## 7 Quantum-Spacetime Phenomenology Beyond the Standard Setup

Most of the ideas for phenomenology I here reviewed are set up following a common strategy. They reflect the expectation that the characteristic scale of quantum-spacetime effects should be within a few orders of magnitude of the Planck scale, and that it should be possible (for studies conducted at scales much below the Planck scale) to analyze quantum-spacetime effects using an expansion in powers of the Planck length. All this is inspired by analogous strategies that have been very fruitful in other areas of physics: many arguments indicate that the Planck scale is the scale where the current theories break down, and usually the breakdown scale is also the scale that governs the magnitude of the effects of the new needed theory. In the case of quantum-spacetime research the expectation of perturbative effects suppressed by a large scale finds further motivation in at least two observations:- The effects we expect from spacetime quantization are rather striking, qualitatively virulent departures from the structure of our current theories. The fact that no trace of such “easily noticeable” effects has ever been seen surely provides further encouragement for the expectation of perturbative effects suppressed by an ultralarge scale.
- I would list the evidence in favor of grand unification as an even more significant source of additional encouragement for the expectation of perturbative effects suppressed by an ultralarge scale. If that evidence is taken at face value (as, I would argue, we should, at least as a natural working assumption) it suggests that particle physics works well on its own up to a scale of about the Planck scale. If quantum-spacetime effects were non-perturbative in ways affecting grand unification or if the scale of spacetime quantization was much lower than the Planck scale, it would then be hard to explain the (preliminary) success of the grand-unification idea.

In light of this, surely quantum-spacetime phenomenologists should continue to focus most of their efforts on applications of the standard strategy, assuming perturbative effects suppressed by a scale in some neighborhood of the Planck scale. However, other scenarios and opportunities should not be completely overlooked. We are clearly presently unable to exclude that the correct quantum picture of spacetime might turn out to be unsuitable to the standard strategy of quantum-spacetime phenomenology. As a way to give some substance to this assessment, I briefly discuss examples of mechanisms that could render ineffective the standard strategy of quantum-spacetime phenomenology.

### 7.1 A totally different setup with large extra dimensions

Can the scale characteristic of quantum-spacetime effects be much lower than the Planck scale? We surely know at least one mechanism by which the quantum-gravity scale can be much lower than the Planck scale, and therefore quantum-gravity models with spacetime quantization affected by this mechanism would describe quantum-spacetime effects at a relatively low scale. I am thinking of the popular scenarios with large extra dimensions.

Through these scenarios one can achieve a sizeable reduction in the quantum-gravity scale with the introduction of extra space dimensions [80, 375, 552, 84] of finite size . Then the fundamental length scale characteristic of quantum gravity in the 3+D+1-dimensional spacetime can be much bigger than the Planck length. The smallness of the Planck length can emerge as the result of the fact that, as deduced from applying Gauss’s law in the 3+D+1-dimensional context, the strength of gravitation at distance scales larger than the size of the extra dimensions in the ordinary (infinite-size) 3+1-dimensional spacetime would be proportional to the square-root of the inverse of the volume of the external compactified space multiplied by an appropriate power of .

These scenarios need to be tuned rather carefully in order to get a phenomenologically-viable picture. Essentially the only truly appealing possibility is the one of 2 extra dimensions of relatively “large” size, somewhere below millimeter size [84] (perhaps or meters). There might be other extra dimensions of smaller (possibly Planckian) size, but for the desired phenomenology one needs two and only two extra dimensions of relatively large size; otherwise one finds effects that either violate known experimental facts or are too small to ever be tested. But with these (however contrived) choices one does end up with a phenomenologically-exciting scenario in which the fundamental length scale of quantum gravity is somewhere in the neighborhood of the length scale, and therefore within the reach of particle-physics experiments (see, e.g., Refs. [260, 258, 207, 82]). Moreover, there are phenomenologically-relevant implications for the behavior of (classical) gravity at submillimeter distances [84, 295].

### 7.2 The example of hard UV/IR mixing

The large-extra-dimension scenario is an example of inapplicability of the standard setup of quantum-spacetime phenomenology due to the fact that, within that scenario, the characteristic scale of quantum gravity is not the Planck scale. There are also scenarios in which one may still assume that quantum-spacetime effects are fundamentally introduced at the Planck scale, but the standard setup of quantum-spacetime phenomenology is inapplicable because the most characteristic effects are not describable in terms of an expansion in powers of the Planck length.

I have already discussed this possibility in Section 5, devoted to soft UV/IR mixing. However, in that context one could fall back on roughly the standard strategy of quantum-spacetime phenomenology, by looking for an IR scale playing the role of characteristic scale of the IR manifestations of the quantum properties of spacetime. Let me just stress that an even more pervasive revision of the standard strategy of quantum-spacetime phenomenology would be required in the case of hard UV/IR mixing, which might take the form of correction terms with the behavior of inverse powers of momentum. With hard UV/IR mixing one should expect that in certain contexts the departures from known physical laws should be dramatic. The most efficacious tests of this hypothesis might not take the shape of searches of small corrections to standard predictions in ordinary contexts, but rather be based on the identification of those peculiar contexts where the implications of UV/IR mixing are large.

### 7.3 The possible challenge of not-so-subleading higher-order terms

Some challenges for the standard setup of quantum-spacetime phenomenology may also be present when the effects are genuinely introduced at the Planck scale and there is nothing peculiar about the IR sector. In particular, just because this standard setup is based on a (truncated) expansion in powers of the Planck length, it can happen that the formally sub-leading terms (higher powers of the Planck length), which are usually neglected in leading-order analyses, are actually not really negligible. The fact that experiments suitable for quantum-spacetime phenomenology must host, as I stressed in several points of this review, some ultralarge ordinary-physics dimensionless “amplifiers” could play a role in these concerns: if some mechanism is allowing the tiny leading-order Planck-length correction to be observably large it would not be so surprising to find that the same (or some other) amplifier is also such that some “formally subleading” Planck-length corrections, neglected in the analysis, are significant.

And another possible source of concern can originate from the fact that some of the contexts of interest for quantum-spacetime phenomenology are characterized by several length scales: expansions in powers of the Planck length actually are expansions in powers of some dimensionless quantity obtained dividing the Planck length by a characteristic length scale of the physical context of interest, and some “pathologies” may be encountered if there are several candidate length scales for the expansion.

While I feel that these issues for the power expansion should not be ignored, it is partly reassuring that the only explicit examples we seem to be able to come up with are rather contrived. For example, in order to illustrate the issues connected with the many length scales available in certain contexts of interest for quantum-spacetime phenomenology, I cannot mention anything more appealing than the following ad hoc formulation of a deformation of the speed-energy relation applicable in the “relativistic regime” ():

At low (but still “relativistic”) energies this would fit within a picture that has been much studied from the quantum-spacetime-phenomenology perspective, that of the speed-energy relation . But whereas in the relevant literature it is assumed that the term , if present, should always be the leading correction, up to particle energies on the order of the Planck scale, , from Eq. (97) one would find that the correction term is already no longer leading at particle energies on the order of , i.e., well below the Planck scale. Here the point is that Eq. (97) is characterized by two distinct small quantities suitable for the “expansion in powers of ”: the quantity and the quantity .