A Broader Perspective on the Phenomenology of Quantum-Gravity-Induced Infrared/Ultraviolet Mixing

Abstract

I revisit some arguments that motivate infrared/ultraviolet (IR/UV) mixing, a mechanism such that ultraviolet quantum-gravity structures produce novel features also in a far-infrared regime. On the conceptual side, I highlight in particular an apparently general connection between IR/UV mixing and departures from the standard relativistic symmetries of classical spacetimes. In addition to its conceptual appeal, interest in IR/UV mixing has also been driven by the availability of some opportunities for experimental testing, and my main focus is on phenomenological models of IR/UV mixing that can provide guidance to the experimental efforts. While usually each formulation of IR/UV mixing is investigated within an isolated research program, some parts of my analysis point to possible connections among different formulations and with other quantum-gravity studies.

1. Introduction

Quantum gravity presents itself as an ultraviolet problem: the need for a unifying more powerful theory (from which general relativity and quantum mechanics would emerge in appropriate limits) arises in investigations of contexts involving very small distances and/or microscopic particles of very high energy. However, several quantum-gravity scenarios predict that some new structures needed to fix things in the ultraviolet also produce modifications in a far-infrared regime. I shall here label as “IR/UV mixing" all such instances in which candidate quantum-gravity effects introduced in the ultraviolet regime are found to have significant consequences also in some relevant infrared regime. It appears likely that this notion of IR/UV mixing is structural to several quantum-gravity approaches, since it characterizes a feature as general as black-hole entropy. Indeed, as observed in [1] (also see [2]), the entropy–area relation is an example of IR/UV mixing: if entropy scales with the boundary area of a region (rather than its volume) there is a mismatch with our current applications of quantum mechanics to field theory that becomes more and more severe as the size of the region increases (simply because the ratio between volume and boundary area grows linearly with the size of the region). It appears to be necessary, in order to model black hole entropy, to assume the existence of a relation between the size of the system and an ultraviolet cutoff [1].

The broad notion of IR/UV mixing, which I am here advocating, can inspire a rich phenomenology (partly discussed here below), and it is plausible that the first observed manifestation of quantum gravity might be in the infrared: even though we still expect that most of the characteristic features of quantum gravity are ultraviolet, it might happen that an infrared effect of quantum gravity is closer to our reach. However, the discovery potential of IR/UV-mixing research, of course, depends also on the efficaciousness of the phenomenological models proving guidance to the experimental plans. Furthermore, a potential limitation of ongoing efforts aimed at developing and analyzing phenomenological models originates from the fact that investigations of IR/UV mixing are split up into a few isolated research programs, mostly unaware of each other and often also missing possible connections with some ultraviolet-focused quantum-gravity proposals. I here attempt to make a first step toward removing these limitations, providing some examples of connections between different scenarios for IR/UV mixing and of connections with ultraviolet-focused quantum-gravity research.

2. Spacetime Fuzziness with IR/UV Mixing

In this manuscript, I shall focus mainly on the possible connection between IR/UV mixing and quantum properties of spacetime, a connection that appears to be rather general (present in different/alternative pictures of spacetime quantization).

An example of the mechanism for IR/UV mixing arises in attempts to formalize the notion of “spacetime fuzziness”, which has been for a long time a hot topic in quantum-gravity research [3,4]. A physical/operative definition of at least one aspect of the fuzziness of a spacetime can be given [5,6] in terms of a contribution to the strain noise of interferometers set up in that spacetime. An interesting result, in relation to IR/UV mixing, is found [5,6] adopting a crude model of spacetime fuzziness, a model such that a distance L between two test masses of an interferometer would be affected by Planck-length fluctuations of random-walk type occurring at a rate of one per Planck time. The fluctuation mechanism is evidently ultraviolet, since it involves fluctuations of Planck-length size occurring with frequency set by the Planck time, but the fact that the fluctuations are of random-walk type produces a rather striking infrared effect: the resulting spacetime-fuzziness contribution to the strain noise power spectrum of interferometers grows as the frequency f decreases [5,6]

$${\rho }_h\left(f\right)={\displaystyle \frac{c{\ell }_P}{L^2{f}^2}},$$

(1)

where c is the speed-of-light scale, ${\ell }_P$ is the Planck length, and L is the length of the arms of the interferometer.

The spacetime-fuzziness-induced strain noise power spectrum of Equation (1) provides a particularly clear example of the general notion of IR/UV mixing, which I am here adopting: the $f^{-2}$ dependence of the strain noise power spectrum is characteristic of all mechanisms with random-walk fluctuations, independently of the scales involved in the random walk. If an observable O accrues random-walk fluctuations over time t, one then finds that the uncertainty in O grows like $\delta O=\chi \sqrt{t}$, where $\chi$ depends only on the scales of the random-walk mechanism, while the dependence on $\sqrt{t}$ is characteristic of all random-walk mechanisms. In turn, from $\delta O\propto \sqrt{t}$, it follows [5,6] that the strain noise power spectrum is $\propto f^{-2}$. In the crude model of spacetime fuzziness that I am considering, some ultraviolet structures set the scales of the random walk, but the random-walk strain noise is nonetheless an infrared effect with $f^{-2}$ dependence.

From the IR/UV-mixing perspective, it is also potentially significant that other attempts to model spacetime fuzziness also produce strain noise for interferometers that grows as the frequency decreases. In particular, a popular scenario for holography in quantum gravity inspired a proposal [7,8] in which spacetime fuzziness produces an uncertainty for distances that grows with the time of observation as $t^{1/3}$, and accordingly, the strain noise power spectrum for interferometers takes the form

$${\rho }_h\left(f\right)={\displaystyle \frac{c^{2/3}{\ell }_P^{4/3}}{L^2{f}^{5/3}}}.$$

(2)

Much lower noise levels are produced if one assumes spacetime fuzziness such that the uncertainty for distances does not grow with the time of observation, remaining Planckian independently of the time of observation. In that scenario, the strain noise power spectrum for interferometers takes the form [6]

$${\rho }_h\left(f\right)={\displaystyle \frac{{\ell }_P^2}{L^{2}f}}.$$

(3)

These models of spacetime fuzziness are crude (most notably for what concerns the estimate of the coefficient of the $f^{-\gamma }$ dependence), but they reflect a plausible feature of quantum gravity, and they inspired a rather rich phenomenology, taking as starting point a generalization of (1), (2), and (3) (see, e.g., ref. [8] and references therein)

$${\rho }_h\left(f\right)=\mu {\displaystyle \frac{{\ell }_P^{3-\gamma }}{f^{\gamma }}},$$

(4)

where $\mu$ is a phenomenological parameter to be determined experimentally, while $\gamma$, assumed [8] to be $1\le \gamma \le 2$, is characteristic of the mechanism of distance fluctuation. Remarkably, in spite of the smallness of the Planck length, strain noise at the level set by (1) and (2) (and for a sizable portion of the parameter space of (4)) is within the reach of modern interferometers [5,6,7,8]. In particular, the crude random-walk model given by Equation (1) is already very tightly constrained, since the sensitivity levels achieved by modern gravity-wave interferometers safely establishes [9], for $\gamma =2$, that $\mu \lesssim {10}^{-6}$.

On the conceptual side, a lot remains to be understood about the implications of these scenarios for quantum-gravity-induced strain noise for relativistic symmetries. No dedicated study has been devoted to this issue (and I am evidently advocating the need for such studies), but we should expect significant implications for relativistic symmetries in light of the connection discussed in the following sections between these scenarios and quantum-gravity proposals known to require departures from standard relativistic symmetries.

In closing this section, I find it appropriate to just bring to the attention of my readers the studies in refs. [10,11,12,13], which are partly connected with the studies of strain noise in quantum gravity on which I focused (but their discussion would take me too far off the main path of this manuscript).

3. Modified On-Shellness with IR/UV Mixing

The most studied mechanism for IR/UV mixing arises in the study of quantum field theories formulated with canonical spacetime noncommutativity, a type of quantum spacetime whose coordinates are such that

$$[x_{\mu },x_{\nu }]=i{\theta }_{\mu \nu },$$

(5)

where ${\theta }_{\mu \nu }$ is a matrix of parameters (with length-squared dimensions and, of course, expected to be small, possibly Planckian) to be determined experimentally.

The commutation relations (5) affect the ultraviolet behavior of the relevant quantum field theories, which, in turn, requires some corresponding modifications of the renormalization procedure. Renormalizability can be recovered, but only at the cost of producing some novel features in the infrared (see, e.g., refs. [14,15,16,17,18] and references therein). For what concerns the Feynman rules, the noncommutativity matrix ${\theta }_{\mu \nu }$ only affects vertices, but, through loop corrections, one then finds significant implications for the propagator. In several field theories formulated with canonical spacetime noncommutativity, one finds loop corrections to the two-point function, leading to both quadratic and logarithmic IR/UV mixing, as a result of the fact that certain loop integrals of the commutative-spacetime version of the field theory are split, in the noncommutative-spacetime case, into a part which is still ${\theta }_{\mu \nu }$-independent and a part which has an explicit dependence on ${\theta }_{\mu \nu }$ and dependence on the momentum $p_{\mu }$ at which the two-point function is being computed [17]. In particular, in the case of a single scalar field, the relevant splitting of the loop integral is given by

$$\int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^4}}{\displaystyle \frac{1}{k^{\mu }{k}_{\mu }+m^2}}⟶{\displaystyle \frac{1}{3}}\int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^4}}{\displaystyle \frac{2+exp\left(i{p}^{\nu }{\theta }_{\mu \nu }{k}^{\mu }\right)}{k^{\mu }{k}_{\mu }+m^2}}$$

and this, in turn, leads (through rather standard algebra [17]) to a regularized propagator of the form

$$\Pi \simeq p^{\mu }{p}_{\mu }+m^2+{\displaystyle \frac{g^2}{96{\pi }^2}}\left({\Lambda }_{eff}^2-m^{2}ln\left({\displaystyle \frac{{\Lambda }_{eff}^2}{m^2}}\right)\right)$$

where g is the coupling constant of the scalar-field theory, and ${\Lambda }_{eff}^2$ is

$${\Lambda }_{eff}^2={\displaystyle \frac{1}{\frac{1}{{\Lambda }^2}-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }}}$$

with $\Lambda$ the standard cutoff regulator and ${\tilde{p}}_{\mu }\equiv p^{\nu }{\theta }_{\mu \nu }$ (notice that $-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }$ is positive definite [17]).

Upon removing the regulator ($\Lambda \to \infty$), one finds that the condition of on-shellness takes the form

$$E^2\simeq {\overrightarrow{p}}^2+m^2+{\displaystyle \frac{g^2}{96{\pi }^2}}\left(-{\displaystyle \frac{1}{{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }}}+m^{2}ln\left(-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }{m}^2\right)\right).$$

(6)

Perhaps most notably, when the quadratic term ${(-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu })}^{-1}$ is present, and if the noncommutativity is space–space (${\theta }_{0\mu }=0$), the IR/UV-mixing correction diverges quadratically as $\overrightarrow{p}\to 0$ even for massive particles.

A softer modification of the on-shell condition for massive particles is found in field theories (for example, $U\left(1\right)$ gauge theories minimally coupled to fermions [16]) where the quadratic term is absent, particularly for the case of “light-like noncommutativity” (${\theta }_{\mu \nu }{\theta }^{\mu \nu }={ϵ}_{\mu \nu \rho \sigma }{\theta }^{\mu \nu }{\theta }^{\rho \sigma }=0$). In such scenarios, the regime $p\ll m$ is characterized by a correction to on-shellness that is linear in momentum. In order to see this within a simple explicit computation, let me focus on the particular case of light-like noncommutativity with ${\theta }_{02}=-{\theta }_{12}\equiv \theta$. Then, one finds $-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }={\theta }^2{(E-p_1)}^2$, which, in turn, when $p\ll m$, gives $-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }\simeq {\theta }^2(m^2-2m{p}_1)$ and

$$m^{2}ln\left(-{\tilde{p}}_{\mu }{\tilde{p}}^{\mu }{m}^2\right)\simeq m^{2}ln\left[m^2{\theta }^2(m^2-2m{p}_1)\right]\simeq m^{2}ln\left(m^4{\theta }^2\right)-2m{p}_1.$$

A somewhat similar scenario for IR/UV mixing arises in studies of some quantum-spacetime models based on discreteness: the spacetime discretization is ultraviolet (characterized by a small length scale), but it leads to a modification of the on-shell relation that is most relevant in the infrared regime $p\ll m$, producing once again a linear correction [19]

$$E=m+{\displaystyle \frac{p^2}{2m}}+\ell mp$$

(7)

where ℓ is a phenomenological parameter, with dimensions of length (in units such that the reduced Planck constant and the speed of light are set to 1), whose modulus is expected to be of the order of the Planck length.

The IR/UV-mixing scenario of (6) proposes a rather broad target to the phenomenology of massive particles with $p\ll m$, with different forms of momentum dependence, different choices of ${\theta }_{\mu \nu }$ being possible, a dependence on the charges of the relevant fields (see the role of g in (6)), and a dependence on the masses of the particles. The IR/UV-mixing scenario of (7) produces effects whose magnitude appears to set a more definite sensitivity target for phenomenology, since it involves no other scale but ℓ: there is a definite dependence on the mass of the particle, and there is no expected dependence on the standard-model charges of the particles. Still, the proportionality of the effect to the mass of the particle invites us to consider a possible effective dependence of ℓ on particle type, especially for the case of atoms (which, as I shall stress, provide an important opportunity for tests of (7)): if one assumes (7) to apply to the constituents of a composite particle, then for the composite particle, one might obtain effectively a value of ℓ different from that of the constituents. One can tentatively motivate this expectation on the basis of a simple heuristic argument, in which a composite particle is made of N identical constituents traveling freely next to each other (i.e., neglecting internal interactions). If the mass, total energy, and total spatial momentum of the composite are given, respectively, by $\mathcal{M}=Nm$, $\mathcal{E}=NE$, and $\mathcal{P}=Np$ (with E and p denoting, respectively, the energy and spatial momentum of the constituents), one finds that applying (7) to the constituents, the resulting effect for the composite particle is

$$\mathcal{E}=N\left(m+{\displaystyle \frac{p^2}{2m}}+\ell mp\right)=\mathcal{M}+{\displaystyle \frac{{\mathcal{P}}^2}{2\mathcal{M}}}+{\displaystyle \frac{\ell }{N}}\mathcal{M}\mathcal{P},$$

i.e., the IR/UV-mixing scale for the composite particle is reduced by a factor of N with respect to that of the constituents.

Evidently a full investigation of IR/UV-mixing scenarios with modified on-shellness will require exploring several branches, allowing for different sizes of the characteristic scale (with a possible particle dependence, possibly connected with the standard-model charges and/or the number of constituents of the particle) and different forms of the spatial-momentum dependence. For such a vast research program, it might be appropriate to start from a balanced initial target for the phenomenology, something analogous to the role played by (4) for the strain noise produced by IR/UV mixing. A natural choice might be

$$E=m+{\displaystyle \frac{p^2}{2m}}+m{(\ell m)}^{\rho }{\left({\displaystyle \frac{p}{m}}\right)}^{\sigma }$$

(8)

with, in light of the indications summarized above, $-2\le \sigma <2$ and setting as initial target the range $0<\rho \le 2$ (this season of quantum-gravity phenomenology is focusing on effects suppressed at most by two powers of the quantum-gravity scale [9,20]). The first target of the phenomenology could naturally be a universal ℓ, but gradually, as more data are accrued, one could explore some forms of particle dependence of ℓ, possibly connected with the standard-model charges and/or the number of fundamental constituents of the particle. Similarly, the dependence on p allowed in (8) could be a natural first target for the phenomenology, but gradually, one could explore possible directional features, such as those described through $p^{\mu }{\theta }_{\mu \nu }$.

As observed in refs. [21,22,23,24,25,26,27,28], cold-atom interferometry represents a promising avenue for testing this sort of IR/UV-mixing scenarios. The relevant interferometers involve several stages in which atoms with momentum much smaller than their mass have their momentum changed through interactions with photons, and clearly, the modifications of on-shellness produced by IR/UV mixing affect the kinematics of atom–photon interactions. Most notably, it was shown in ref. [29] that the IR/UV-mixing scenario of Equation (7) provides a solution for a puzzling discrepancy [30,31,32] between Cesium-based and Rubidium-based atom-interferometric measurements of the fine structure constant, leading to the estimate $\ell =(9.8\pm 1.9)·{10}^{-36}\mathrm{m}$ (strikingly close to the Planck length).

Other phenomenological avenues based on the type of IR/UV mixing considered in this section could be inspired by the fate of relativistic symmetries in theories such that (6) or (7) holds. Both (6) and (7) are clearly incompatible with standard relativistic symmetries [17,19], and, at least in principle, they could also be tested by probing the passive aspects of relativistic-symmetry transformations, i.e., through experimental setups allowing a comparison of the properties of a given system, as described by different observers.

4. IR/UV Mixing from Spacetime Noncommutativity

As discussed in the previous section, there is already an established connection between spacetime noncommutativity and IR/UV mixing, since it was shown that quantum field theories formulated in a spacetime with canonical noncommutativity (5) are characterized by on-shellness with IR/UV mixing. In this section, I observe that some semiquantitative heuristic arguments suggest that the connection between spacetime noncommutativity and IR/UV mixing might be broader and may deserve being explored in greater depth.

For what concerns canonical noncommutativity (5), an aspect of IR/UV mixing is present already in the commutation relations among spacetime coordinates since they imply, in particular,

$$\delta x_1\ge {\displaystyle \frac{|{\theta }_{12}|}{\delta x_2}}.$$

From this, it follows that there is no pure-ultraviolet regime in these spacetimes: when $\delta x_2$ is small (a requirement for the pure-ultraviolet regime), then $\delta x_1$ is inevitably large.

It is also intriguing to consider the same aspect when the time coordinate is involved,

$$\delta x_1\delta x_0\ge |{\theta }_{10}|,$$

in the context of the Einstein procedure for operative definition of spatial distances, given in terms of the travel time of a particle for a two-way journey between two points of space. Assuming for simplicity that the measurement procedure is confined to the $x_1$ direction, a distance D measured in that way would have uncertainty receiving contributions both from $\delta x_1$ and $\delta x_0$,

$$\delta D\ge \delta x_1+\delta x_0,$$

and in light of $\delta x_1\delta x_0\ge |{\theta }_{10}|$, one finds that

$$\delta D\ge {\displaystyle \frac{|{\theta }_{10}|}{\delta x_0}}+\delta x_0\ge \sqrt{|{\theta }_{10}|}.$$

This crude exploratory argument suggests tentatively that the limitations on $\delta D$ produced by canonical spacetime noncommutativity are not dependent on the time of observation of the system, which, in turn, leads me to conjecture that the associated strain noise power spectrum for interferometers should be $f^{-1}$, as in the case of Equation (3).

Next, let me consider spacetime noncommutativity of Lie-algebra type, and specifically $\kappa$-Minkowski noncommutativity, with its spatial coordinates that commute among themselves but do not commute with the time coordinate [33,34,35]

$$[x_j,x_0]={\displaystyle \frac{i}{\kappa }}{x}_j,[x_j,x_i]=0$$

(9)

Investigations of quantum field theory in $\kappa$-Minkowski are still in an embryonic phase [36,37,38], but in light of the implications of (9) for

$$\delta x_1\delta x_0\ge {\displaystyle \frac{|<x_1>|}{\kappa }},$$

it is natural to expect that when properly understood, these quantum field theories will be affected by IR/UV mixing.

An alternative way to investigate the implications of $\kappa$-Minkowski noncommutativity for IR/UV mixing can be suggested by contemplating the possible implications of $\delta x_1\delta x_0\ge |<x_1>|/\kappa$ for Einstein’s distance-measurement procedure, when combined with the requirement $\delta D\ge \delta x_1+\delta x_0$ motivated above. This leads to

$$\delta D\ge \delta x_1+\delta x_0\ge \delta x_1+{\displaystyle \frac{|<x_1>|}{\delta x_1\kappa }}\ge \sqrt{|<x_1>|/\kappa }.$$

The formal manipulations leading to this result leave us, however, wanting for an understanding of the interpretation of $|<x_1>|$ in the measurement of the distance D. For the purpose of the crude exploratory estimate I am attempting to provide, it might be acceptable to assume that the sought time of observation t of the system might be proportional $|<x_1>|$, in which case, one ends up with $\delta D\propto \sqrt{t}$, suggesting that $\kappa$-Minkowski noncommutativity might involve spacetime fuzziness characterized by a strain noise power spectrum for interferometers like $f^{-2}$, as in the case of the random-walk scenario of Equation (1). Further encouragement for conjecturing the presence of random-walk noise in theories with $\kappa$-Minkowski noncommutativity can be found by analyzing the implications of the most studied candidate, modified on-shellness for the description particle propagation in $\kappa$-Minkowski, but I postpone that observation to the next section, where I offer a few different observations on ultraviolet-focused studies of modified on-shellness. While at present not yet understood in relation to IR/UV mixing, the new approach introduced in ref. [39], whose core structure is a noncommutative space of worldlines, might provide new tools for the investigation of the type of spacetime fuzziness produced by $\kappa$-Minkowski noncommutativity. Interestingly, the implications of $\kappa$-Minkowski noncommutativity for IR/UV mixing would also provide another path for a connection between IR/UV mixing and the fate of relativistic symmetries in quantum spacetimes, since $\kappa$-Minkowski is a much-studied example of spacetime governed by deformed relativistic symmetries [35,40].

5. IR/UV Mixing and UV Scenarios for Modified On-Shellness

The most active area of quantum-gravity phenomenology has been focusing for several years on ultraviolet-regime tests of modifications of the on-shell relation of the type (see, e.g., refs. [9,20,40,41,42,43,44,45])

$$E^2=m^2+p^2+{\alpha }_0{\ell }_P{p}^3+{\alpha }_1{\ell }_{P}E{p}^2+{\alpha }_2{\ell }_P{E}^{2}p+{\alpha }_3{\ell }_P{E}^3\simeq m^2+p^2+{\alpha }_{\ast }{\ell }_P{p}^3$$

(10)

where ${\ell }_P$ denotes the Planck length, ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$ are dimensionless parameters to be determined experimentally, and ${\alpha }_{\ast }\equiv {\alpha }_0+{\alpha }_1+{\alpha }_2+{\alpha }_3$. The equality on the right-hand side of Equation (10) applies in the ultraviolet regime $E\simeq p$, and indeed, the rich phenomenology motivated by Equation (10) has only been sensitive (For these remarks, I am setting aside preliminary results (such as those in refs. [46,47,48,49]), which suggests that the interplay between modifications of on-shellness and the presence of macroscopic spacetime curvature (e.g., FRW macroscpopic geometry) might depend individually on the parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$.) to ${\alpha }_{\ast }$ (rather than individually to ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$) since it was based on observations of astrophysical sources in photons (see, e.g., refs. [41,50]) or in high-energy neutrinos (see, e.g., refs. [51,52]), and in both cases, $E\simeq p$. However, of course, quantum-gravity phenomenology must set at least as one of its long-term targets the objective of performing experimental investigations individually of the parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$, and in this section, I offer some observations aimed toward properties that depend individually on (some of) the parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$, with intriguing roles for IR/UV mixing.

Let me start by contemplating the case of Equation (10) with only the ${\alpha }_1$ parameter (i.e., the case ${\alpha }_0={\alpha }_2={\alpha }_3=0$): this gives the most studied candidate (The debate on what is the correct description of on-shellness in $\kappa$-Minkowski noncommutative spacetime has kept busy an interested community for more than two decades (see, e.g., refs. [53,54]). While a majority of studies have adopted and motivated the on-shellness given by (10) with only the ${\alpha }_1$ parameter, several alternatives have received substantial interest. This is connected to the fact that the $\kappa$-Minkowski noncommutativity implies curvature of momentum space (see, e.g., ref. [55]), which renders it natural to consider non-linear redefinitions of momenta (whose description at the level of the relativistic symmetries of $\kappa$-Minkowski amount to changes in basis [34,53]).) modification of on-shellness for the description particle propagation in $\kappa$-Minkowski noncommutative spacetime. In part of the previous section, I offered a crude heuristic argument suggesting that $\kappa$-Minkowski noncommutative spacetime might be affected by spacetime fuzziness producing random-walk noise for interferometers, and interestingly, it turns out that Equation (10) with only the ${\alpha }_1$ parameter also appears to be connected with the random-walk noise for interferometers. To see this possible connection, I contemplate again Einstein’s distance-measurement procedure, now focusing on the quantum properties of the particle probe appearing as agent in that measurement procedure. Such a quantum particle probe can be prepared with uncertainties $\delta x$ and $\delta v$, respectively, on its initial position and velocity, and then after traveling for a time t, the position of the particle has uncertainty

$$\delta x+\delta vt.$$

In relation to Einstein’s distance-measurement procedure, the travel time t here of interest is the order of the distance D being measured ($t=2D$ in units with the speed-of-light scale set to 1). For a particle probe governed by Equation (10) with only the ${\alpha }_1$ parameter, one finds that for $p\gg m$, the velocity has momentum dependence given by

$$v={\displaystyle \frac{dE}{dp}}=1-{\displaystyle \frac{p^2}{2m^2}}+{\alpha }_1{\ell }_{P}p.$$

The connection, within the quantum state used to prepare the probe, between the velocity uncertainty of the probe $\delta v$ and its momentum uncertainty $\delta p$ includes then a Planckian contribution, taking the form

$$\delta v\sim {\alpha }_1{\ell }_P\delta p.$$

Combining these observations with the Heisenberg-principle relation $\delta x\delta p\ge \hslash$, we find a contribution to the uncertainty of Einstein’s distance-measurement procedure given by

$$\delta D=\delta x+{\alpha }_1{\ell }_P\delta pt\ge \delta x_0+{\alpha }_1{\ell }_P{\displaystyle \frac{\hslash }{\delta x_0}}t\ge \sqrt{{\alpha }_1{\ell }_P\hslash t}.$$

(11)

This line of reasoning suggests that Equation (10) with only the ${\alpha }_1$ parameter might be connected with spacetime distances whose fuzziness has the characteristic $\sqrt{t}$ dependence of random-walk noise, thereby providing further motivation for studies of a corresponding $f^{-2}$ component of the strain noise power spectrum for interferometers.

Also interesting from the IR/UV-mixing perspective is the case of Equation (10) with only the ${\alpha }_2$ parameter (i.e., the case ${\alpha }_0={\alpha }_1={\alpha }_3=0$), for which I want to stress a clear connection with the type of IR/UV mixing here discussed in Section 3, with modified dependence of the energy of a massive particle on its spatial momentum in the regime such that the spatial momentum p is much smaller than the particle’s mass m. When the spatial momentum p is much smaller than the particle’s mass m, one has that $\ell E^2\simeq \ell m^2$ and then Equation (10) with only the ${\alpha }_2$ parameter can be rewritten as

$$E^2\simeq m^2+p^2+{\alpha }_2{\ell }_P{m}^{2}p,$$

which, in turn, gives

$$E\simeq m+{\displaystyle \frac{p^2}{2m}}+{\displaystyle \frac{{\alpha }_2}{2}}{\ell }_{P}mp.$$

(12)

So the infrared-regime approximation of Equation (10) with only the ${\alpha }_2$ parameter gives (replacing ℓ with ${\alpha }_2{\ell }_P/2$) the IR/UV-mixing scenario here described in Equation (7) of Section 3. This shows a particularly clear and simple example of the IR/UV-mixing mechanism: the term ${\alpha }_2{\ell }_P{E}^{2}p$ in Equation (10) can be "of order 1" only at very high momenta $p>1/\left({\alpha }_2{\ell }_P\right)$ (so that ${\alpha }_2{\ell }_P{E}^{2}p>p^2$) and at very small momenta $p<{\alpha }_2{\ell }_P{m}^2$ (so that ${\alpha }_2{\ell }_P{m}^{2}p>p^2$). Instead, the term ${\alpha }_2{\ell }_P{E}^{2}p$ gives a very small contribution to Equation (10) in the whole intermediate range of momenta most accessible to our observations, given by ${\alpha }_2{\ell }_P{m}^2\ll p\ll 1/\left({\alpha }_2{\ell }_P\right)$.

For what concerns phenomenological applications, the present experimental situation provides an opportunity for me to illustrate how (as a long-term objective) it might be possible to investigate individually the parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$, by combining observations performed in the ultraviolet regime and in the infrared regime. The present status of observations of neutrinos from distant astrophysical sources provides preliminary encouragement [56,57,58,59,60,61,62,63] for the hypothesis ${\alpha }_{\ast }\ne 0$. Furthermore, as already stressed in Section 3, a puzzling discrepancy between Cesium-based and Rubidium-based atom-interferometric measurements of the fine structure constant might provide preliminary encouragement [29] for the IR/UV-mixing scenario of Equation (7), i.e., the case ${\alpha }_2\ne 0$. If one day we were able to actually determine the values of ${\alpha }_{\ast }$ and ${\alpha }_2$, we would then also know about ${\alpha }_{\ast }-{\alpha }_2$, which is ${\alpha }_0+{\alpha }_1+{\alpha }_3$.

6. Closing Remarks

At present, IR/UV mixing is not really one of the directions that compose quantum-gravity research: quantum-gravity research remains composed of several alternative directions to tackle the ultraviolet challenges that characterize quantum gravity, and then IR/UV mixing is contemplated only when it is found in pursuing one of the ultraviolet research directions. Perhaps the interconnections I have here highlighted might motivate a change in attitude such that IR/UV mixing is by itself one of the central topics of quantum-gravity research.

The opportunities for testing candidate quantum-gravity effects are still very few, even combining ultraviolet and infrared opportunities, but surely by combining an ultraviolet quantum-gravity phenomenology and an infrared quantum-gravity phenomenology, the chances for success increase. Moreover, cases like the phenomenological parameters ${\alpha }_1$ and ${\alpha }_2$, in which the same candidate with new features can have tangible manifestations both in the deep ultraviolet and the deep infrared, might have an important role in those efforts.

On the theory side, among those here reported, perhaps the observations that might motivate more investigative efforts in the future concern spacetime noncommutativity. There appears to be a rather general connection between spacetime noncommutativity and (interestingly different) IR/UV-mixing mechanisms, but the arguments that support it are either merely heuristic or, when they are more robust, lack a satisfactory interpretation (i.e., IR/UV mixing is established robustly without any deep understanding of its emergence in the relevant quantum-gravity scenario). It is natural to expect that a lot could be learned by improving our understanding of the connection between spacetime noncommutativity and IR/UV mixing, both by establishing the links more robustly and by investigating in greater depth which aspects of a spacetime-noncommutativity scenario are responsible for the emergence of IR/UV mixing.