The equity principle, the minimum time principle and the fundamental length

Another argument in favour of the use of the space of frames $\mathcal{S}$ in the formulation of field theories has been pointed out by Lurçat [49,50] in 1964. At that time a large part of the research in elementary particle physics was devoted to the relation between mass and spin described by ``Regge trajectories'' [66] and it was realized that spin is not an ``unessential complication'' and has a dynamical role.

It was suggested that energy-momentum and spin are equally important and should be treated on the same footing. As a consequence, a unified treatment of spacetime translations and rotations was desirable. This argument suggested a formalism in which quantum fields are defined on the Poincaré group, that, as we have shown in Section 1.9, can be identified with the bundle of the Lorentz frames of the Minkowski spacetime.

These ideas were suggested by strong interaction physics, but we apply them to the microscopic physics of space-time and gravitation. Also the ideas of string theory, one of the most popular candidates for the solution of the problems of quantum gravity, had their origin in the theory of strong interactions.

From the point of view of the space $\mathcal{S}$, vertical and horizontal vectors, which represent Lorentz transformations and (parallel) translations, should be treated on an equal footing, but this is not compatible with the structure of principal bundle. The treatment given in Chapter 1 shows that a considerable part of the geometric concepts presented there does not distinguish between vertical and horizontal vectors and we have employed some effort in order to put in evidence these aspect. This is particularly evident in the compact formalism of Section 1.7.

We propose the term equity principle for the idea that mass and spin, translations and rotations, horizontal and vertical subspace should be treated on an equal footing. There is no operational justification for this principle as, for instance, for the relativity principle (see Section 2.2). In fact, spacetime translations and Lorentz transformations have different operational interpretations.

Also the space and time translations have different operational definitions, but relativity theory treats them in a symmetric way, in the sense that the Lorentz group has an irreducible action on the four-vector space. In Chapter 3 we implement the equity principle by introducing a group that acts irreducibly on the space $\mathcal{T}$. This means, in particular, that the vertical and the horizontal subspaces cannot be invariant.

The equity principle has mainly an heuristic value, namely it is useful for the construction of new theories, to be compared with experiments. It is known that after pruning a fruit tree, it grows stronger and it gives more and better fruits. A similar occurrence has been observed several times in the history of physics, the best known examples being the absolute time in relativity and the electron orbits in quantum atomic physics. It is possible that dropping the assumptions that spoil the symmetry between vertical and horizontal vectors can help in the treatment of the problems that affect the theories of spacetime and quantum gravity.

As it is discussed in Section 2.3, the vertical subspace is strictly related to the idea of spacetime coincidence. If the equity principle is accepted, the absolute character of the space-time coincidence, a fundamental assumption of General Relativity, has to be abandoned.

One should also remark that translations and rotations are described, respectively, by a length and by an adimensional parameter (angle or velocity). It follows that a symmetric treatment requires the introduction of a fundamental lenght $\ell$, in the same way as the introduction of a fundamental velocity $c$ is necessary for a symmetric treatment of space and time in special relativity. The introduction of a fundamental length in the classical theory of gravitation should help to eliminate the singularities that appear in the black hole and in the big bang solutions.

We have to discuss the relation between the fundamental length $\ell$ appearing in a (still hypothetic) modified classical theory of gravity and the Planck length

$$\ell_P = (\hbar G)^{1/2},$$ (2.9)

which appears in all the attempts to quantize General Relativity. If $\ell \gg \ell_P$, some (not yet observed) effects of quantum gravity could be masked by the effects of $\ell$. However, we think that one should find some reason why $\ell = \nu \ell_P$, where $\nu$ is a constant of the order of one.

One could consider the modified classical theory as an approximation of the quantum theory in a situation in which the effects of $\hbar$ can be disregarded, but the effects of $\ell_P$ are still present. For instance, if one considers particles with an extremely large energy-momentum dispersion, the spacetime uncertainty introduced by $\ell$ could be more important than the uncertainty introduced by the Heisenberg relations.

From a more formal point of view, one could remark that in a classical theory $\ell$ and $G$ are two independent parameters and that in many cases a classical system can be quantized only if a relation between its parameters and $\hbar$ is satisfied [67,68]. The simplest example is given by a classical spin system described by a phase space given by 2-dimensional sphere with a symplectic form proportional to the rotation invariant element of area and normalized in such a way that the total area is $a$. The classical system makes sense for any positive value of the constant $a$, which has the dimension of an action, but quantization is possible only if $a$ is an integral multiple of $2 \pi \hbar$. The relation $\ell = \nu \ell_P$ could have a similar origin. Further suggestions are given in Section 4.5.

In another scenario, the parameters $\ell$, $G$ and $\hbar$ are independent and the quantized theory has a perturbative expansion in the adimensional parameter $G \hbar \ell^{-2}$ (of course for $\ell > 0$), but a singularity appears when the parameter reaches a value of the order of one. We are not proposing a quantum theory of gravitation, but we only want to show that a classical theory of gravitation containing a fundamental length is not a priori wrong. With a bit of imagination, one could find in this way an explanation for the difficulties found in the quantization of a normal gravitational theory with $\ell = 0$.

We have assumed that $G$ is a fundamental constant and that the classical theory contains another fundamental constant $\ell$ with the dimension of a length. A function of these constants has the dimension of an action and, as we have discussed above, it must have some connection with the fundamental constant $\hbar$ introduced by quantization. In this situation, we say that the classical theory is ``prepared'' for quantization.

However, several authors [69,70,71] have proposed theories with variable $G$. These theories have not been confirmed experimentally, but they have several interesting features and deserve a serious consideration. If $G$ is a variable, and $\ell$ is the only fundamental constant (besides the velocity of light), the classical theory is not ``prepared'' for quantization unless the fundamental constant $\ell$, or some power of it, has the dimension of an action. We shall discuss further this argument in Section 6.3.

The analogy between translations, rotations and Lorentz boosts implies the analogy between velocity, angular velocity and acceleration. Since in relativistic theories there is an upper bound to the velocity, the equity principle suggests the existence of upper bounds of the order of $\ell^{-1}$ to the angular velocity and the acceleration. A maximal acceleration has been suggested, with various motivations, by many authors [7,14,72,73,74,75,76,77]. A more detailed discussion is given in Section 3.1 and in Chapter 8.

The same concepts can be formulated in a different way. The existence of a maximal velocity means that it takes a minimum time to perform a space translation. The equity principle implies that it also takes a minimum time to perform rotations and boosts. We have already suggested in Section 2.2 that it takes a minimum time to perform measurements. It is natural to formulate a minimum time principle requiring that it takes a minimum time to perform any physical operation.

We assume that mathematical operations can be performed in an arbitrarily short time. In particular, we do not apply the minimum time principle to gauge transformations, usually considered as a purely mathematical change in the description of physical phenomena. An investigation of problems of this kind requires an interaction between physics and inforamtion theory, that we are not prepared to tackle.