holds in the sense of Gromov–Hausdorff–Prokhorov, where $\mathscr{M}^{0}$ is
the Brownian map and $\mathscr{M}^{\varrho}$ is the Brownian disk with
perimeter $\varrho$ if the latter is non-zero.

  3. (iii)

Suppose that $\nu_{\mathbf{q}}$ satisfies $\mathrm{(H_{1}^{loc})}$. Assume
that $\limsup_{n\to\infty}a_{n}^{-1}\varrho_{n}<\infty$ and for every
$n\geqslant 1$, let $M^{\varrho_{n}}_{E=n}$ have the law
$\mathbf{P}^{\varrho_{n}}_{E=n}$. Then the convergence in distribution

$\left(V(M^{\varrho_{n}}_{E=n}),|2b_{n}|^{-1/2}d_{\mathrm{gr}},p_{\mathrm{unif}}\right)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(\mathscr{T}_{X^{0}},\mathscr{d}_{X^{0}},\mathscr{p}_{X^{0}})$

holds in the sense of Gromov–Hausdorff–Prokhorov, where $X^{0}$ is the
standard Brownian excursion.

  4. (iv)

Suppose that $\nu_{\mathbf{q}}$ satisfies $\mathrm{(H_{1})}$. For every
$n\geqslant 1$, let $M_{E\geqslant n}$ have the law
$\mathbf{P}^{1}_{E\geqslant n}$. Then the convergence in distribution

$\left(V(M_{E\geqslant
n}),|2b_{n}|^{-1/2}d_{\mathrm{gr}},p_{\mathrm{unif}}\right)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(\mathscr{T}_{I^{1/2}X^{0}},\mathscr{d}_{I^{1/2}X^{0}},\mathscr{p}_{I^{1/2}X^{0}})$

holds in the sense of Gromov–Hausdorff–Prokhorov, where $X^{0}$ is the
standard Brownian excursion, and $I$ is independently sampled from the law
$\mathbf{P}(I\geqslant x)=x^{-1}$ for all $x\geqslant 1$.

Finally, the same results hold when the maps are obtained by forgetting the
distinguished vertex in the pointed versions of the laws.

Let us first discuss the last statement since we shall in fact prove the
theorem for pointed maps, thanks to the representation with labelled forests.
Let $\phi:\mathbf{PM}\to\mathbf{M}$ be the projection $\phi((M,\star))=M$
which ‘forgets the distinguished vertex’. We stress that, unless in the case
$S=V$ for which there is no bias, the laws $\mathbf{P}^{\varrho}_{S=n}$ and
$\phi_{\ast}\mathbf{P}^{\varrho,\star}_{S=n}$ on $\mathbf{M}^{\varrho}$ differ
at $n$ fixed. Nonetheless, this bias disappears at the limit as shown in
several works [Abr16, BJM14, BM17, Mar18b, Mar18a].

######  Proposition 4.

Let $\|\cdot\|_{TV}$ denotes the total variation norm; if $\mathbf{q}$, $S$,
and $\varrho_{n}$ are as in Theorem 6, then

$\left\|\mathbf{P}^{\varrho}_{S=n}-\phi_{*}\mathbf{P}^{\varrho,\star}_{S=n}\right\|_{TV}\enskip\mathop{\longrightarrow}_{n\to\infty}\enskip
0.$

The same holds for the laws conditioned to $E\geqslant n$ in the last case of
Theorem 6.

###### Proof.

This result generalises Proposition 12 in [Mar18b] whose proof extends readily
here: for notational convenience, suppose that we are in one of the first
three cases of the theorem (the last case is similar). Let
$\Lambda(T^{\varrho_{n}}_{n,B_{S}})$ denote the number of leaves of the forest
$T^{\varrho_{n}}_{n,B_{S}}$, which equals the number of vertices minus one in
the associated map. Then the above total variation distance is bounded above
by

$\mathbf{E}\left[\left|\frac{1}{\Lambda(T^{\varrho_{n}}_{n,B_{S}})-1}\frac{1}{\mathbf{E}[\frac{1}{\Lambda(T^{\varrho_{n}}_{n,B_{S}})-1}]}-1\right|\right].$

see e.g. the proof of Proposition 12 in [Mar18b]. It thus suffices to prove
that this expectation tends to $0$, which is the content of [Mar18b, Lemma 8].
Again, the proof of this lemma is fairly general once we have the exponential
concentration of the proportion of leaves we obtained in the proof of Lemma 3:
take the sets $A$ and $B$ in (23) to be respectively $A$ and $\\{-1\\}$ here.
∎

We may now easily prove Theorem 6 for pointed maps.

###### Proof of Theorem 6.

The statement (i) for $\alpha\in(1,2)$ follows easily from [Mar19, Theorem 1]
and Theorem 5(ii). Indeed, according to the latter, the factor
$\sigma_{n}^{1/2}$ in [Mar19, Theorem 1] for the random degree sequence of
$M^{\varrho_{n}}_{S=n}$ is of order $a_{n}^{1/2}$; since furthermore
conditional on these degrees the map has the uniform distribution on the sets
of all possible maps, we may then apply [Mar19, Theorem 1] conditional on the
degrees, with $a_{n}^{1/2}$ instead of $\sigma_{n}^{1/2}$, and then average
with respect to the degrees. The statement (ii) for $\alpha=2$ follows
similarly from Theorem 1 and Theorem 5(i); here the scaling constant
$(3/(2\sigma_{n}))^{1/2}$ is given by
$((9\mu_{\mathbf{q}}(B_{S}))/(8a_{n}^{2}))^{1/4}$. The statements (iii) and
(iv) finally follow from Theorem 2 and Theorem 5(iii) and 5(iv) respectively.
∎

###  6.4. Sub-critical maps

Let us end this paper with a similar condensation result for _sub-critical_
maps, that is, when the weight sequence $\mathbf{q}$ is chosen so that the law
$\mu_{\mathbf{q}}$ has mean smaller than one. In order to directly use the
results available in the literature, we shall only work with maps without
boundary, so $\varrho_{n}=1$. The first convergence below, for pointed maps,
is the main result of Janson & Stefánsson [JS15].

######  Theorem 7.

Assume that $\mathbf{q}$ is an admissible sequence such that the law
$\mu_{\mathbf{q}}$ has mean $m_{\mu}=\sum_{k\geqslant
0}k\mu_{\mathbf{q}}(k)<1$.

  1. (i)

Suppose that there exists a slowly varying function $L$ and an index $\beta>1$
such that we have

$\mu_{\mathbf{q}}(k)=k^{-(1+\beta)}L(k),\qquad k\geqslant 1.$

For every $n\geqslant 1$, let $M^{\varrho_{n}}_{E=n}$ have the law
$\mathbf{P}_{E=n}$. Then the convergence in distribution

$\left(V(M_{E=n}),(2n)^{-1/2}d_{\mathrm{gr}},p_{\mathrm{unif}}\right)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(\mathscr{T}_{\gamma^{1/2}X^{0}},\mathscr{d}_{\gamma^{1/2}X^{0}},\mathscr{p}_{\gamma^{1/2}X^{0}})$

holds in the sense of Gromov–Hausdorff–Prokhorov, where $X^{0}$ is the
standard Brownian excursion and $\gamma=1-m_{\mu}$.

  2. (ii)

Under the weaker assumption that there exists a slowly varying function $L$
and an index $\beta>1$ such that

$\mu_{\mathbf{q}}([k,\infty))=k^{-\beta}L(k),\qquad k\geqslant 1,$

if now $M_{E\geqslant n}$ has the law $\mathbf{P}_{E\geqslant n}$, then the
convergence in distribution

$\left(V(M_{E\geqslant
n}),(2n)^{-1/2}d_{\mathrm{gr}},p_{\mathrm{unif}}\right)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(\mathscr{T}_{I^{1/2}X^{0}},\mathscr{d}_{I^{1/2}X^{0}},\mathscr{p}_{I^{1/2}X^{0}})$

holds in the sense of Gromov–Hausdorff–Prokhorov, where $X^{0}$ is the
standard Brownian excursion, and $I$ is independently sampled from the law
$\mathbf{P}(I\geqslant x)=(\frac{1-m_{\mu}}{x})^{\beta}$ for all $x\geqslant
1-m_{\mu}$.

Finally, the same results hold when the maps are obtained by forgetting the
distinguished vertex in the pointed versions of the laws.

###### Proof.

The proof goes exactly as that of Theorem 6 in the case $\alpha=1$. First, the
analog of Theorem 5(iii) is provided by Kortchemski [Kor15]: according to
Theorem 1 there, with the notation of Theorem 5, we have in the first case
that

$n^{-1}\left(\Delta(W_{n}),\Delta^{\prime}(W_{n})\right)\enskip\mathop{\longrightarrow}^{\mathbf{P}}_{n\to\infty}\enskip(1-m_{\mu},0).$

This implies that
$n^{-2}\sum_{k\leqslant\Delta^{\prime}(W_{n})}k(k+1)J_{k}(W_{n})\leqslant
n^{-1}(1+\Delta^{\prime}(W_{n}))^{2}$ converges in probability to $0$. The
analog of Theorem 5(iv) in the second case is provided by Kortchemski &
Richier [KR18]: the convergence in distribution

$n^{-1}\left(\Delta(W_{n}),\Delta^{\prime}(W_{n})\right)\enskip\mathop{\longrightarrow}^{(d)}_{n\to\infty}\enskip(I,0)$

follows from Proposition 9 there, and again this implies that
$n^{-2}\sum_{k\leqslant\Delta^{\prime}(W_{n})}k(k+1)J_{k}(W_{n})$ converges in
probability to $0$. Theorem 2 allows us to conclude in the case of pointed
maps.

In order to consider un-pointed maps, we need an analogous result to
Proposition 4, which holds as soon as the proportion of leaves in the tree
coded by $W_{n}$ or $W_{\geqslant n}$ satisfies an exponential concentration
as in (23). The proof is easily adapted: we only need the analog of Equation
(22) and (21) with $\varrho_{n}=1$ on the hitting time of $-1$; the former, in
the case of the tail conditioning, is given by [KR18, Equation 9] and
references therein, and the latter, in the case of the local conditioning, is
given by [Kor15, Equation 14]. This completes the proof. ∎

## References

  * [Abr16] Céline Abraham.  Rescaled bipartite planar maps converge to the Brownian map.  Ann. Inst. H. Poincaré Probab. Statist., 52(2):575–595, 2016\.
  * [Ald85] David J. Aldous.  Exchangeability and related topics.  In École d’été de probabilités de Saint-Flour, XIII—1983, volume 1117 of Lecture Notes in Math., pages 1–198. Springer, Berlin, 1985.
  * [Ald93] David Aldous.  The continuum random tree. III.  Ann. Probab., 21(1):248–289, 1993.
  * [AMP04] David Aldous, Grégory Miermont, and Jim Pitman.  The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity.  Probab. Theory Related Fields, 129(2):182–218, 2004.
  * [BCM19] Olivier Bernardi, Nicolas Curien, and Grégory Miermont.  A Boltzmann approach to percolation on random triangulations.  Canadian Journal of Mathematics (online first), pages 1–43, 2019\.
  * [BCP03] Jean Bertoin, Loïc Chaumont, and Jim Pitman.  Path transformations of first passage bridges.  Electron. Comm. Probab., 8:155–166, 2003.
  * [BDFG04] Jérémie Bouttier, Philippe Di Francesco, and Emmanuel Guitter.  Planar maps as labeled mobiles.  Electron. J. Combin., 11(1):Research Paper 69, 27, 2004.
  * [Ber18] Quentin Berger.  Notes on random walks in the Cauchy domain of attraction.  Probab. Theory Related Fields (online first), 2018.
  * [Bet10] Jérémie Bettinelli.  Scaling limits for random quadrangulations of positive genus.  Electron. J. Probab., 15:no. 52, 1594–1644, 2010.
  * [Bet15] Jérémie Bettinelli.  Scaling limit of random planar quadrangulations with a boundary.  Ann. Inst. Henri Poincaré Probab. Stat., 51(2):432–477, 2015\.
  * [Bet16] Jérémie Bettinelli.  Geodesics in Brownian surfaces (Brownian maps).  Ann. Inst. Henri Poincaré Probab. Stat., 52(2):612–646, 2016\.
  * [BJM14] Jérémie Bettinelli, Emmanuel Jacob, and Grégory Miermont.  The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection.  Electron. J. Probab., 19:no. 74, 16, 2014.
  * [BM] Jérémie Bettinelli and Grégory Miermont.  Compact Brownian surfaces II: The general case.  _In preparation_.
  * [BM14] Nicolas Broutin and Jean-François Marckert.  Asymptotics of trees with a prescribed degree sequence and applications.  Random Struct. Alg., 44(3):290–316, 2014.
  * [BM17] Jérémie Bettinelli and Grégory Miermont.  Compact Brownian surfaces I: Brownian disks.  Probab. Theory Related Fields, 167(3-4):555–614, 2017.
  * [Bud16] Timothy Budd.  The peeling process of infinite Boltzmann planar maps.  Electron. J. Combin., 23(1):Paper 1.28, 37, 2016.
  * [CLGM13] Nicolas Curien, Jean-François Le Gall, and Grégory Miermont.  The Brownian cactus I. Scaling limits of discrete cactuses.  Ann. Inst. Henri Poincaré Probab. Stat., 49(2):340–373, 2013\.
  * [Fel71] William Feller.  An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney, 1971\.
  * [JS15] Svante Janson and Sigurður Örn Stefánsson.  Scaling limits of random planar maps with a unique large face.  Ann. Probab., 43(3):1045–1081, 2015.
  * [Kal02] Olav Kallenberg.  Foundations of Modern Probability.  Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002.
  * [Kor12] Igor Kortchemski.  Invariance principles for Galton–Watson trees conditioned on the number of leaves.  Stochastic Process. Appl., 122(9):3126–3172, 2012.
  * [Kor15] Igor Kortchemski.  Limit theorems for conditioned non-generic Galton-Watson trees.  Ann. Inst. Henri Poincaré Probab. Stat., 51(2):489–511, 2015\.
  * [Kor17] Igor Kortchemski.  Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees.  Probab. Theory Related Fields, 168(1-2):1–40, 2017.
  * [KR18] Igor Kortchemski and Lo\̈mathrm{i}c Richier.  The boundary of random planar maps via looptrees.  To appear in Ann. Fac. Sci. Toulouse Math. Preprint available at arXiv:1802.00647, 2018\.
  * [KR19] Igor Kortchemski and Lo\̈mathrm{i}c Richier.  Condensation in critical Cauchy Bienaymé–Galton–Watson trees.  Ann. Appl. Probab., 29(3):1837–1877, 2019.
  * [Lei17] Tao Lei.  Scaling limit of random forests with prescribed degree sequences.  Preprint available at arXiv:1704.02064, 2017.
  * [LG07] Jean-François Le Gall.  The topological structure of scaling limits of large planar maps.  Inventiones mathematicae, 169(3):621–670, 2007.
  * [LG13] Jean-François Le Gall.  Uniqueness and universality of the Brownian map.  Ann. Probab., 41(4):2880–2960, 2013.
  * [LGM11] Jean-François Le Gall and Grégory Miermont.  Scaling limits of random planar maps with large faces.  Ann. Probab., 39(1):1–69, 2011.
  * [LGP08] Jean-François Le Gall and Frédéric Paulin.  Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere.  Geom. Funct. Anal., 18(3):893–918, 2008.
  * [Mar18a] Cyril Marzouk.  On scaling limits of planar maps with stable face-degrees.  ALEA Lat. Am. J. Probab. Math. Stat., 15:1089–1122, 2018.
  * [Mar18b] Cyril Marzouk.  Scaling limits of random bipartite planar maps with a prescribed degree sequence.  Random Struct. Alg., 53(3):448–503, 2018.
  * [Mar19] Cyril Marzouk.  On the growth of random planar maps with a prescribed degree sequence.  Preprint available at arXiv:1902.04539, 2019.
  * [Mie08] Grégory Miermont.  On the sphericity of scaling limits of random planar quadrangulations.  Electron. Commun. Probab., 13:248–257, 2008.
  * [Mie09] Grégory Miermont.  Tessellations of random maps of arbitrary genus.  Ann. Sci. Éc. Norm. Supér. (4), 42(5):725–781, 2009.
  * [Mie13] Grégory Miermont.  The Brownian map is the scaling limit of uniform random plane quadrangulations.  Acta Math., 210(2):319–401, 2013.
  * [MM03] Jean-François Marckert and Abdelkader Mokkadem.  The depth first processes of Galton–Watson trees converge to the same Brownian excursion.  Ann. Probab., 31(3):1655–1678, 2003.
  * [MM06] Jean-François Marckert and Abdelkader Mokkadem.  Limit of normalized quadrangulations: The Brownian map.  Ann. Probab., 34(6):2144–2202, 2006.
  * [MM07] Jean-François Marckert and Grégory Miermont.  Invariance principles for random bipartite planar maps.  Ann. Probab., 35(5):1642–1705, 2007.
  * [Pit06] Jim Pitman.  Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2006.  Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard.


# Phenomenology of the dispersion law in a three-dimensional quantum space

Samuel Kováčik,11footnotetext: Corresponding author. Michaela Ďuríšková  and
Patrik Rusnák

###### Abstract

The concept of a quantum structure underlying space has been proposed for
quite some time, yet it has largely eluded direct observation. There is a
prevailing notion that the minuscule effects stemming from quantum space
could, under certain astronomical conditions, accumulate to an observable
magnitude. Typically, these effects are perturbative, limited to the first one
or two orders. This study presents a full dispersion law derived from a three-
dimensional quantum space model. Our primary focus is investigating the in-
vacuo dispersion phenomenon and the threshold anomaly. Furthermore, we briefly
explore the implications of additional spatial dimensions within this
framework and suggest a way of distinguishing between various theories that
agree in the leading order approximation.

##  1 Introduction

The idea of quantum space followed shortly after the formulation of quantum
mechanics; Snyder described the first mathematical model in 1947 [1]. From the
Planck units we can estimate that the quantum structure of space will manifest
at the scale $\lambda\approx 10^{-35}\mbox{m}$, which is many orders away from
the current reach of particle accelerators or laboratory experiments. The
generality of the minimal length scale feature has been reviewed by
Hossenfelder [2], and possible phenomenological scenarios were gathered by
Amelino-Camelia [3] and a growing community recently [4].

In brief, a quantum structure of space seems to be a rather robust prediction
and a common feature across different models. Still, it leads to effects that
are exceedingly difficult to observe. There are, importantly, a few examples
that seem to be within the reach of current technologies. Probably the aspect
most worthy of attention is the wavelength dependant vacuum velocity that can
affect measurably particles moving at the speed of light [5, 3, 6, 7, 8] or
close to it [6, 9, 10]. Recently, observation of TeV photons from the GRB
221009A brought attention to the threshold anomaly; the quantum structure of
space, or some other cause, reduces the interaction between gamma and low-
energy background photons. This would explain the large number of detected TeV
photons since their flux should be within the standard model attenuated [11].
There are some other possible phenomenological aspects which are, at the
current technological level, of smaller importance. Also, perhaps even more
importantly, there are other possible explanations for the observed
discrepancy; for example, axion-like particles are a possible solution for the
threshold anomaly [12, 13, 14]. In summary, we are entering an era in which
astronomical observations might drive the search for physics (well) beyond the
standard model.

Since we expect effects to be considerable for large energies, keeping only
the leading corrections in $E/E_{\mbox{\scriptsize{c}}}$ is often justified.
Here, $E_{\scriptsize{c}}$ is the energy cut-off scale, which might be the
Planck energy scale or can be of a different order. This work presents a full
dispersion law obtained from studying three-dimensional quantum space. A
series of studies showed that the rotationally invariant model can lead to
exact results instead of perturbative results obtained in other descriptions
[15, 16, 17]. Also, this model was analyzed in the context of microscopic
black holes [18, 19, 20], and recently, its matrix formulation has been
proposed, which allows for a wider range of astrophysical studies [21].
Therefore, we discuss this model in the context of quantum-space
phenomenology.

Here is the outline of the paper. Section 2 introduces the model of the
mentioned three-dimensional quantum space. In Section 3, we summarize some
aspects of nontrivial vacuum dispersion. In Section 4, we discuss the
robustness of the predictions and in Section 5, we conclude our findings.

##  2 Vacuum dispersion from an exact formulation of three-dimensional quantum
space

It is often assumed that space has a structure on a microscopic scale. Planck
units give this scale a natural value, but in principle, different options
should be considered unless we have data to pick the correct one. It can be
argued, see [2], that we can expect that on this scale, one cannot measure the
position of a particle with arbitrary precision. This means that in quantum
formalism, we have two position operators that do not commute:

$[\hat{x}_{i},\hat{x}_{j}]=i\hat{\theta}_{ij}.$ (2.1)

Both sides of this equation have the dimension of length squared. A reasonable
choice for the r.h.s. of this equation that leads to a rotationally invariant
model is given by

$\hat{\theta}_{ij}=2\lambda\varepsilon_{ijk}\hat{x}_{k},$ (2.2)

where $\varepsilon$ is the Levi-Civita tensor, and $\lambda$ is a constant
with the dimension of length that dictates the scale of space quantumness. An
obvious feature of this relation is that in the limit $\lambda\rightarrow 0$,
the ordinary quantum mechanics relation is restored, and the model describes
the ordinary three-dimensional space. Therefore, $\lambda$ can be considered a
deformation parameter, usually related to the Planck length. The rotational
symmetry of this relation leads to the exact formula for the Coulomb problem
spectrum in [15].

Models that satisfy the relation 2.2 can be constructed using the
noncommutative start product [22], auxiliary bosonic operators [15, 17] or
matrices [21]. Let us briefly discuss one way to construct a quantum space
that provides insight and intuition. To quantize a theory of something, one
needs to replace Poisson brackets with a commutator, for example,
$\\{x,p\\}\rightarrow\frac{1}{i\hbar}[x,p]$. This is usually done on phase
spaces, which are by default even-dimensional; however, the space
$\textbf{R}^{3}$ is odd-dimensional. There is a workaround — one can first
embed $\textbf{R}^{3}$ into a higher dimensional space with a natural Poisson
structure $\\{z_{\scriptsize{\alpha}},\bar{z}_{\scriptsize{\beta}}\\}$,
quantize it there and return to $\textbf{R}^{3}$. As shown in [23], one can
formulate ordinary $\textbf{R}^{3}$ quantum mechanics in a space with two
complex dimensions, $\textbf{C}^{2}$, quantize it and then consider only
states independent of the extra dimensions. As a result, one is left with a
three-dimensional theory of quantum mechanics in a quantized space. It can be
shown that this theory reproduces the ordinary quantum mechanics in the
$\lambda\rightarrow 0$ limit, justifying the construction.

This yields the starting kit of quantum mechanics: the Hilbert space of states
$\Psi$, the position operators $\hat{x}_{i}$ and the Hamiltonian $\hat{H}$
obtained from the Poisson structure. From this, the velocity was defined and
studied in [24], where the following relation between the free Hamiltonian and
the velocity operator, $\hat{v}_{i}$, was found:

$\hat{H}_{0}=\frac{1-\sqrt{1-m^{2}\lambda^{2}\hat{v}^{2}}}{m\lambda^{2}}=\frac{m\hat{v}^{2}}{2}+\frac{m^{3}\hat{v}^{4}}{8}\lambda^{2}+{\cal{O}}(\lambda^{4}).$
(2.3)

The usual relation is recovered in the $\lambda\rightarrow 0$ limit. We can
also utilize the Legendre transformation,
$\hat{p}=\frac{\partial\hat{H}_{0}}{\partial\hat{v}}=\frac{\hat{mv}}{\sqrt{1-m^{2}\hat{v}^{2}\lambda^{2}}}$,
to find:

$\hat{H}_{0}=\frac{1}{m\lambda^{2}}\left(1-\sqrt{1-\frac{\lambda^{2}\hat{p}^{2}}{1+\lambda^{2}\hat{p}^{2}}}\right)=\frac{\hat{p}^{2}}{2m}-\frac{3\hat{p}^{4}}{8m}\lambda^{2}+{\cal{O}}(\lambda^{4}).$
(2.4)

This expression was derived within nonrelativistic quantum mechanics, and
additional effects might be present in the full theory. However, it is
plausible that the dispersion law sensitive to the scale of
$E_{\mbox{\scriptsize{c}}}$ will be of a similar form, and one can replace
$p^{2}/2m\rightarrow pc$, arguing the universality of the modification as in
[3]. The maximal achievable energy is
$E_{\scriptsize{c}}=\frac{1}{m\lambda^{2}}$, and we will fix this to be the
energy cut-off scale by taking
$\lambda\rightarrow\frac{1}{\sqrt{mE_{\scriptsize{c}}}}$. We will also
consider only energy (momentum) eigenstates satisfying $\hat{H}_{0}\Psi=E\psi$
(or $\hat{p}\Psi=p\Psi$) as it was shown in [24] that these operators commute
and, therefore, can be diagonalized simultaneously. By doing these
modifications and putting the expression in the common form, we obtain the
square of the energy:

$E^{2}(p)=E_{\scriptsize{c}}^{2}\left(1-\sqrt{1-\frac{2pc}{E_{\scriptsize{c}}+2pc}}\right)^{2}=p^{2}c^{2}\left(1-\frac{3pc}{E_{\scriptsize{c}}}+{\cal{O}}\left(E_{\scriptsize{c}}^{-2}\right)\right).$
(2.5)

This form agrees, up to a numerical factor in the leading correction term,
which is ambiguous due to the unfixed value of $E_{\mbox{\scriptsize{c}}}$,
with dispersions laws commonly used in previous works, for example, in [4, 11,
25]. The leading correction is of the order of ${\cal
O}\left(E/E_{\mbox{\scriptsize{c}}}\right)$, which was expected. The particle
accelerator LHC is currently probing the energies of the order of $10\
\mbox{TeV}$, which is 15 orders below the Planck energy scale — we are not
hoping to see signals of quantum space there; that is, unless the existence of
extra spatial dimensions changes the value of Planck scale considerably as we
discuss later. Or, as also possible, the effective scale of space structure is
some orders away from the Planck scale. However, the minuscule effect can
build up for particles that travel across the universe for billions of years
and can lead to measurable effects. Mixing UV and IR physics can apply a lever
that enhances these phenomena. We will now discuss both of these effects in
more detail.

##  3 Phenomenological aspects

###  3.1 Flight time delay

One of the obvious consequences of the nontrivial dispersion law, such as 2.4,
is a wavenumber-dependent wave propagation velocity. Depending on the sign in
front of the first correction term, waves with higher frequencies travel
either faster or slower than the low-frequency ones — and this happens even in
an empty space. Gamma-ray bursts therefore represent an efficient tool for
studying the in-vacuo dispersion, one of the potential observable consequences
of the quantum theory of gravity. Some of its (effective) models predict that
quantum space-time behaves like a dispersive medium. One can envision such a
situation as two photons of different energies emitted from the same source
simultaneously but detected with a time delay. The amplitude of the time delay
is usually negligible, but the effect accumulates along cosmological
distances. If we take the dispersion law for massless particles to be of the
form as assumed in [3]:

$p^{2}\simeq E^{2}+\eta p^{2}\left(E/E_{\mbox{\scriptsize{c}}}\right)^{n},$
(3.1)

the arrival-time difference relation between particles of various energies is,
according to [3], given as:

$\Delta
t\simeq\eta\frac{n+1}{2H_{0}}\frac{p^{n}}{E_{\mathrm{c}}^{n}}\int_{0}^{z}dz^{\prime}\frac{(1+z^{\prime})^{n}}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}.$
(3.2)

Here $\Omega_{m}$, $\Omega_{\Lambda}$, $H_{0}$ are cosmological parameters and
$E_{\mathrm{c}}$ is usually taken as the energy cut-off scale — perhaps the
Planck energy — or the scale of Lorenz invariance violation (LIV). Also, two
options appear in the literature, $n=1,2$, where $n=1$ leads to stronger
effects. While working with redshifts $z\ll 1$, one can simplify the relation
(3.2) into the following form [26]:

$\Delta
t|_{\mathrm{small-z}}\simeq\eta\left(E/E_{\mbox{\scriptsize{c}}}\right)^{n}d,$
(3.3)

where $d=\frac{z}{H_{0}}$ is the distance of the source and $\Delta E$ is the
energy difference between the two considered wavelengths.

We can now follow the strategy of [25] to derive a similar formula for our
dispersion law 2.5. We replace the momentum with the comoving momentum $p/a$,
transform back to velocities by $v=\frac{\partial H}{\partial p}$:

$v(E)=ca^{-1}\left(\frac{aE_{\mbox{\scriptsize{c}}}}{aE_{\mbox{\scriptsize{c}}}+2pc}\right)^{3/2}.$
(3.4)

We can now replace the scale factor $a=\frac{1}{1+z}$ and integrate over to
find that a received signal travelled:

$x(z,E)=\frac{c}{H_{0}}\int\limits_{0}^{z}\left(\frac{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})}{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})+2pc}\right)^{3/2}\frac{dz^{\prime}}{\sqrt{\Omega_{m}\left(1+z^{\prime
3}\right)+\Omega_{\Lambda}}}.$ (3.5)

Importantly, if we have two photons with energies $E_{1}$ and $E_{2}$, the
time difference of their arrivals will be (neglecting subleading effects)

$\displaystyle\Delta t(E_{1},E_{2},z)$ $\displaystyle=$
$\displaystyle\frac{c}{H_{0}}\int\limits_{0}^{z}\left(\left(\frac{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})}{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})+2p_{1}c}\right)^{3/2}-\left(\frac{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})}{E_{\mbox{\scriptsize{c}}}/(1+z^{\prime})+2p_{2}c}\right)^{3/2}\right)$
$\displaystyle\times\frac{dz^{\prime}}{\sqrt{\Omega_{m}\left(1+z^{\prime
3}\right)+\Omega_{\Lambda}}}$ $\displaystyle=$ $\displaystyle-\frac{3c\Delta
E}{H_{0}E_{\mbox{\scriptsize{c}}}}\int\limits_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{m}\left(1+z^{\prime
3}\right)+\Omega_{\Lambda}}}+{\cal{O}}((E/E_{\mbox{\scriptsize{c}}})^{2}),$

where we denoted $E=pc$. This result is consistent, up to a small numerical
factor, with values appearing in the literature. An interesting side note is
that the integral can be expanded in $E$, and each term can be expressed
exactly using hypergeometric functions.

###  3.2 Threshold anomaly

The important aspect of the modified dispersion law is that it affects
particles propagating in the vacuum of intergalactic space. However, this
medium is a vacuum only within the first approximation since it is filled,
among other things, with background photons. A reversed process to electron-
positron annihilation can happen as long as the energy of the photon with
larger energy satisfies, assuming the trivial dispersion law, the following:

$E\geq E_{\mathrm{th}}=\frac{m_{e}^{2}}{E_{b}},$ (3.7)

here $m_{e}$ is the electron mass, $E_{b}$ is the energy of the background
photon and $E_{\mathrm{th}}$ is the threshold energy. This follows from the
conservation of four-momentum as was described, for example, in [27]; however,
we will show a slightly different description. In [28], one can find
reasonable estimates and distributions for energies of background photons of
extragalactic origin, $E_{b}$; the other option is to consider CMB photons
with $E_{b}\sim 10^{-3}\mbox{eV}$. In what follows, we will consider various
values. This interaction causes attenuation of energetic photons travelling on
Mpc and Gpc scales. Photons with energies on TeV scales and from distant
sources have been observed, which left a paradox to be explained; a possible
solution is the discussed correction to the dispersion law [3, 29, 30, 31,
32]. A recent example is the observation of GRB 221009A, for which the
distance was estimated to be 724 Mpc [33] with peak energies reported at least
at 18 TeV [34].

One can think of this process in the following way. The interaction of two
photons with energies $E$ and $E_{\scriptsize{B}}$ can create a particle-
antiparticle pair with particle mass $m=\sqrt{EE_{\scriptsize{B}}}/c^{2}$. As
long as $m$ is smaller than the mass of any particle allowed by conservation
laws, this interaction does not occur. Here, $E_{\scriptsize{B}}$ is
considered fixed and taken to be the energy of the background photons. By
increasing $E$, the electron mass is reached, and pair production becomes
possible; therefore, photons of sufficiently large energies disappear
gradually from the flux. With the considered form of the dispersion law, 2.5,
and other choices with a negative leading correction, the obtainable mass is,
due to the finiteness of $E_{\mbox{\scriptsize{c}}}$, reduced. That means that
for sufficiently large $E$ and $1/E_{\scriptsize{B}}$ is the production
suppressed and the intergalactic space remains transparent, contrary to the
classical prediction.

The minimal mass of a particle produced by a high-energy photon with the
momentum $p$ is

$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{2c^{2}}\sqrt{E^{2}(p)-\left(pc\right)^{2}+2E_{\scriptsize{B}}\left(E(p)+pc\right)})$
$\displaystyle=$
$\displaystyle\frac{1}{2c^{2}}\sqrt{E_{\mbox{\scriptsize{c}}}^{2}\left(1-\sqrt{1-\frac{2pc}{E_{\mbox{\scriptsize{c}}}+2pc}}\right)^{2}-p^{2}c^{2}+2E_{\scriptsize{B}}\left(E_{\mbox{\scriptsize{c}}}\left(1-\sqrt{1-\frac{2pc}{E_{\scriptsize{c}}+2pc}}\right)+pc\right)}$
$\displaystyle\approx$ $\displaystyle
m_{0}\left(1-\frac{3\left(pc\right)^{2}}{8E_{\mbox{\scriptsize{c}}}E_{\scriptsize{B}}}\right)+...$

where $m_{0}=\sqrt{pcE_{\scriptsize{B}}}/c^{2}$ would be the unmodified
threshold of the pair production result. We can observe that modifying the
dispersion law indeed reduces the obtainable mass and suppresses the
production. The result is sensitive to the chosen value of
$E_{\scriptsize{B}}$. Results for different values of the background photon
energy, $E_{\scriptsize{B}}$ and the cut-off energy
$E_{\mbox{\scriptsize{c}}}$, are in Table 1.

$E_{\scriptsize{B}}\setminus E_{\mbox{\scriptsize{c}}}$ [MeV] | $E_{\scriptsize{pl}}$ | $10^{-1}E_{\scriptsize{pl}}$ | $10^{-2}E_{\scriptsize{pl}}$
---|---|---|---
$1\ \mbox{eV}$ | $4.2$ | $3.8$ | $0.019$
$10^{-1}\ \mbox{eV}$ | $1.2$ | $0.0061$ | $<0$
$10^{-2}\ \mbox{eV}$ | $0.0019$ | $<0$ | $<0$
$10^{-3}\ \mbox{eV}$ | $<0$ | $<0$ | $<0$
Table 1: Mass of a particle produced by an interaction between an $18\
\mbox{TeV}$ photon with a background photon of energy $E_{\scriptsize{B}}$
given the relation 3.2 with various values of the fundamental energy scale
$E_{\mbox{\scriptsize{c}}}$ as measured in $\mbox{MeV}/c^{2}$. We can see that
for theories with the fundamental scale away from the Planck length, pair
production is suppressed for all background photons. For the case
$E_{\mbox{\scriptsize{c}}}\sim E_{\scriptsize{pl}}$, only background photons
of lower energies are prevented from the interaction, as for larger values of
$E_{\scriptsize{B}}$, the producible mass is greater than that of electrons
$m_{\scriptsize{e}}=0.511\ \mbox{MeV}/c^{2}$ Figure 1: Maximal achievable mass
for the classical relation from the equation 3.2 with
$E_{\mbox{\scriptsize{c}}}=10^{22}\mbox{MeV}$ and fixed
$E_{\scriptsize{B}}=3\times 10^{-2}\mbox{eV}$. Different lines correspond to
different orders of expansion of 3.2 in $E_{\mbox{\scriptsize{c}}}$. The
uppermost line is the classical (zeroth order) result and the bottom line is
the expansion up to the fourth order; different orders have different grades
of black/grey. The dashed line is the mass of the electron; if it is not
reached, pair production is forbidden. As we can see if
$E_{\mbox{\scriptsize{c}}}\approx E_{\scriptsize{pl}}$, the reaction is
forbidden for background photons with minuscule energies; for smaller values
of $E_{\mbox{\scriptsize{c}}}$, it is prohibitive for the larger part of the
background photon spectrum, even all of it.

###  3.3 Constraints on extra dimensions

The theme of our discussion so far is that there is a good reason to hope that
evidence of new physics could come from astrophysical sources, even though
particle accelerators were the driving force in fundamental physics. A nice
example is the conjectured existence of extra spatial dimensions. Nearly a
decade before the launch of the Large Hadron Collider, it was suggested in
[35, 36, 37] that the existence of extra dimensions would mean that the Planck
scale is shifted — maybe even to electro-weak scale. This hypothesis has been
ruled out, but it is still possible that some extra dimensions are present.
The true Planck scale might still be far from the reach of particle
accelerators — but perhaps not from the observations of the effects discussed
earlier in this paper.

The basic idea is the following. While on a macroscopic scale, the ordinary
Newton’s gravity with force proportional to $1/r^{2}$ would be observed, with
$D=3+d$ dimensions, one would observe $1/r^{2+d}$ dependency when measuring on
a scale proportional to the extra dimension radius, $R$. The deviation from
Newton’s law of gravity has been searched for and not found down to the scale
$52\ \mu\mbox{m}$, see [38, 39].

The existence of these dimensions would mean that the gravitational constant
$G$ observed now is only apparent and not fundamental. This, in turn, shifts
the value of other fundamental scales. For that purpose, we give the
expression between the apparent Planck length, $l_{\scriptsize{Pl}}$, and
fundamental Planck length, $l_{d}$, for $d$ extra spatial dimensions of size
$R$:

$l_{d}=\left(l_{\scriptsize{Pl}}\ R^{d}\right)^{\frac{1}{d+2}}.$ (3.9)

In this notation $l_{\scriptsize{Pl}}=l_{0}$. Research on the effects
discussed in this paper puts constraints on the possible value of the
fundamental energy scale — either the LIV scale, the scale of the quantum
structure of space or something else. The existence of extra spatial
dimensions traversable by gravity would also lead to a different Planck
energy. We can define the ratio of the phenomenological energy scale and the
currently assumed Planck energy scale as:

$\varepsilon=E_{\mbox{\scriptsize{c}}}/E_{\scriptsize{pl}}.$ (3.10)

Expressed using this, the size of uniform extra dimensions is

$R=l_{\scriptsize{Pl}}\ \varepsilon^{-\frac{d+2}{d}}.$ (3.11)

The naive choice would be $\varepsilon\sim 1$, which would keep the scale
unchanged, $R=l_{\scriptsize{Pl}}$, while the aforementioned research suggests
the value of $\varepsilon\gtrsim 10^{-4}$. We can also consider a significant
deviation, say of the order of $\varepsilon\sim 10^{-10}$, for curiosity. One
can then ask what the size of circular extra dimensions would lead to such a
value of $\varepsilon$. The results are gathered in Table 2. We stop at six
extra dimensions, as this is the value considered in superstring theories.

The bottom line is that if our universe had extra spatial dimensions, the
fundamental scales would be different, leading to stronger dispersion effects
we have discussed. By putting constraints on the energy cut-off scale by not
observing them above some magnitude, we can put rather severe constraints on
the size and number of extra spatial dimensions.

$\varepsilon\setminus n$ | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
$10^{-2}$ | -28.8 | -30.8 | -31.5 | -31.8 | -32 | -32.1
$10^{-3}$ | -25.8 | -28.8 | -29.8 | -30.3 | -30.6 | -30.8
$10^{-4}$ | -22.8 | -26.8 | -28.1 | -28.8 | -29.2 | -29.5
$10^{-10}$ | -4.8 | -14.8 | -18.1 | -19.8 | -20.8 | -21.5
Table 2: The table shows the value of $\log_{10}R$ expressed in meters and
given in 3.11. If observations confirmed the value of $\varepsilon\sim
10^{-4}$, we could put constraints on the size of extra dimensions stricter
than direct laboratory experiments on the order of a micrometre. If the
laboratory observed a deviation on that scale, we would get $\varepsilon\sim
10^{-10}$, which would contradict predictions of current models of vacuum
dispersion with a linear correction.

##  4 Observational considerations

The starting point of our analysis was a model of three-dimensional quantum
space and a Hamiltonian operator that was the direct result of the
quantisation process. Naturally, this led to a dispersion relation, which we,
with some assumptions, turned into the form 2.5. When taking only the leading
correction term, our expression agrees with some other values used in
literature, [3, 40]. Even in the case of nontrivial vacuum dispersion, it was
unclear whether to consider a leading correction linear or quadratic in
$E/E_{\mbox{\scriptsize{c}}}$; our analysis favours the former. This is
important because with it, 3.1 produces a measurable effect. Going to the
next-to-leading term leads, however, to corrections that are below the
detection threshold.

Let us compare our result with the one reported from the analysis of the
recent event GRB 221109A [41]. The source of this signal was close,
$z=0.1505$, but due to unprecedented energies, this observation is also
relevant in the context of threshold anomaly, which will be discussed shortly.
The mentioned work operated with the dispersion relation:

$\displaystyle
v(E)=c\left(1-\frac{n+1}{2}\left(\frac{E}{E_{\mbox{\scriptsize{c}}}}\right)^{n}\right),$
(4.1)

where the case close to ours is $n=1$. The leading order correction differs by
a factor of $3$, which, being ${\cal{O}}(1)$, is negligible as it comes with
the undetermined energy scale $E_{\mbox{\scriptsize{c}}}$. The value used for
energy difference in this reference was $99.3\ \mbox{GeV}$, which led to the
prediction of $\delta t\approx 20\ \mbox{s}$. Using the same values of
$E_{\mbox{\scriptsize{c}}},E$ and cosmological parameters $H_{0}=67.3\
\mbox{km}\ \mbox{s}^{-1}\mbox{Mpc}^{-1},\
\Omega_{m}=0.315,\Omega_{\Lambda}=0.685$ we obtain $\delta t\approx 60\
\mbox{s}$. We could obtain the same value by considering a slightly different
value of $E_{\mbox{\scriptsize{c}}}$. If we use
$E_{\mbox{\scriptsize{c}}}\approx E_{\scriptsize{pl}}$, the time-delay value
would be $\Delta t\approx 1.83\mbox{s}$, which is easily measurable if we can
identify a low-energy photon that escaped from the source simultaneously with
a high-energy one. Some efforts along those lines were made in the
aforementioned reference.

For a small satellite detector with the upper energy limit on the order of $1\
\mbox{MeV}$, the time shift with $E_{\mbox{\scriptsize{c}}}\approx
E_{\scriptsize{pl}}$ would be on the order of $10^{-5}\mbox{s}$, which is
below the temporal resolution of $\approx 1\ \mbox{s}$. That means that not
observing the effect across multiple channels restricts the value of the
fundamental energy scale,
$E_{\mbox{\scriptsize{c}}}/E_{\scriptsize{pl}}>10^{-5}$. We can compare this
with the values in Table 2 and conclude that under the aforementioned
assumptions, even this small satellite disfavours extra spatial dimensions
larger than $\approx 10^{-20}\ \mbox{m}$, a stronger constrain than the one
obtained in LHC [42]. Also note in Table 3 that some current and future
telescopes are well prepared to observe this effect, especially by cross-
correlation across multiple channels.

| Temporal resolution | Energy range
---|---|---
GRBAlpha [2023AA...677L...2R] | 1 s | 80 - 950 keV
Fermi GBM [43] | 2 $\mu$s | 8 keV - 40 MeV
Fermi LAT [43] |  $<$ 1 $\mu$s | 20 MeV - 300 GeV
Hermes [44] | fraction of microseconds | keV - MeV
Table 3: Technical limits of some GRB detectors.

The second important effect is the reduced attenuation of energetic photons
due to pair production. Here, the effect is inversely proportional to the
energy of background photon $E_{\scriptsize{B}}$, which is tantalizing as it
can make event next-to-leading terms relevant. In Figure 1, we compare the
minimal producible mass with the expression truncated to one and two levels
––– the difference is obvious. This means that the effect of non-trivial
dispersion law is observable beyond the leading order, given one has access to
the interaction of two photons of vastly different energies; this can be
either relevant for astrophysical observations or maybe even in some future
laboratory settings [45, 46]. The laboratory experiments would be preferable
because the distribution of background photon energies across cosmological
scales might be challenging. Theoretically, one needs a stable source of gamma
photons to traverse an isolated box filled with low-frequency radiation to
make the threshold 3.7 important.

Another interesting option is to analyse pulsar sources that might not be so
distant but are stable and allow for signal accumulation. For example, it has
been reported that the Vela pulsar is a source of $~{}20\ \mbox{TeV}$ photons
[47]. The time it takes for light to travel from this source is $~{}3\times
10^{10}\mbox{s}$, and the spinning frequency of the source is $~{}10\
\mbox{Hz}$. Assuming $E_{\mbox{\scriptsize{c}}}$ is on the Planck scale and
taking $\Delta E\sim 10\ \mbox{TeV}$, this translates into flight time delays
of $10^{-5}\mbox{s}$ — which should be with observational limits. Given the
stability of the source, one could hope for a robust dataset on which a cross-
correlation analysis could be done to observe an energy-dependent time delay.
In this regard, we would not be limited by the flux sensitivity of smaller
detectors.

##  5 Conclusion

We have touched upon several open questions in physics, with the most
significant one being whether space possesses a quantum structure. If so, what
structural equation does describe it? And what is the value of the parameters
in it? We assumed the answer to the first question is positive and have
derived a dispersion law from a robust model of three-dimensional quantum
space. Some dispersion laws in the literature have a linear correction term,
and some have a quadratic correction term. Our result agrees with the linear
form, which leads to stronger observational effects.

Their scale depends on the energy value at which the underlying physics
becomes relevant. The natural value of this would be the Planck energy scale,
but other values have been considered in the literature, somewhere between 3
and 4 orders below the Planck scale. For example, some approaches [48] to the
string theory give the value of $10^{11}\mbox{GeV}$, eight orders below the
Planck scale. If the energy cut-off scale was also of this order, the flight-
time delays would be not relevant only for GRB photons but also for photons of
energy multiple orders lower. For larger energies, this would lead to delays
on the order of years instead of seconds. Therefore, it is safe to assume we
should be operating roughly in the interval
$10^{-8}E_{\scriptsize{pl}}<E_{\mbox{\scriptsize{c}}}\leq
E_{\scriptsize{pl}}$, perhaps somewhere in the upper half (on a logarithmic
scale).

Observing some phenomenological aspects of modified dispersion law can, in
principle, be expected with reasonable confidence as various theories contain
this feature. It should also be noted that some theories, for example, quantum
spaces created by a $*$-product, have a trivial dispersion law [49]. The
distinction line between those remains elusive. That means that confirming the
nontrivial corrections to the dispersion law would suggest some of the novel
theories are correct; not observing the deviation from the dispersion law
would not disprove them all. This might be a frustrating state of matter, but
on the contrary, it is astonishing that we have a window into physics well
behind the reach of particle accelerators.

The novel aspect of our research is that we have obtained a result in
agreement with others used in the literature from a well-formulated and
studied model of three-dimensional quantum space. The mathematical structures
of this model have been studied thoroughly [15, 16, 23] and so have been some
phenomenological aspects, such as the behaviour of regular black holes [18,
20] or confined particles [50, 51]. In this regard, this paper adds new
phenomenological aspects to this mode. From this point of view, the results
presented connect to other areas in physics and help build a foundation of an
interconnected phenomenological web that is eventually needed for establishing
or disproving a new theory.

The flight-time delay we have derived agrees with previous works and depends
mostly on the energy scale on which the modification to the dispersion law
becomes important. Our considered correction is first-order, and the
phenomenon manifests without invoking artificially low energy scales. However,
in the case of threshold anomalies, we have observed different results by
including various levels of approximation, which means this effect can be used
to distinguish between various theories with nontrivial vacuum dispersion.
Perhaps, in the future, laboratory tests of the interaction of high and low-
energy photons with the required precision might be feasible.

Regarding observations and experiments, we are in a state where brinks of new
physics have been suggested by data [6], and scheduled satellite missions
should solidify — or disprove — this evidence. Therefore, attention should be
paid to all models of quantum space and other new physics features that can be
verified in the near future.

The levers of astronomical data — either the accumulation of minuscule effects
or interaction with low-energy background photons — offer better hope of
finding signals of Planck-scale physics than particle accelerators. To prove
this point, we have made a little detour to verify what the known results
infer about the extra spatial dimensions under similar assumptions when
discussed in the context of particle accelerators. We have seen that if the
presence of extra dimensions indeed caused a different value of the Planck
scale, and this, in turn, would affect the scale of deviation from nontrivial
dispersion law, the coherency of typical gamma signals can put severe
constraints on the number and size of extra dimensions — order of magnitude
more restrictive than particle accelerators.

Astronomical searches for physics well behind the standard model are
impressive. With this research, we must have added a new piece to the overall
mosaic.

## Acknowledgement

This research was supported by VEGA 1/0025/23 and the MUNI Award for Science
and Humanities, funded by the Grant Agency of Masaryk University. We want to
thank Benedek Bukor, Juraj Tekel, Veronika Gáliková, Peter Prešnajder,
Wladimiro Leone, and Norbert Werner for their valuable comments and
discussion.

## References

  * [1] H.S. Snyder, _Quantized space-time_ , _Phys. Rev._ 71 (1947) 38.
  * [2] S. Hossenfelder, _Minimal Length Scale Scenarios for Quantum Gravity_ , _Living Rev. Rel._ 16 (2013) 2 [1203.6191].
  * [3] G. Amelino-Camelia, _Quantum-Spacetime Phenomenology_ , _Living Rev. Rel._ 16 (2013) 5 [0806.0339].
  * [4] A. Addazi et al., _Quantum gravity phenomenology at the dawn of the multi-messenger era—A review_ , _Prog. Part. Nucl. Phys._ 125 (2022) 103948 [2111.05659].
  * [5] G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, _Tests of quantum gravity from observations of gamma-ray bursts_ , _Nature_ 393 (1998) 763 [astro-ph/9712103].
  * [6] G. Amelino-Camelia, G. D’Amico, G. Rosati and N. Loret, _In-vacuo-dispersion features for GRB neutrinos and photons_ , _Nature Astron._ 1 (2017) 0139 [1612.02765].
  * [7] G. Amelino-Camelia, _Testable scenario for relativity with minimum length_ , _Phys. Lett. B_ 510 (2001) 255 [hep-th/0012238].
  * [8] G. Amelino-Camelia, D. Frattulillo, G. Gubitosi, G. Rosati and S. Bedic, _Phenomenology of DSR-relativistic in-vacuo dispersion in FLRW spacetime_ , 2307.05428.
  * [9] G. Amelino-Camelia, M.G. Di Luca, G. Gubitosi, G. Rosati and G. D’Amico, _Could quantum gravity slow down neutrinos?_ , _Nature Astron._ 7 (2023) 996 [2209.13726].
  * [10] G. Amelino-Camelia, L. Barcaroli, G. D’Amico, N. Loret and G. Rosati, _IceCube and GRB neutrinos propagating in quantum spacetime_ , _Phys. Lett. B_ 761 (2016) 318 [1605.00496].
  * [11] H. Li and B.-Q. Ma, _Searching Lorentz invariance violation from cosmic photon attenuation_ , _Eur. Phys. J. C_ 83 (2023) 192 [2210.05563].
  * [12] D.J.E. Marsh, _Axion Cosmology_ , _Phys. Rept._ 643 (2016) 1 [1510.07633].
  * [13] M. Meyer, D. Horns and M. Raue, _First lower limits on the photon-axion-like particle coupling from very high energy gamma-ray observations_ , _Phys. Rev. D_ 87 (2013) 035027 [1302.1208].
  * [14] M.A. Sanchez-Conde, D. Paneque, E. Bloom, F. Prada and A. Dominguez, _Hints of the existence of Axion-Like-Particles from the gamma-ray spectra of cosmological sources_ , _Phys. Rev. D_ 79 (2009) 123511 [0905.3270].
  * [15] V. Gáliková, S. Kováčik and P. Prešnajder, _Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space_ , _J. Math. Phys._ 54 (2013) 122106 [1309.4614].
  * [16] P. Prešnajder, V. Gáliková and S. Kováčik, _Laplace-runge-lenz Vector in Quantum Mechanics in Noncommutative Space_ , _Acta Polytech._ 54 (2014) 149.
  * [17] V. Gáliková and P. Prešnajder, _Coulomb Scattering in Non-commutative Quantum Mechanics_ , _Acta Polytech._ 53 (2013) 427.
  * [18] S. Kováčik, _$R^{3}_{\Lambda}$ -inspired black holes_, _Mod. Phys. Lett. A_ 32 (2017) 1750130.
  * [19] S. Kováčik, _Microscopic black holes as probes for quantum gravity_ , _PoS_ CORFU2021 (2022) 300 [2203.12332].
  * [20] S. Kováčik, _Hawking-radiation recoil of microscopic black holes_ , _Phys. Dark Univ._ 34 (2021) 100906 [2102.06517].
  * [21] S. Kováčik and J. Tekel, _Fuzzy Onion as a Matrix Model_ , 2309.00576.
  * [22] J.M. Gracia-Bondia, F. Lizzi, G. Marmo and P. Vitale, _Infinitely many star products to play with_ , _JHEP_ 04 (2002) 026 [hep-th/0112092].
  * [23] S. Kováčik and P. Prešnajder, _Alternative Description of Magnetic Monopoles in Quantum Mechanics_ , _Eur. Phys. J. C_ 78 (2018) 745 [1804.04015].
  * [24] S. Kováčik and P. Prešnajder, _The velocity operator in quantum mechanics in noncommutative space_ , _J.Math.Phys._ 54 (2013) 102103 [1309.4592].
  * [25] U. Jacob and T. Piran, _Lorentz-violation-induced arrival delays of cosmological particles_ , _JCAP_ 01 (2008) 031 [0712.2170].
  * [26] G. Amelino-Camelia and L. Smolin, _Prospects for constraining quantum gravity dispersion with near term observations_ , _Phys. Rev. D_ 80 (2009) 084017.
  * [27] H. Li and B.-Q. Ma, _Threshold anomalies of ultra-high energy cosmic photons due to Lorentz invariance violation_ , _JHEAp_ 32 (2021) 1 [2105.06647].
  * [28] A. Franceschini and G. Rodighiero, _The extragalactic background light revisited and the cosmic photon-photon opacity_ , _Astron. Astrophys._ 603 (2017) A34 [1705.10256].
  * [29] T. Kifune, _Invariance violation extends the cosmic ray horizon?_ , _Astrophys. J. Lett._ 518 (1999) L21 [astro-ph/9904164].
  * [30] R. Aloisio, P. Blasi, P.L. Ghia and A.F. Grillo, _Probing the structure of space-time with cosmic rays_ , _Phys. Rev. D_ 62 (2000) 053010 [astro-ph/0001258].
  * [31] G. Amelino-Camelia and T. Piran, _Planck scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV gamma paradoxes_ , _Phys. Rev. D_ 64 (2001) 036005 [astro-ph/0008107].
  * [32] R.J. Protheroe and H. Meyer, _An Infrared background TeV gamma-ray crisis?_ , _Phys. Lett. B_ 493 (2000) 1 [astro-ph/0005349].
  * [33] E. Burns et al., _GRB 221009A: The BOAT_ , _Astrophys. J. Lett._ 946 (2023) L31 [2302.14037].
  * [34] S. Sahu, B. Medina-Carrillo, G. Sánchez-Colón and S. Rajpoot, _Deciphering the $\sim$18 TeV Photons from GRB 221009A_, _Astrophys. J. Lett._ 942 (2023) L30 [2211.04057].
  * [35] N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, _The Hierarchy problem and new dimensions at a millimeter_ , _Phys. Lett. B_ 429 (1998) 263 [hep-ph/9803315].
  * [36] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, _New dimensions at a millimeter to a Fermi and superstrings at a TeV_ , _Phys. Lett. B_ 436 (1998) 257 [hep-ph/9804398].
  * [37] P.C. Argyres, S. Dimopoulos and J. March-Russell, _Black holes and submillimeter dimensions_ , _Phys. Lett. B_ 441 (1998) 96 [hep-th/9808138].
  * [38] W.-H. Tan, S.-Q. Yang, C.-G. Shao, J. Li, A.-B. Du, B.-F. Zhan et al., _New Test of the Gravitational Inverse-Square Law at the Submillimeter Range with Dual Modulation and Compensation_ , _Phys. Rev. Lett._ 116 (2016) 131101.
  * [39] J.G. Lee, E.G. Adelberger, T.S. Cook, S.M. Fleischer and B.R. Heckel, _New Test of the Gravitational $1/r^{2}$ Law at Separations down to 52 $\mu$m_, _Phys. Rev. Lett._ 124 (2020) 101101 [2002.11761].
  * [40] Y. Huang and B.-Q. Ma, _Lorentz violation from gamma-ray burst neutrinos_ , _Communications Physics_ 1 (2018) 62 [1810.01652].
  * [41] J. Zhu and B.-Q. Ma, _Light speed variation from GRB 221009A_ , _Journal of Physics G Nuclear Physics_ 50 (2023) 06LT01 [2210.11376].
  * [42] N. Deutschmann, T. Flacke and J.S. Kim, _Current LHC Constraints on Minimal Universal Extra Dimensions_ , _Phys. Lett. B_ 771 (2017) 515 [1702.00410].
  * [43] D.J. Thompson and C.A. Wilson-Hodge, _Fermi Gamma-Ray Space Telescope_ , in _Handbook of X-ray and Gamma-ray Astrophysics_ , p. 29 (2022), DOI.
  * [44] F. Fiore, L. Burderi, M. Lavagna, R. Bertacin, Y. Evangelista, R. Campana et al., _The HERMES-technologic and scientific pathfinder_ , in _Space Telescopes and Instrumentation 2020: Ultraviolet to Gamma Ray_ , J.-W.A. den Herder, S. Nikzad and K. Nakazawa, eds., vol. 11444 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ , p. 114441R, Dec., 2020, DOI [2101.03078].
  * [45] V.M. Budnev, I.F. Ginzburg, G.V. Meledin and V.G. Serbo, _The Two photon particle production mechanism. Physical problems. Applications. Equivalent photon approximation_ , _Phys. Rept._ 15 (1975) 181.
  * [46] L. Esnault, E. d’Humières, A. Arefiev and X. Ribeyre, _Electron-positron pair production in the collision of real photon beams with wide energy distributions_ , _Plasma Phys. Control. Fusion_ 63 (2021) 125015 [2103.09099].
  * [47] H.E.S.S. collaboration, _Discovery of a radiation component from the Vela pulsar reaching 20 teraelectronvolts_ , _Nature Astron._ 7 (2023) 1341 [2310.06181].
  * [48] I. Antoniadis and B. Pioline, _Low scale closed strings and their duals_ , _Nucl. Phys. B_ 550 (1999) 41 [hep-th/9902055].
  * [49] R.J. Szabo, _Quantum field theory on noncommutative spaces_ , _Phys. Rept._ 378 (2003) 207 [hep-th/0109162].
  * [50] B. Bukor and J. Tekel, _On quarkonium masses in 3D non-commutative space_ , _Eur. Phys. J. Plus_ 138 (2023) 499 [2209.09028].
  * [51] S. Kováčik and P. Prešnajder, _NC plane waves, Casimir effect and flux tube potential with Lüscher terms_ , 1603.00292.


* [CK12] Andrew M. Childs and Robin Kothari.  Quantum query complexity of minor-closed graph properties.  SIAM Journal on Computing, 41(6):1426–1450, 2012.
  * [CLSW17] Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong.  Subquadratic submodular function minimization.  In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 1220–1231, 2017.
  * [DHHM06] Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla.  Quantum query complexity of some graph problems.  SIAM J. Comput., 35(6):1310–1328, 2006.
  * [FK99] Alan Frieze and Ravi Kannan.  Quick approximation to matrices and applications.  Combinatorica, 19(2):175–200, 1999.
  * [FMV11] Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák.  Maximizing non-monotone submodular functions.  SIAM J. Comput., 40(4):1133–1153, 2011.
  * [Gal14] François Le Gall.  Improved quantum algorithm for triangle finding via combinatorial arguments.  In 55th IEEE Annual Symposium on Foundations of Computer Science, pages 216–225, 2014.
  * [GK98] Vladimir Grebinski and Gregory Kucherov.  Reconstructing a hamiltonian cycle by querying the graph: Application to DNA physical mapping.  Discret. Appl. Math., 88(1-3):147–165, 1998.
  * [GK00] Vladimir Grebinski and Gregory Kucherov.  Optimal reconstruction of graphs under the additive model.  Algorithmica, 28(1):104–124, 2000.
  * [GLS81] Martin Grötschel, László Lovász, and Alexander Schrijver.  The ellipsoid method and its consequences in combinatorial optimization.  Combinatorica, 1(2):169–197, 1981.
  * [GLS88] Martin Grötschel, László Lovász, and Alexander Schrijver.  Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics.  Springer, 1988.
  * [Har08] Nicholas J. A. Harvey.  Matchings, matroids and submodular functions.  PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2008.
  * [HMT88] András Hajnal, Wolfgang Maass, and György Turán.  On the communication complexity of graph properties.  In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 186–191, 1988.
  * [HRRS19] Yassine Hamoudi, Patrick Rebentrost, Ansis Rosmanis, and Miklos Santha.  Quantum and classical algorithms for approximate submodular function minimization.  Quantum Inf. Comput., 19(15&16):1325–1349, 2019.
  * [IKL+12] Gábor Ivanyos, Hartmut Klauck, Troy Lee, Miklos Santha, and Ronald de Wolf.  New bounds on the classical and quantum communication complexity of some graph properties.  In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, pages 148–159, 2012.
  * [Jia20] Haotian Jiang.  Minimizing convex functions with integral minimizers.  CoRR, abs/2007.01445, 2020.
  * [JKM13] Stacey Jeffery, Robin Kothari, and Frédéric Magniez.  Nested quantum walks with quantum data structures.  In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1474–1485, 2013.
  * [Kre95] I. Kremer.  Quantum communication.  Technical report, Hebrew University of Jerusalem, 1995.
  * [LMS17] Troy Lee, Frédéric Magniez, and Miklos Santha.  Improved quantum query algorithms for triangle detection and associativity testing.  Algorithmica, 77(2):459–486, 2017.
  * [LSW15] Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong.  A faster cutting plane method and its implications for combinatorial and convex optimization.  In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1049–1065. IEEE Computer Society, 2015.
  * [LW19] Anita Liebenau and Nick Wormald.  Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph.  CoRR, abs/1702.08373, 2019.
  * [Maz10] Hanna Mazzawi.  Optimally reconstructing weighted graphs using queries.  In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 608–615. SIAM, 2010.
  * [MN20] Sagnik Mukhopadhyay and Danupon Nanongkai.  Weighted min-cut: sequential, cut-query, and streaming algorithms.  In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 496–509, 2020\.
  * [MSS07] Frédéric Magniez, Miklos Santha, and Mario Szegedy.  Quantum algorithms for the triangle problem.  SIAM Journal on Computing, 37(2):413–424, 2007.
  * [RSW18] Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg.  Computing exact minimum cuts without knowing the graph.  In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, pages 39:1–39:16, 2018\.
  * [RWZ20] Cyrus Rashtchian, David P. Woodruff, and Hanlin Zhu.  Vector-matrix-vector queries for solving linear algebra, statistics, and graph problems.  In Proceedings of the 24th International Conference on Randomization and Computation, to appear, 2020.
  * [Sto83] Larry J. Stockmeyer.  The complexity of approximate counting (preliminary version).  In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 118–126, 1983\.
  * [SYZ04] Xiaoming Sun, Andrew Chi-Chih Yao, and Shengyu Zhang.  Graph properties and circular functions: How low can quantum query complexity go?  In 19th Annual IEEE Conference on Computational Complexity, pages 286–293, 2004.
  * [Zha05] Shengyu Zhang.  On the power of Ambainis lower bounds.  Theoretical Computer Science, 339(2-3):241–256, 2005.