we are dealing with classical field theories, we can freely define a state as a solutioon of the field equations, as we have done in Section 2.5. The outcomes of all the observables measured in arbitrary local frames are given by the values taken by the fields. This is possible in a classical theory, since all the observables are compatible and their measurement does not affect the state of the system.

We think that it is correct to develop, as we are doing in the present notes, the drastically simplified classical scheme based on the manifold $\mathcal{S}$, even if it can be criticized from a methodological point of view, as soon as it can be used to formulate useful physical ideas [58]. A similar and perhaps more severe criticism applies to the classical field theories based on the spacetime manifold $\mathcal{M}$.