The operational interpretation and the relativity principle

Important ideas about the geometric structure of the space $\mathcal{S}$ and the role it plays in physical theories follow from an operational analysis of the geometric concepts of physics given in refs. [1,2]. The operational point of view has been discussed by P. W. Bridgman [56] and an accurate presentation, which has strongly influenced our considerations, is given by R. Giles in ref. [57].

Of course, nobody is obliged to adopt the operational point of view. Any methodological choice is valid as soon as it helps to put some order in the physical experience [58]. We shall see that the operational analysis suggests several very interesting physical ideas.

According to ref. [57], a physical theory is a mathematical theory with an operational interpretation of some (not necessarily all) of its concepts (terms and relations). It is important to remember that some mathematical concepts may have no direct operational interpretation. For instance, spinor and charged fields are not observable [59,60], but they are very useful in the formulation of a field theory.

An operational interpretation is based on physical (laboratory) operations. What is relevant, however, is not the single, concrete, operation, but a set of prescriptions, called a procedure, clearly stated in a specific document, which describes exhaustively how the operations have to be performed. For instance, in the description of a procedure it is not allowed to point the finger at some physical object. When this point of view is is rigorously accepted, we speak of strictly operational interpretation.

In order to specify the spacetime conditions, namely where and when the operation is performed and which has to be the velocity and the orientation of the instruments, the procedure must refer to some pre-existent physical object, chosen by the experimenter in any single case, which determines a ``reference frame''. In order to avoid confusion with the mathematical concept of reference frame, we use the term situation. A procedure does not specify how the situation has to be chosen. As we shall discuss in Section 2.6, it may be difficult to separate the geometric meaning of a situation from other physical information it necessarily contains.

The simplest kinds of procedures are the measurement procedures, which give a numerical result, and the transformation procedures which have the aim of building a new situation starting from a pre-existent one. More complicated procedures will be discussed in Section 3.4. It is convenient to call a measurement a class of equivalent measurement procedures that give (statistically) the same results in all the situations and, similarly, to call a transformation a class of equivalent transformation procedures that act (statistically) in the same way on the situations (a more precise definition is given in [2]).

One can define in a natural way the composition of two transformations, which is again a transformation, and of a transformation and a measurement, which is a new measurement. In agreement with our previous conventions, we write on the right the transformation performed later. In other words, the transformations form a semigroup acting on the right on the situations and on the left on the measurements. Other algebraic properties of the spaces of measurements and transformations are discussed in ref. [2].

It is important to remark that there is no possible strictly operational prescription for choosing a situation, unless a preceding situation is available. This means that the situations have no strict operational interpretation. The transformations have an operational interpretation and it is proposed in ref. [2] that the geometric concepts of physics should be defined in terms of transformations.

We see that the operational point of view leads to a relational geometry, in which the important concepts are not the frames (or the events), but the relations between them.

In the classical formalism we are considering, the manifold $\mathcal{S}$ is a model for the space of all the situations. Some problems raised by this definition are discussed in Section 2.6. We are considering the case $n = 0$, since the extended manifold requires a more delicate discussion.

Since the elements of $\mathcal{S}$ have no strict operational interpretation, the physical laws cannot privilege, and not even single out, any of them. It follows that all the points of $\mathcal{S}$, namely all the local inertial frames, have to be treated, a priori in the same way. This is a statement of the relativity principle, that appears as a consequence of the requirement that a physical theory must have a strict operational interpretation.

If one accepts this requirement, the relativity principle has to be considered as a part of the very definition of physics. Statements that privilege a particular local inertial frame do not belong to physics, but, possibly, to other sciences [23].

It is necessary to specify that only a priori equivalence of the inertial local frames is required. After the measurement of a field, the frames in which it takes a certain value may be privileged. This means that if one finds a seeming violation of the relativity principle, one has to find a field responsible for it. Of course, this field must have its own dynamics.

Originally, the relativity principle was restricted to pairs of frames which have different velocities. The formulation given above extends the principle to pairs of frames with different location in spacetime and different orientation in space. This general interpretation agrees with the ideas discussed in Section 2.4.

In the present notes we deal mainly with classical field theories, modelled on the Maxwell's and Einstein's theories, with the spacetime $\mathcal{M}$ replaced by $\mathcal{S}$. In particular, we assume that the measurement and the transformation procedures do not affect the state of the system. Some remarks about the limitations of this approach are given in Section 2.6. Then we can define a state of the system (including its time evolution, as in the Heisenberg picture of quantum mechanics), as a given solution of the field equations.

A measurement defines a scalar fields on $\mathcal{S}$, the value of the field at the point $s$ being the outcome of the measurement performed in the local frame $s$. We are not assuming that all the scalar fields represent operationally defined measurements, but it seems natural to assume that the structure coefficients $F_{\alpha \beta}^{\gamma}$ are measurable fields.

A transformation induces a mapping of $\mathcal{S}$ into itself, that is assumed to be differentiable. We also assume the existence of one-parameter semigroups of transformations corresponding to mappings of the kind $s \to s \exp(\tau B)$ with $\tau \geq 0$. The vector field $B$ describes an infinitesimal transformation.

Our main assumption is that the vector fields that describe infinitesimal transformations generate the $10$-dimensional linear subspace $\mathcal{T}$ of the much larger space of all the vector fields in $\mathcal{S}$. The elements of $\mathcal{T}$ are called fundamental vector fields and define the absolute parallelism of $\mathcal{S}$, as it is explained in Section 2.1.

Only the vector fields belonging to a subset $\mathcal{T}^+ \subset \mathcal{T}$ generate a semigroup of transformations as we discuss in Section 3.1. The vector fields $A_{\alpha}$ form a basis of $\mathcal{T}$, but do not necessarily belong to $\mathcal{T}^+$. Of course, one can introduce a (global) change of basis, but a local (gauge) change of basis, different in the various tangent spaces $T_s \mathcal{S}$, is not admitted by our operational interpretation (see Section 3.8).

In order to justify our assumptions, we have to explain why a semigroup of diffeomorphisms of the kind

$$\exp(\tau b^{\alpha}(s) A_{\alpha}(s)),$$ (2.1)

where $b^{\alpha}(s)$ are suitably chosen nonconstant measurable scalar fields, cannot describe semigroup of transformations. Otherwise, we could define infinitesimal transformations represented by the vector field $b^{\alpha} A_{\alpha}$ not belonging to $\mathcal{T}$ and the absolute parallelism could not be defined in an unique way.

One may try to consider the transformation (2.1) as the composition of many transformations corresponding to small values of $\tau$; in every step one measures the values of $b^{\alpha}(s)$ and then one performs the transformation as if these quantities were constant. One must remark, however, that it takes some minimum time to measure the values of the scalar fields $b^{\alpha}(s)$ and therefore the steps cannot be too small. As a consequence, we are authorized to assume that in the limit $\tau \to 0$ the mapping (2.1) can represent a transformation only if the fields $b^{\alpha}(s)$ are numerical constants known in advance.

Note that we had to assume that it takes some minimum time to perform a measurement. The generalization of this important principle to all the physical operations is discussed in Sections 2.4 and 3.1.