eqn

$\displaystyle\begin{split}\BODY\end{split}$ (1)

eqn*

$\displaystyle\begin{split}\BODY\end{split}$

# Discreteness of Space from GUP in Strong Gravitational Fields

Ashmita Das<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Saurya Das<EMAIL_ADDRESS>Theoretical Physics Group and Quantum Alberta, Department of Physics and
Astronomy, University of Lethbridge, 4401 University Drive, Lethbridge,
Alberta T1K 3M4, Canada  Elias C. Vagenas<EMAIL_ADDRESS>Theoretical
Physics Group, Department of Physics, Kuwait University, P.O. Box 5969, Safat
13060, Kuwait

###### Abstract

A large class of quantum theories of gravity show that the Heisenberg’s
uncertainty principle is modified to the “Generalised Uncertainty Principle”
(GUP) near the Planckian scale. It has also been shown that the GUP induces
perturbative corrections to all quantum mechanical Hamiltonians, even at low
energies, and thereby introduces Planck scale corrections to the Schrödinger
equation and to the relativistic quantum mechanical equations. Some of these
corrections give rise to potentially measurable effects in the low-energy
laboratory. Another prediction of these corrections is that a measured length
must be quantized, as seen by studying the solutions of the GUP modified
Schrödinger, Klein-Gordon, and Dirac equations in a one, two, and three
dimensional box. This result was subsequently extended to spacetimes with weak
gravitational fields. In this work, we further extend this length quantization
to spacetimes with strong gravitational fields and show that this result
continues to hold, thereby showing that it is robust.

##  I Introduction

It is well known that the standard quantum field theory (QFT) is successful up
to a certain energy scale $\Lambda\ll M_{Pl}c^{2}$, where $M_{Pl}$ is the
Planck mass and $M_{Pl}c^{2}\sim 10^{19}~{}GeV$ is the Planck energy. As a
result, QFT generally excludes the energy scale associated with quantum
gravitational interactions. A related issue is that in quantum gravity (QG),
one expects quantum fluctuations of the background spacetime, whereas standard
QFT assumes a fixed background spacetime on which quantum fluctuation of
matter and gauge fields are studied. As a result, a complete and consistent
theory of QG is yet to be formulated, although there are promising candidate
theories such as String Theory, Loop Quantum Gravity, Causal Dynamical
Triangulations, Doubly Special Relativity (DSR) etc. A consistent theory of QG
may be a gateway to the unification of all fundamental forces of nature. One
shortcoming of all such theories is the complete absence of direct or indirect
experimental evidence in support of or contradicting them. This being clearly
undesirable, it is important to explore potential signatures of these theories
in current or future experiments. This has been the subject of study of
Quantum Gravity Phenomenology AmelinoCamelia:2008qg ; Hossenfelder:2012jw .

In this respect, one has often exploited a robust prediction of candidate
theories of QG, namely a minimum uncertainty in position measurement to
$\mathcal{O}$($l_{Pl}$) with $l_{Pl}\sim 10^{-35}~{}m$ to be the Planck
length, the corresponding existence of a minimum measurable length
Amati:1988tn ; Garay:1994en ; Kempf:1994su , and the modification of the
Heisenberg uncertainty principle (HUP) to the so-called Generalised
Uncertainty Principle (GUP) Maggiore:1993rv ; Maggiore:1993zu ;
Maggiore:1993kv ; Scardigli:1999jh .

The GUP implies a modification of the standard Heisenberg commutator, i.e.,
$[x_{i},p_{j}]=i\hbar\delta_{ij}$, by terms induced from QG. Although these
terms imply significant contributions near the Planck length or energy scale,
they give rise to potentially measurable effects at much lower energy scales.
A generalization of the GUP was proposed in Refs. Das:2008kaa ; Das:2010sj ;
Das:2010sj ; Das:2009hs ; Ali:2009zq ; Ali:2010yn ; Das:2010sj  of the form

$\displaystyle[x_{i},\,p_{j}]=i\hbar\bigg{[}\delta_{ij}-$
$\displaystyle\alpha\bigg{(}p\,\delta_{ij}+\frac{p_{i}p_{j}}{p}\bigg{)}+\alpha^{2}\bigg{(}p^{2}\,\delta_{ij}+3\,p_{i}p_{j}\bigg{)}\bigg{]}$
(2a) $\displaystyle[p_{i},\,p_{j}]=[x_{i},\,x_{j}]=0~{}.$ (2b)

This modification of the standard Heisenberg commutator corresponds to the
following modified position-momentum uncertainty relation

$\displaystyle\Delta x\,\Delta p$ $\displaystyle\geqslant$
$\displaystyle\,\frac{\hbar}{2}\bigg{[}1-2\alpha\braket{p}+4\alpha^{2}\braket{p^{2}}\bigg{]}$
(3) $\displaystyle\geqslant$
$\displaystyle\,\frac{\hbar}{2}\bigg{[}1+\bigg{(}\frac{\alpha}{\sqrt{\braket{p^{2}}}}+4\alpha^{2}\bigg{)}\,\Delta
p^{2}+4\alpha^{2}\braket{p}^{2}-2\alpha\sqrt{\braket{p^{2}}}\bigg{]}$

where $i$ and $j$ take values $1,2,3$, the magnitude of the squared momentum
is $p^{2}=\sum_{j=1}^{3}\,p_{j}p_{j}$, and
$\alpha=\alpha_{0}/M_{Pl}c=\alpha_{0}l_{Pl}/\hbar$ with $\alpha_{0}$ to be a
dimensionless constant, sometimes assumed to be ${\cal O}(1)$111Recently, in
Ref. Scardigli:2016pjs , using a GUP with only a quadratic term in momentum, a
numerical value of the dimensionless GUP parameter $\beta_{0}$ (with
$\beta_{0}\sim\alpha^{2}_{0}$) was obtained, namely $82\pi/5$. Furthermore, in
Ref. Vagenas:2018zoz  a similar analysis with linear $+$ quadratic GUP yielded
$\alpha_{0}$ to be proportional to powers of the dimensionless ratio
($M_{Pl}$/M).. since GUP as formulated allows for a general $\alpha_{0}$ and
one does not yet have a direct measurement of quantum gravity parameters, we
leave it arbitrary. This in turn implies the existence of an intermediate
length scale and imposes meaningful bounds on the quantum gravity parameters,
which can in principle be measurable in future experiments Das:2008kaa ;
Das:2010sj ; Das:2009hs ; Ali:2009zq .

The above form of GUP is compatible with the modification of uncertainty
principle, as proposed in String Theory, DSR, and black hole (BH) physics. For
alternate forms of GUP, we refer our readers to Refs. Kempf:1994su ;
Kempf:1996fz ; Brau:1999uv ; Hossenfelder:2003jz ; Das:2008kaa ; Adler:1999bu
; Scardigli:2003kr ; Jizba:2009qf  and for phenomenological implications to
Refs. Ali:2010yn ; Ali:2011fa ; Basilakos:2010vs ; Scardigli:2014qka . Note
that having been derived as a consequence of DSR theories, although the form
of GUP as given in Eq. (3) is not manifestly Lorentz invariant, it is
covariant under the non-linear Lorentz transformations in DSR theories
Magueijo:2001cr ; AmelinoCamelia:2000mn ; AmelinoCamelia:2000ge ;
Magueijo:2002am ; Cortes:2004qn . The aforementioned forms of the modified
commutation relation, namely Eq. (2a), and of the GUP, namely Eq. (3), imply a
minimum measurable length and a maximum measurable momentum of the form

$\displaystyle\Delta x\,\geqslant\,(\Delta x)_{{\rm
min}}\,\approx\,\alpha_{0}\,l_{Pl}$ (4a) $\displaystyle\Delta
p\,\leqslant\,(\Delta p)_{{\rm
max}}\,\approx\,\frac{M_{Pl}\,c}{\alpha_{0}}~{}.$ (4b)

For ease of calculations, one defines the physical position and momentum
operators (which are no longer canonically conjugate) in terms of auxiliary
and “canonical” variables $x_{0i}$ and $p_{0i}$, such that

$\displaystyle x_{i}\,=\,x_{0i},\,\,\,\,\,\,\,\,p_{i}=p_{0i}\left(1-\alpha
p_{0}+2\alpha^{2}p_{0}^{2}\right)$ (5)
$\displaystyle[x_{0i},p_{0j}]\,=\,i\hbar\,\delta_{ij}$ (6)

with $p_{0i}=-i\hbar\frac{\partial}{\partial x_{0i}}$ to be interpreted as the
components of the low-energy momentum $p_{0}$, where the QG effects are
negligible, and $p_{0}^{2}\equiv\sum_{i=1}^{3}p_{0i}p_{0i}$. On the other
hand, $p_{i}$ may be regarded as the momentum at “high energies”. Using the
above equations, one can indeed show that Eq. (2a) is satisfied.

In the current paper, we re-examine an interesting aspect of GUP, which is the
prediction that not only is there a minimum length, but that all measurable
lengths are quantized and discrete. Note that the former does not
automatically imply the latter. A similar set of results follows for area and
volume measurements as well. It has been shown earlier that this discreteness
holds not only for flat background spacetimes, but also in the presence of
weak gravity. Specifically, in Ref. Ali:2009zq , the authors have shown that
the one-dimensional space confining an elementary particle must be discrete,
implying all measurable lengths are quantized in units of a fundamental length
scale, which can be the Planck length. Subsequently, in Ref. Das:2010zf , this
was extended for a relativistic particle confined in a rectangular as well as
in a spherical box in one, two, and three dimensional space by solving the
GUP-modified Klein Gordon and Dirac equations. It was shown that the length,
area, and volume of the box are quantized in units of a fundamental length
scale. Extending the results of flat spacetime to weakly curved spacetime, the
authors of Ref. Deb:2016psq  have shown that the quantization of lengths,
areas, and volumes continue to hold in the weak gravitational regime. These
results support the fundamentally discrete nature of space and a breakdown of
the spacetime continuum picture near the Planck scale. Here we will explore
the extent to which the length quantization holds for strong gravity, i.e., in
the background of spacetimes of high curvature. As a concrete example of a
spacetime with strong gravity, we first consider the Schwarzschild BH metric
in its high curvature region. Without exploiting any other BH property such as
its horizon or singularity, we show that our result for the quantization of
the length of the box continues to hold. Then, we obtain similar results for
the case of Reissner-Nordström (RN) BH and for the case of the cosmological
Friedmann-Lema$\hat{{\rm i}}$tre-Robertson-Walker (FLRW) spacetime. In the
above background spacetimes, we apply the aforementioned GUP, which is
covariant under the non-linear Lorentz transformation. Following our earlier
works, our strategy would be to solve the scalar field equation in the
background of the aforementioned spacetimes, impose “box boundary conditions”
such that the field is confined to a finite region of space and show that such
a region can only be of quantized or discrete size. In other words, the length
measurements, which require the confinement of one or multiple particles
(described by the scalar field) within a boundary, signifying the end points
of the length to be measured, imply in turn that the measurable lengths are
discrete as opposed to continuous. Deriving this in the context of strong
gravity completes this program. Although discreteness of lengths, areas, and
volumes have been derived from other approaches in the past, our approach here
for the length quantization is relatively simpler and does rely neither on any
particular approach of quantum gravity, nor does it require additional
assumptions Thiemann:1996at ; Rovelli:1994ge ; Nicolai:2005mc .

The remainder of the paper is organized as follows. In sections II, III, and
IV we present the analysis for the quantum character of space in the strong
gravity regions, as produced by the Schwarzschild BH, RN BH, and FLRW metrics,
respectively. Finally, in section V, we discuss our findings and conclude.
Appendices are provided for the paper to be self-sufficient. Throughout the
paper, we consider the metric signature to be $(-,+,......+)$.

##  II GUP in Schwarzschild BH spacetime

In this section we study the scalar field equation in the strong gravitational
field of the Schwarzschild BH spacetime. We start working in $D$ spacetime
dimensions with coordinates ($t,r,\chi_{1},\chi_{2},...\chi_{D-2}$) and, thus,
the Schwarzschild BH metric reads Myers:1986un

$\displaystyle
ds^{2}=-f(r)c^{2}dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega^{2}_{D-2}$ (7)

with $f(r)=\bigg{(}1-\frac{2\,GM}{c^{2}\,r^{D-3}}\bigg{)}$ and
$d\Omega_{D-2}^{2}=d\chi_{1}^{2}+{\rm sin}^{2}\chi_{1}\,d\chi_{2}^{2}+{\rm
sin}^{2}\chi_{1}\,{\rm sin}^{2}\chi_{2}\,d\chi_{3}^{2}+{\rm
sin}^{2}\chi_{1}.....{\rm sin}^{2}\chi_{D-3}\,d\chi^{2}_{D-2}$, where
$\Omega_{D-2}$ is the area of a unit sphere $S^{D-2}$. The radius of the event
horizon is given as $r_{H}=\big{(}\frac{2\,GM}{c^{2}}\big{)}^{1/(D-3)}$ with
$M$ to be the BH mass. It should be stressed that the above BH spacetime
consists of three distinct regions: (i) the asymptotically flat region, i.e.,
$r\to\infty$, (ii) the region near the event horizon of the BH, i.e., $r\sim
r_{H}$, and (iii) the region near the BH curvature singularity, i.e., $r\to
0$. Region (iii) is the strong gravity region in which the metric given by Eq.
(7) reduces to

$ds^{2}=\frac{2\,GM}{c^{2}\,r^{D-3}}c^{2}dt^{2}-\frac{c^{2}\,r^{D-3}}{2\,GM}dr^{2}+r^{2}d\Omega^{2}_{D-2}~{}.$
(8)

At this point it should be stressed that we will keep the initial formalism
general for any $D$ since this is straightforward and it may be useful for
future generalizations of our work. We will then specialize to $D=4$.
It is well known that the scalar field or Klein-Gordon (KG) equation in any
$D$-dimensional curved spacetime is written as (□+m2c2ℏ2)Φ=0 . Assuming
$\Phi(t,r,\chi_{1},\chi_{2},\ldots,\chi_{D-2})=\tau(t)\psi(r)\,Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})$
and separating variables, Eq. (8) now reads (see Appendix A)

$\displaystyle\underbrace{-\frac{\hbar^{2}}{r^{D-2}}\frac{d}{dr}\bigg{[}r^{D-2}\frac{d\psi(r)}{dr}\bigg{]}}_{p_{0}^{2}}\,\,f(r)-\hbar^{2}\,f^{\prime}(r)\frac{d\psi}{dr}+\bigg{[}\frac{\hbar^{2}\,l(l+1)}{r^{2}}-\,\frac{E^{2}}{c^{2}\,f(r)}+m^{2}c^{2}\bigg{]}\psi(r)=0$
(9)

where $l=0,\,1,\,2\,\ldots$ is the orbital quantum number and $E$ the energy
of the scalar field. It is easily seen that Eq. (9) portrays the radial
equation of motion for a scalar field in the Schwarzschild BH background. At
this point we need to modify the squared low-energy 3-momentum, i.e.,
$p_{0}^{2}$, due to the GUP and for this reason we utilize Eq. (5)222The
angular part can be GUP modified as well Bosso:2016frs , although since it
does not contain any black hole parameters, it is insensitive to them.
Therefore, the result, i.e., discreteness of space in those directions, would
not differ from those found in flat spacetimes as well as in weak
gravitational backgrounds Ali:2009zq ; Ali:2011fa ; Das:2010zf ; Deb:2016psq
.. Therefore, the squared low-energy 3-momentum, i.e., $p_{0}^{2}$, will be
transformed to the high-energy 3-momentum, i.e., $p_{i}^{2}$, as follows
Ali:2009zq ; Das:2010zf

$p_{0}^{2}\to\,p_{i}^{2}=-\hbar^{2}\,\nabla^{2}\,\psi(r)-2\,i\alpha\hbar^{3}\,\nabla^{2}(\nabla\psi(r))+5\alpha^{2}\hbar^{4}\,\nabla^{2}(\nabla^{2}\psi(r))$
(10)

where the Laplace operator is written in the Cartesian coordinates. Hence, we
transform the operator to the spherical polar coordinates and so we use
$p_{i}^{2}$ in Eq. (9). Thus, we obtain the GUP modified scalar field equation

$\displaystyle-\frac{\hbar^{2}}{r^{D-2}}\frac{d}{dr}\bigg{[}r^{D-2}\frac{d\psi(r)}{dr}\bigg{]}-\frac{2\,i\alpha\hbar^{3}}{r^{D-2}}\,\bigg{(}\frac{d}{dr}\bigg{[}r^{D-2}\frac{d}{dr}\bigg{]}\frac{d\psi}{dr}\bigg{)}+\frac{5\alpha^{2}\hbar^{4}}{r^{D-2}}\frac{d}{dr}\bigg{[}r^{D-2}\frac{d}{dr}\bigg{]}\bigg{[}\frac{1}{r^{D-2}}\frac{d}{dr}\bigg{\\{}r^{D-2}\frac{d\psi(r)}{dr}\bigg{\\}}\bigg{]}$
$\displaystyle-\frac{\hbar^{2}\,f^{\prime}(r)}{f(r)}\frac{d\psi}{dr}+\bigg{[}\frac{\hbar^{2}\,l(l+1)}{f(r)\,r^{2}}-\,\frac{E^{2}}{c^{2}\,f^{2}(r)}+\frac{m^{2}c^{2}}{f(r)}\,\bigg{]}\psi(r)=0~{}.$
(11)

The above equation can be viewed as the master equation for the massive KG
field in the background of a stationary and spherically symmetric spacetime.

Now we focus on a $4$-dimensional spacetime, namely $D=4$, for which Eq. (11)
reduces to

$\displaystyle-\frac{2\hbar^{2}}{r}\psi^{\prime}(r)-\hbar^{2}\,\psi^{\prime\prime}(r)-$
$\displaystyle\frac{4i\alpha\hbar^{3}}{r}\,\psi^{\prime\prime}(r)-2i\alpha\hbar^{3}\,\psi^{\prime\prime\prime}(r)+\frac{20\alpha^{2}\hbar^{4}}{r}\,\psi^{\prime\prime\prime}(r)+5\alpha^{2}\hbar^{4}\,\psi^{\prime\prime\prime\prime}(r)$
(12)
$\displaystyle-\frac{\hbar^{2}\,f^{\prime}(r)}{f(r)}\,\psi^{\prime}(r)+\bigg{[}\frac{\hbar^{2}\,l(l+1)}{f(r)\,r^{2}}-\,\frac{E^{2}}{c^{2}f^{2}(r)}+\frac{m^{2}c^{2}}{f(r)}\,\bigg{]}\psi(r)=0~{}.$

Utilizing the following approximations in the $r\rightarrow 0$ limit

$\displaystyle\lim_{r\to
0}f(r)\,\approx\,\bigg{(}-\frac{2GM}{rc^{2}}\bigg{)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lim_{r\to
0}f^{\prime}(r)\approx\,\bigg{(}\frac{2GM}{c^{2}r^{2}}\bigg{)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lim_{r\to
0}\frac{f^{\prime}(r)}{f(r)}\approx\,-\frac{1}{r}$ (13)

and omitting ${\cal O}(\alpha^{2})$ terms, Eq. (12) after taking the limit
$r\to 0$, reduces to

$-\hbar^{2}\,\psi^{\prime}(r)-4i\alpha\hbar^{3}\,\psi^{\prime\prime}(r)+\bigg{[}\frac{c^{2}\hbar^{2}\,l(l+1)}{-2GM}\bigg{]}\,\psi(r)=0~{}.$
(14)

The above equation has a solution of the form 333For $\alpha\to 0$, Eq. (14)
becomes a first order differential equation and its solution contains one
unknown constant. It can be shown by imposing $\alpha\to 0$ limit in Eq. (15),
which is the solution for second order differential Eq. (14), that only the
term associated with constant $C_{1}$ matches exactly with the solution of the
first order differential equation.

$\displaystyle\psi(r)=C_{1}\,{\rm
exp}\bigg{[}\frac{ir}{8\alpha\hbar}\bigg{\\{}1-\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}+C_{2}\,{\rm
exp}\bigg{[}\frac{ir}{8\alpha\hbar}\bigg{\\{}1+\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}~{}.$
(15)

Now we impose the boundary conditions on solution given by Eq. (15). To
determine the boundary conditions near the $r\to 0$ region of the
Schwarzschild BH, we adopt a similar setup as for a particle confined in a
box. Due to the spherical symmetry of the Schwarzschild BH, we imagine a black
sphere of radius $L$ inside the event horizon of BH, namely $L<\,r_{H}$. Thus,
the boundary conditions turned out to be

$\psi(r=0)\,=\,0\,\,\,\,\,\mbox{and}\,\,\,\,\,\psi(r=L)=0~{}.$ (16)

At this point it should be stressed that at $r=0$ a physical (curvature)
singularity occurs. However, normalizable wavefunctions should remain finite
there and therefore the above boundary condition at $r=0$ is justified.
Furthermore, as we show in Appendix B, if one sets $\psi(r=\epsilon)=0$ with
$\epsilon\approx 0$ and then take the limit $\epsilon\rightarrow 0$, the
obtained quantization condition is identical with the one as obtained here by
setting $\psi(r=0)=0$.

The boundary condition at $r=0$ on the wave function Eq. (15) yields

$C_{1}=\,-\,C_{2}$ (17)

which now reads

$\displaystyle\psi(r)=\,\tilde{C}\,{\rm
exp}\bigg{(}\frac{i\rho}{\alpha}-\frac{i\pi}{2}\bigg{)}\,\sin\bigg{[}\frac{\rho}{\alpha}\,(1-i\alpha\beta)^{1/2}\bigg{]}$
(18)

where $\tilde{C}=2C_{1}\equiv|\tilde{C}|\,{\rm exp}(-i\,\theta_{\tilde{c}})$,
$\beta=\frac{16\hbar\,l(l+1)}{r_{H}}$, and $\rho=\frac{r}{8\hbar}$. Next, we
impose the boundary condition at $r=L$ on the wave function given by Eq. (18)
and this yields

$\psi(r=L)=|\tilde{C}|\,{\rm
exp}\bigg{(}\frac{iL}{8\alpha\hbar}-\frac{i\pi}{2}-i\theta_{\tilde{c}}\bigg{)}\,\sin\bigg{[}\frac{L}{8\alpha\hbar}\,(1-i\alpha\beta)^{1/2}\bigg{]}=0~{}.$
(19)

This leads to two possibilities:

$\displaystyle{\rm Case\,\,1\,:}\,\,\,\,\,\,\,\,\,\,\,\,\,$ $\displaystyle{\rm
exp}\bigg{(}\frac{i\,L}{8\alpha\hbar}-\frac{i\pi}{2}-i\theta_{\tilde{c}}\bigg{)}=0$
$\displaystyle{\rm Case\,\,2\,:}\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\sin\bigg{[}\frac{L}{8\alpha\hbar}\,(1-i\alpha\beta)^{1/2}\bigg{]}=0~{}.$
(20)

Case 1 does not give rise to any consistent condition while, from Case 2, we
obtain

$\displaystyle\bigg{[}\frac{L}{8\alpha\hbar}-\,\frac{L}{8\hbar}\frac{i\beta}{2}+\,\frac{L}{8\hbar}\frac{\alpha\beta^{2}}{8}+O(\alpha^{2})\bigg{]}\,=\,n\pi$
$\displaystyle\implies\,\frac{L}{8\alpha\hbar}\,\bigg{[}1+\,\frac{\alpha^{2}\beta^{2}}{8}\bigg{]}\,=\,n\pi$
$\displaystyle\implies\,\frac{L}{8\alpha\hbar}\,\approx\,n\pi\,\bigg{[}1-32\,\frac{\alpha_{0}^{2}\,l_{pl}^{2}\,l^{2}(l+1)^{2}}{r_{H}^{2}}\bigg{]}$
$\displaystyle\implies\,\frac{L}{8\alpha\hbar}\,\approx\,n\pi~{}.$ (21)

Therefore, omitting $O(\alpha^{2})$ terms, we end up with a length
quantization condition of the form

$L=8n\pi\alpha_{0}l_{pl}$ (22)

with $n\,\in\,\mathbb{N}$.

A number of comments are in order. First, the length quantization condition is
independent of the curvature of spacetime and the mass of the test field. This
makes the length quantization condition robust and trustworthy. Second,
starting from Eq. (14), we have been mostly working only with the linear order
of the GUP parameter, i.e., $\alpha$. For Case 2 here, we retained the
$O(\alpha^{2})$ term until we obtain the condition in Eq. (21) to demonstrate
that the term includes gravitation/curvature in terms of the Schwarzschild BH
radius $r_{H}$. Third, our result suggests that the space in a strong gravity
region of the Schwarzschild BH spacetime is indeed quantized in units of a
fundamental length scale and the discrete nature of space continues to hold.

##  III GUP in RN BH spacetime

In this section we study the scalar field equation in the strong gravitational
field of the RN spacetime. The $4$-dimensional RN metric in spherical polar
coordinates can be described as

$\displaystyle
ds^{2}=-f(r)c^{2}dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}\,d\Omega^{2}$ (23)

with

$\displaystyle
f(r)=\bigg{(}1-\frac{r_{s}}{r}+\frac{r_{Q}^{2}}{r^{2}}\bigg{)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r_{s}=\frac{2GM}{c^{2}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r_{Q}^{2}=\frac{Q^{2}\,G}{4\pi\epsilon_{0}\,c^{4}}~{}.$
(24)

Using the following approximations in the $r\to 0$ limit for the RN metric, we
obtain

$\displaystyle\lim_{r\to
0}f(r)\,\approx\,-\frac{r_{s}}{r}\bigg{(}1-\frac{r_{Q}^{2}}{r_{s}\,r}\bigg{)}\,\approx\,\,\frac{r_{Q}^{2}}{r^{2}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lim_{r\to
0}f^{\prime}(r)\approx\,-\bigg{(}\frac{2r_{Q}^{2}}{r^{3}}\bigg{)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lim_{r\to
0}\frac{f^{\prime}(r)}{f(r)}\approx\,-\,\frac{2}{r}~{}.$ (25)

For the scalar field in RN background one can indeed use the master equation
for a 4-dimensional spacetime as given by Eq. (12). Thus, using Eq. (25) into
the master equation and implementing the approximations for $r\to 0$ limit as
adopted in case of Schwarzschild BH, we get444Since we study the high
curvature region of the black hole spacetimes, namely $r\rightarrow 0$, in
this regime, the Reissner-Nordström metric (which goes as $1/r^{2}$) does not
smoothly go over to the Schwarzschild metric (which goes as $1/r$). This is
the reason why Eqs.(14) and (26) are different and one cannot smoothly go from
one to the other.

$\displaystyle
4i\,\alpha\hbar^{3}\psi^{\prime\prime}(r)\,-\,20\alpha^{2}\,\hbar^{4}\,\psi^{\prime\prime\prime}(r)\,=\,0~{}.$
(26)

It is evident that in Eq. (26) if we keep terms up to the linear order in
$\alpha$, no GUP modifications will show up. This leads us to keep terms up to
the quadratic order of $\alpha$ in the GUP modified KG equation. Therefore, we
write

$\displaystyle
i\,\psi^{\prime\prime}(r)\,-\,5\alpha\,\hbar\,\psi^{\prime\prime\prime}(r)\,=\,0$
(27)

and solving the above equation, we obtain

$\displaystyle\psi(r)=\,(-25\,\alpha^{2}\,\hbar^{2}\,C_{1})\,e^{ir/5\,\alpha\,\hbar}\,+\,C_{2}\,r\,+\,C_{3}~{}.$
(28)

Therefore, in case of the KG equation in the background of the strong gravity
region of the RN spacetime, the corrections due to GUP start to emerge from
the quadratic order of $\alpha$. This feature distinguishes the strong gravity
region of RN from that of Schwarzschild spacetime.

Following a similar analysis as in section II, we implement the boundary
condition at $r=0$ and obtain the condition

$\displaystyle C_{3}=\,25\,C_{1}\alpha^{2}\,\hbar^{2}~{}.$ (29)

Then, implementing the boundary condition at $r=L$, we obtain

$\displaystyle C_{2}\,L+\,25\,C_{1}\alpha^{2}\,\hbar^{2}\,\bigg{[}1-\,{\rm
cos}\bigg{(}\frac{L}{5\,\alpha\hbar}\bigg{)}\bigg{]}+i\,25\,C_{1}\alpha^{2}\,\hbar^{2}\,{\rm
sin}\bigg{(}\frac{L}{5\,\alpha\hbar}\bigg{)}\,=\,0~{}.$ (30)

Equating the imaginary parts from both sides of Eq. (30), we obtain a
quantization condition similar to Eq. (21)

$\displaystyle{\rm sin}\bigg{(}\frac{L}{5\,\alpha\hbar}\bigg{)}\,=\,0$
$\displaystyle\implies\,\frac{L}{5\,\alpha\hbar}\,=\,n_{1}\pi$ (31)

where $n_{1}\,\in\,\mathbb{N}$.

Equating the real parts from both sides of Eq. (30), we obtain the following
condition

$\displaystyle{\rm
cos}\bigg{(}\frac{L}{5\,\alpha\hbar}\bigg{)}\,=\,1+\frac{C_{2}\,L}{25\,C_{1}\alpha^{2}\,\hbar^{2}}~{}.$
(32)

The LHS of the above equation contains the even powers of $L$, thus, two
conditions emerge of the following form

$\displaystyle{\rm condition}\,\,1:\,\,$ $\displaystyle{\rm
cos}\bigg{(}\frac{L}{5\,\alpha\hbar}\bigg{)}\,=\,1\implies\,\frac{L}{5\,\alpha\hbar}\,=\,2\,n_{2}\,\pi$
(33) $\displaystyle{\rm condition}\,\,2:\,\,$
$\displaystyle\,\frac{C_{2}\,L}{25\,C_{1}\alpha^{2}\,\hbar^{2}}\,=\,0$ (34)

where, $n_{2}\,\in\,\mathbb{N}$. From condition 2, $C_{2}$ turns out to be
zero as $L\neq 0$.

Combining the conditions Eqs. (31) and (33) which have to be satisfied
simultaneously, we end up with a length quantization condition of the form

$L=10n\pi\alpha_{0}l_{pl}$ (35)

with $n\,\in\,\mathbb{N}$. Our result suggests that the space in a strong
gravity region of the RN BH spacetime is indeed quantized in units of a
fundamental length scale and the discrete nature of space continues to hold.

##  IV GUP in FLRW spacetime

In this section we study the scalar field equation in the background of the
FLRW spacetime. Studying the KG equation in the background of FLRW spacetime
will lead us to the discrete nature of space as well, as we shall see.

We consider the observed spatially flat FLRW metric in spherical polar
coordinates

$\displaystyle
ds^{2}=-c^{2}dt^{2}+\,a^{2}(t)\bigg{[}dr^{2}+\,r^{2}\,d\Omega^{2}\bigg{]}$
(36)

where $a(t)$ is the scale factor of the Universe. As is well known, the above
spacetime correctly describes the Universe at large scales for all epochs
except the very early Planck epoch. This includes the inflationary epoch,
radiation (RD) and matter dominated (MD) phases, as well as the current
accelerated expanding phase. To identify the strong gravity regime for the
FLRW spacetime, we first do a comparative study of the various phases starting
from the inflationary epoch to the MD epoch and estimate the scalar curvature
in each epoch.

###  IV.1 Identification of strong gravity domain in FLRW background

The Ricci scalar for the FLRW metric is given by

$\displaystyle\mathcal{R}\,=\,\frac{6}{a^{2}\,c^{2}}\,\bigg{(}a\ddot{a}+\,\dot{a}^{2}\bigg{)}$
(37)

where $\dot{}\equiv d/dt$. Using the Einstein’s equation for FLRW metric one
obtains the Friedmann equations as

$\left.\begin{aligned}
&\frac{\ddot{a}}{a}\,=\,-\,\frac{4\pi\,G}{3\,c^{2}}\,\,(\rho+\,3\,p)\\\
&H^{2}=\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}\,=\,\frac{8\pi\,G\rho}{3\,c^{2}}\end{aligned}\,\,\,\,\,\,\,\,\right\\}$
(38)

where $\omega=\frac{p}{\rho}$ is the parameter specifying the equation of
state, $\rho$ is the proper energy density, $p$ the pressure in the rest frame
of the proper fluid, and $H$ the Hubble parameter. Using the above equations,
we write the curvature scalar as following

$\displaystyle\mathcal{R}\,=\,\frac{8\pi\,G\,\rho}{c^{4}}\,\,(1-3\,\omega)~{}.$
(39)

It is well known that $\rho_{inf}(\sim\,10^{60}\,{\rm
Gev}^{4})\gg\,\rho_{RD}\,(\sim\,{\rm MeV}^{4})\,>\,\rho_{MD}\,(\sim\,{\rm
eV}^{4})$, where $\rho_{inf},\,\rho_{RD},\,\rho_{MD}$ are the proper energy
densities of inflationary, RD, and MD phases of the Universe, respectively
Baumann:2009ds  . Now using $\rho_{inf},\,\rho_{RD},\,\rho_{MD}$ and
($\omega=\,-1,\,1/3,2/3$) for inflationary, RD and MD phases, respectively,
one can perceive that the curvature scalar corresponding to the inflationary
epoch is much larger than the curvature scalar for the RD and MD eras.

###  IV.2 GUP modified KG equation in the FLRW background

In the background of the FLRW metric (36), the scalar field equation (8) takes
the form

$\displaystyle-\frac{1}{c^{2}\,a^{3}(t)}\,\partial_{t}\bigg{[}a^{3}(t)\,\partial_{t}\,\Phi\bigg{]}+\frac{1}{a^{2}(t)r^{2}}\partial_{r}\bigg{[}r^{2}\,\partial_{r}\,\Phi\bigg{]}+\frac{1}{a^{2}(t)r^{2}\,{\rm
sin}\theta}\partial_{\theta}\bigg{[}{\rm
sin}\theta\,\partial_{\theta}\,\Phi\bigg{]}$
$\displaystyle+\frac{1}{a^{2}(t)r^{2}{\rm
sin}^{2}\theta}\partial_{\phi}^{2}\,\Phi\,-\,\frac{m^{2}\,c^{2}}{\hbar^{2}}\Phi\,=\,0~{}.$
(40)

Furthermore, assuming
$\Phi(t,r,\theta,\phi)=\tau(t)\psi(r)Y_{l,m}(\theta,\phi)$ the separated time
and space parts are

$\displaystyle\hbar^{2}\,\frac{d}{dt}\bigg{[}a^{3}(t)\,\frac{d}{dt}\,\tau(t)\bigg{]}+m^{2}c^{4}\,a^{3}(t)\,\tau(t)+\,E^{2}\,a(t)\tau(t)=0$
(41)
$\displaystyle-\frac{\hbar^{2}}{r^{2}}\,\frac{d}{dr}\bigg{[}r^{2}\,\frac{d}{dr}\,\psi(r)\bigg{]}+\frac{\hbar^{2}}{r^{2}}\,l(l+1)\,\psi(r)-\frac{E^{2}}{c^{2}}\,\psi(r)=0~{}.$
(42)

As before, we focus on the radial part (42) and incorporate the GUP to linear
order in $\alpha$ to get

$\displaystyle\frac{2\hbar^{2}}{r}\,\frac{d\psi(r)}{dr}\,+\hbar^{2}\,\frac{d^{2}\psi(r)}{dr^{2}}+\frac{4\,i\alpha\,\hbar^{3}}{r}\,\frac{d^{2}\psi(r)}{dr^{2}}+2\,i\alpha\,\hbar^{3}\,\frac{d^{3}\psi(r)}{dr^{3}}-\frac{\hbar^{2}}{r^{2}}\,l(l+1)\,\psi+\frac{E^{2}}{c^{2}}\,\psi(r)\,=\,0~{}.$
(43)

It is noteworthy that the above equation holds for all the epochs of the
Universe including strong and weak gravity regimes. This implies that any
space quantization obtained from the above will also hold for strong and weak
gravity regimes. Now, we study Eq.(43) under the small and large $r$ condition
in our subsequent analysis.

For small values of $r$, i.e, $r\to 0$, Eq.(43) reads

$\displaystyle\,2\,i\alpha\,\hbar\,\frac{d^{2}\psi(r)}{dr^{2}}\,+\frac{d\psi(r)}{dr}\,-\,\frac{l(l+1)}{2r}\,\psi(r)\,=\,0~{}.$
(44)

For large values of $r$, i.e., $r\to\infty$, Eq. (43) reads

$\displaystyle
2\,i\alpha\,\hbar^{3}\,\frac{d^{3}\psi(r)}{dr^{3}}+\hbar^{2}\,\frac{d^{2}\psi(r)}{dr^{2}}+\frac{E^{2}}{c^{2}}\,\psi(r)\,=\,0~{}.$
(45)

Solving Eq.(44), we get

$\displaystyle\psi(r)=\,C_{1}\frac{i\,_{1}F_{1}\left(1-\frac{l^{2}}{2}-\frac{l}{2}\,;\,\,2\,\,;\,\,\frac{ir}{2\alpha\hbar}\right)}{2\alpha\hbar}\,r\,+\,C_{2}\,\,G_{1,2}^{2,0}\left(-\frac{ir}{2\alpha\hbar}\bigg{|}\begin{array}[]{c}\frac{1}{2}\left(l^{2}+l+2\right)\\\
0,1\\\ \end{array}\right)\,$ (48)

where
${}_{1}F_{1}\left(1-\frac{l^{2}}{2}-\frac{l}{2}\,;\,\,2\,\,;\,\,\frac{ir}{2\alpha\hbar}\right)$
is a confluent hypergeometric function and $G_{1,2}^{2,0}$ denotes a Meijer-G
function. We set the boundary conditions for small r to be

$\psi(r\,\to 0)\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\psi(r=L)=0~{}.$ (49)

The first condition demands that

$\displaystyle\frac{C_{2}}{\Gamma\bigg{[}\frac{1}{2}\left(l^{2}+l+2\right)\bigg{]}}\,=\,0\,\,\implies\,\,C_{2}\,\,=\,\,0~{}.$
(50)

Implementing the second boundary condition, we obtain

$\displaystyle\psi(L)\,=\,\,_{1}F_{1}\left(1-\frac{l^{2}}{2}-\frac{l}{2}\,;\,\,2\,\,;\,\,\frac{iL}{2\alpha\hbar}\right)\,\,=\,\,0~{}.$
(51)

Restricting ourselves to the test field $\Phi$ (and hence $\psi(r)$)
preserving the homogeneity and isotropy of the background spacetime, i.e. for
$l=0$, we have
${}_{1}F_{1}\left(1-\frac{l^{2}}{2}-\frac{l}{2}\,;\,\,2\,\,;\,\,\frac{iL}{2\alpha\hbar}\right)\,=\,\frac{2i\,\alpha\hbar}{L}\left(e^{\frac{iL}{2\alpha\hbar}}-1\right)$,
so Eq. (51) becomes

$\displaystyle\frac{2i\,\alpha\hbar}{L}\left(e^{\frac{iL}{2\alpha\hbar}}-1\right)\,=\,0$
$\displaystyle\implies\,$ $\displaystyle\bigg{[}\bigg{\\{}{\rm
cos}\bigg{(}\frac{L}{2\alpha\hbar}\bigg{)}-1\bigg{\\}}+i\,{\rm
sin}\bigg{(}\frac{L}{2\alpha\hbar}\bigg{)}\bigg{]}\,=\,0$ (52)

and we end up with a length quantization condition of the form

$L=4n\pi\alpha_{0}l_{pl}$ (53)

with $n\,\in\,\mathbb{N}$.

We now turn our attention to large $r$. It is easily seen that Eq. (45) is
identical to that for a particle in a box in flat spacetime as in Ref.
Ali:2009zq . For a detailed analysis, we refer the reader to Eq. (11) of Ref.
Ali:2009zq  and the subsequent discussion. Solving Eq. (45) and keeping terms
up to the leading order in $\alpha$, we obtain a solution for $\psi(r)$ of the
form

$\displaystyle\psi(r)=\,A\,e^{ik^{\prime}\,r}+\,B\,e^{-ik^{\prime\prime}\,r}+\,Ce^{\frac{i\,r}{2\alpha\hbar}}$
(54)

where, $k^{\prime}=k(1+k\alpha\hbar)$, $k^{\prime\prime}=k(1-k\alpha\hbar)$,
and $k^{2}=\frac{E^{2}}{c^{2}\hbar^{2}}$. It should be pointed out that in the
limit $\alpha\to 0$, Eq. (45) reduces to the one-dimensional Schr$\ddot{{\rm
o}}$dinger equation. The appearance of the first term in Eq. (45) is due to
the GUP and as a consequence, an additional oscillatory solution appears in
Eq. (54). For this to go away in the $\alpha\rightarrow 0$ limit, we require
$\lim_{\alpha\to 0}\,|C|\,=\,0$. Furthermore, for simplification we consider
$A$ to be real and absorb any phase factor of $A$ within $\psi(r)$.

Next we set the boundary conditions in the form
$\psi(r=L_{1})\,=\,\psi(r=L_{2})\,=\,0$. We impose these boundary conditions
in Eq. (54) and consider $L=\,L_{2}\,-\,L_{1}$, with $L$ to be the
characteristic length of the space where the particle resides and is a small
quantity. We also expand ${\rm exp}[i(\ldots)L)]$ and keep terms up to the
leading order in $L$. Thus, we obtain

$\displaystyle
e^{\frac{iL_{2}}{2\alpha\hbar}}\,(i\,L)\bigg{[}A\,e^{ik^{\prime}L_{1}}\,\bigg{(}\frac{1}{2\alpha\hbar}-\,k^{\prime}\bigg{)}+B\,e^{-ik^{\prime\prime}L_{1}}\,\bigg{(}\frac{1}{2\alpha\hbar}+\,k^{\prime\prime}\bigg{)}\bigg{]}\,=\,0~{}.$
(55)

From the above equation, three possibilities emerge

$\left.\begin{aligned} &{\rm
Case\,1:}\,\,\,\,\,\,\,\,\,\,\,\,\,&&e^{\frac{iL_{2}}{2\alpha\hbar}}=0\\\
&{\rm Case\,2:}\,\,\,\,\,\,\,\,\,\,\,\,\,&&L=0\\\ &{\rm
Case\,3:}\,\,\,\,\,\,\,\,\,\,\,\,\,&&A\,e^{ik^{\prime}L_{1}}\,\bigg{(}\frac{1}{2\alpha\hbar}-\,k^{\prime}\bigg{)}+B\,e^{-ik^{\prime\prime}L_{1}}\,\bigg{(}\frac{1}{2\alpha\hbar}+\,k^{\prime\prime}\bigg{)}=0\end{aligned}\right\\}~{}.$
(56)

As can be seen, Case 1 does not give a condition on $L$. Case 2 does not turn
out to be useful either because we chose the boundaries such that $L\neq 0$.
Finally, Case 3 yields

$\displaystyle
B\,=\,A\,e^{2i\,k\,L_{1}}\,\,\bigg{[}\,\,\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,\,\bigg{]}~{}.$
(57)

Now we explore Case 3 by substituting Eq. (57) in Eq.(54), and using $r=L_{1}$
and $C=\,|C|\,e^{-i\,\theta_{C}}$, we obtain

$\displaystyle\frac{2\,A\,k}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,=\,e^{i\bigg{(}\frac{L_{1}}{2\alpha\hbar}-kL_{1}-\theta_{C}\bigg{)}}(|C|\,i\alpha\,\hbar\,L_{1}\,k^{2}-|C|)$
(58)

where in the above equation, we have expanded $e^{i\,k^{2}\alpha\hbar L_{1}}$
and kept terms up to the linear order in $\alpha$, i.e.,
$e^{-ik^{2}\alpha\hbar L_{1}}\approx(1-ik^{2}\alpha\hbar L_{1})$ since
$k^{2}\alpha\hbar L_{1}$ is small. Then, in Eq. (58), this term is multiplied
with $|C|$ which is also small (as explained three (3) lines below Eq. (54))
and, thus, we can neglect the first term in the RHS of Eq. (58) compared to
the second one.

$\displaystyle|C|\,{\rm
cos}\bigg{[}\frac{L_{1}}{2\alpha\hbar}-kL_{1}-\theta_{C}\bigg{]}+i\,|C|\,{\rm
sin}\bigg{[}\frac{L_{1}}{2\alpha\hbar}-kL_{1}-\theta_{C}\bigg{]}\,=\,-\,\,\frac{2\,k\,A}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}~{}.$
(59)

Equating the imaginary parts from both sides of the above equation, we obtain

$\displaystyle\frac{L_{1}}{2\alpha\hbar}-kL_{1}-\theta_{C}\,=\,n_{1}\,\pi$
(60)

where $n_{1}\,\in\,\,\mathbb{N}$.

Now, following similar analysis as above, we substitute Eq. (57) in Eq. (54)
and use $r=L_{2}$. Thus, we obtain

$\displaystyle
A\,\bigg{[}\,\,1+\,e^{-2i\,kL}\,\bigg{\\{}\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\bigg{\\}}\,\,\bigg{]}=\,-|C|e^{i\bigg{(}\frac{L_{2}}{2\alpha\hbar}\,-k\,L_{2}-\,\theta_{C}\bigg{)}}\,\,e^{-ik^{2}\alpha\hbar\,L_{2}}~{}.$
(61)

It can be noted that for $\alpha\to 0$ the factor
$\bigg{\\{}\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\bigg{\\}}\,\to\,-1$.
Therefore, in the $\alpha\to 0$ limit both sides of the above equation vanish
when $kL=p\pi$ and $|C|\,=0$, where $p$ $\in\,\mathbb{N}$. Thus, when
$\alpha\neq 0$, we can expect that $kL=p\,\pi+\delta$, where $\delta$
$\in\,\mathbb{R}$, and $\lim_{\alpha\to 0}\delta=0$. This implies that
$\delta$ has to be proportional to $\alpha^{q}$ where $q\in\,\mathbb{R}$ with
$q\,>\,0$. Now, following the previous analysis, Eq. (61) further reduces to

$\displaystyle-\frac{2\,k\,A}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,=\,|C|\,e^{i\bigg{(}\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\bigg{)}}\,-\,2ik\,A\,L\,\bigg{[}\,\,\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,\,\bigg{]}~{}.$
(62)

Equating the RHS of Eq. (62) with the LHS of Eq. (59), we obtain

$\displaystyle|C|\,e^{i\big{(}\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\big{)}}\,\bigg{[}1-e^{-iL\big{(}\frac{1}{2\alpha\hbar}-k\big{)}}\bigg{]}\,=\,2ikA\,L\,\bigg{[}\,\,\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,\,\bigg{]}~{}.$
(63)

Expanding $e^{-iL\big{(}\frac{1}{2\alpha\hbar}-k\big{)}}$ with respect to $L$
and keeping terms up to the leading order in $L$, we obtain

$\displaystyle|C|\,{\rm
cos}\bigg{[}\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\bigg{]}+i\,|C|\,{\rm
sin}\bigg{[}\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\bigg{]}\,=\,\,\frac{2k\,A}{\big{(}\frac{1}{2\alpha\hbar}-k\big{)}}\,\,\bigg{[}\,\,\frac{k+k^{2}\,\alpha\hbar-\frac{1}{2\alpha\hbar}}{k-k^{2}\,\alpha\hbar+\frac{1}{2\alpha\hbar}}\,\,\bigg{]}~{}.$
(64)

Equating the imaginary parts from both sides of the above equation, we obtain
the following condition

$\displaystyle\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\,=\,n_{2}\,\pi$
(65)

where $n_{2}\,\in\,\mathbb{N}$. Subtracting Eq. (60) from Eq. (65), we get

$\displaystyle\frac{L}{2\alpha\hbar}\,=\,(n_{2}-n_{1})\pi\,+\,k\,L$ (66)

where $n_{2}>\,n_{1}$ since $L_{2}>\,L_{1}$. Following the previous
discussion, we use $kL=p\,\pi+\delta$ in the above equation and end up with a
length quantization condition of the form

$\displaystyle\frac{L}{2\alpha\hbar}\,=\,\underbrace{(n_{2}-n_{1}+p)}_{n}\,\pi+\delta$
$\displaystyle L\,=\,2\,n\,\alpha\,\hbar\pi+2\alpha\hbar\,\delta$ (67)

where $n$ $\in\,\,\mathbb{N}$. It was mentioned earlier that $\delta$ ought to
be proportional to $\alpha^{q}$ which leads the last term of Eq. (67) to be of
the order of a higher power in $\alpha$. Therefore, we discard this term and
obtain a length quantization condition of the form

$L=2n\pi\alpha_{0}l_{pl}~{}.$ (68)

From the imaginary parts of Eqs. (59) and (64), the sine functions are
determined, and this leads to ${\rm
cos}\big{[}\frac{L_{1}}{2\alpha\hbar}-kL_{1}-\theta_{C}\big{]}={\rm
cos}\big{[}\frac{L_{2}}{2\alpha\hbar}-kL_{2}-\theta_{C}\big{]}=\pm\,1$, by the
well known trigonometric identity. Therefore, the cosine functions in Eqs.
(59) and (64) will also give rise to similar quantization condition as in Eq.
(67). Furthermore, the real parts of Eqs. (59) and (64) will determine the
relation between the two constants $A$ and $C$. It should be noted that the
length quantization condition given in Eq. (68) is consistent with the length
quantization condition given in Eq. (53).

The results of this section show that despite the time dependence of the
background spacetime, the discreteness of space in terms of the same
fundamental unit remains in all epochs of the Universe. This shows the
robustness of our results as well as the fundamental nature of the scale of
discreteness.

##  V Discussion

In this paper we have derived the discreteness of measured lengths in strong
gravitational fields in a number of BH and cosmological spacetimes. In
particular, we worked with the Schwarzschild and RN BH spacetimes as well as
the FLRW cosmological spacetime. For the BH spacetime, the strong gravity
region is close to the singular region of the BH whereas for the FLRW
spacetime it is in the inflationary epoch of the Universe. Remarkably, the
derived discreteness turns out to be independent of parameters such as the
Schwarzschild mass, the RN charge, the KG mass or for that matter any
particular epoch during the evolution of our Universe. In addition, it agrees
with the corresponding results derived for zero gravity (flat spacetime) and
weak gravity (weakly curved spacetime) regimes and employing test fields of
various spins Ali:2009zq ; Ali:2011fa ; Deb:2016psq . Furthermore, all length
quantization conditions independently of the gravitational background are of
the form $L\sim n\pi\alpha_{0}l_{pl}$ modulo a natural number that depends on
the specific gravitational background under study. Finally, as in many
previous works Das:2008kaa ; Das:2010sj ; Das:2009hs ; Ali:2009zq ; Das:2010zf
; Deb:2016psq , our results suggest the existence of a “new” length scale
$l_{new}\equiv\alpha_{0}l_{Pl}$. A recent theoretical work suggests
$\alpha_{0}={\cal O}(1)-{\cal O}(10)$ Scardigli:2016pjs . However, since no
values of $\alpha_{0}$ are experimentally ruled out, we leave the possibility
open for $\alpha_{0}$ to assume higher values, resulting in an intermediate
length scale between the Planck and the electroweak length scales. This would
lead to length quantization conditions of the form $L\sim n\pi l_{new}$. This
points towards the fundamental nature and in fact the potential universality
of the discreteness of measured spaces. As we saw this follows directly from
the application of GUP, which is another robust predictions of most candidate
theories of QG. As for the concrete computation to arrive at the above
results, we studied the massive KG equation in the background of a spacetime
with strong gravity. Since QG effects cannot be ignored in this regime, we
included this by implementing GUP in the KG equation. The form of GUP that we
employed in our analysis is not manifestly Lorentz invariant. Therefore one
may ask as to why we did not try to implement a relativistic version of GUP,
e.g. as in Refs. Hossenfelder:2006cw ; Kober:2010sj ; Capozziello:1999wx ;
Quesne:2006is ; Todorinov:2018arx ; Bosso:2020fos ; Bosso:2020jay . It should
be stressed that while the aforesaid relativistic version of GUP is Lorentz
covariant in flat spacetimes, it still needs to be extended to curved
spacetimes. Furthermore, the form of GUP utilized here can be viewed as an
“effective theory” from a fully covariant theory as described in Ref.
Bosso:2018ufr . Of course, our results may undergo further modifications due
to higher order QG effects arising from the higher order terms in $\alpha$
which we have ignored. We hope to explore the implications of our results in
the future.

##  VI Acknowledgement

The work was supported by the Natural Sciences and Engineering Research
Council of Canada. A. Das would like to thank D. Kothawala for useful
discussions. The authors would like to thank the Referees for their
constructive comments. This research was supported by the Quantum Major
Innovation Fund Project, funded by the Government of Alberta.

##  Appendix A Equation of motion for a scalar field in D-dimensional
spacetime

The Klein-Gordon equation in a curved background is
□Φ=-1-g(D)∂_M[-g^(D)g^MN∂_NΦ]

with -g^(D)=r^D-2 (sinχ_1)^D-3 (sinχ_2)^D-4…..(sinχ_D-3) . In the above
equation $(M,\,N)$ runs from $(0,1,...D-1)$. We write the mode solution for
the scalar field as following

$\displaystyle\Phi(t,r,\chi_{1},\chi_{2},....,\chi_{D-2})=\tau(t)\psi(r)\,Y_{l,m,n..}(\chi_{1},\chi_{2},\chi_{3},...,\chi_{D-2})~{}.$
(69)

The scalar field equation given by Eq. (8) can be explicitly written as
&1c2-g(D)∂_t[-g^(D)g^tt∂_tΦ]+1-g(D)∂_r[-g^(D)g^rr∂_rΦ]
+1-g(D)∂_χ_1[-g^(D)g^χ_1χ_1∂_χ_1Φ]+1-g(D)∂_χ_2[-g^(D)g^χ_2χ_2∂_χ_2Φ]+…-m2c2ℏ2Φ=0
. Employing Eq. (A) and utilizing the metric as given in Eq. (7), we obtain

$\displaystyle-\frac{1}{c^{2}f(r)}\partial_{t}^{2}\Phi+\frac{1}{r^{D-2}}\partial_{r}\bigg{[}r^{D-2}f(r)\partial_{r}\Phi\bigg{]}+\frac{1}{r^{2}({\rm
sin}\chi_{1})^{D-3}}\,\partial_{\chi_{1}}\bigg{[}({\rm
sin}\chi_{1})^{D-3}\,\partial_{\chi_{1}}\Phi\bigg{]}$
$\displaystyle+\frac{1}{r^{2}({\rm sin}^{2}\chi_{1})}\,\frac{1}{({\rm
sin}\chi_{2})^{D-4}}\,\partial_{\chi_{2}}\bigg{[}({\rm
sin}\chi_{2})^{D-4}\,\partial_{\chi_{2}}\Phi\bigg{]}+\ldots-\frac{m^{2}c^{2}}{\hbar^{2}}\Phi=0~{}.$
(70)

Then, we can write the solution for the scalar field mode in the form

$\displaystyle\Phi(t,r,\chi_{1},\chi_{2},\ldots,\chi_{D-2})=\tau(t)\psi(r)\,Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})$
(71)

and, thus, Eq. (70) becomes

$\displaystyle-\frac{1}{c^{2}f(r)}\partial_{t}^{2}[\tau(t)]\psi(r)Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})+\frac{1}{r^{D-2}}\partial_{r}\bigg{[}r^{D-2}f(r)\partial_{r}\psi(r)\bigg{]}\tau(t)Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})+$
$\displaystyle\frac{1}{r^{2}({\rm
sin}\chi_{1})^{D-3}}\,\partial_{\chi_{1}}\bigg{[}({\rm
sin}\chi_{1})^{D-3}\,\partial_{\chi_{1}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}\psi(r)\tau(t)$
$\displaystyle+\frac{1}{r^{2}({\rm sin}^{2}\chi_{1})}\,\frac{1}{({\rm
sin}\chi_{2})^{D-4}}\,\partial_{\chi_{2}}\bigg{[}({\rm
sin}\chi_{2})^{D-4}\,\partial_{\chi_{2}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}\psi(r)\tau(t)+$
$\displaystyle\ldots-\frac{m^{2}c^{2}}{\hbar^{2}}\,\tau(t)\psi(r)Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})=0~{}.$
(72)

By dividing both sides of the above equation with
$\tau(t)\,\psi(r)\,Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})$,
we obtain

$\displaystyle-\frac{1}{c^{2}f(r)\tau(t)}\partial_{t}^{2}[\tau(t)]+\frac{1}{r^{D-2}}\partial_{r}\bigg{[}r^{D-2}f(r)\partial_{r}\psi(r)\bigg{]}\frac{1}{\psi(r)}+$
$\displaystyle\frac{1}{r^{2}({\rm
sin}\chi_{1})^{D-3}Y_{l,m,n,\ldots}}\,\partial_{\chi_{1}}\bigg{[}({\rm
sin}\chi_{1})^{D-3}\,\partial_{\chi_{1}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}$
$\displaystyle+\frac{1}{r^{2}({\rm sin}^{2}\chi_{1})}\,\frac{1}{({\rm
sin}\chi_{2})^{D-4}}\frac{1}{Y_{l,m,n,\ldots}}\,\partial_{\chi_{2}}\bigg{[}({\rm
sin}\chi_{2})^{D-4}\,\partial_{\chi_{2}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}+\ldots-\frac{m^{2}c^{2}}{\hbar^{2}}=0$
(73)

which is rewritten as

$\displaystyle-\frac{r^{2}\hbar^{2}}{c^{2}f(r)\tau(t)}\partial_{t}^{2}[\tau(t)]+\frac{\hbar^{2}r^{2}}{r^{D-2}}\partial_{r}\bigg{[}r^{D-2}f(r)\partial_{r}\psi(r)\bigg{]}\frac{1}{\psi(r)}-m^{2}\,r^{2}\,c^{2}=$
$\displaystyle-\hbar^{2}\,\bigg{[}\frac{1}{({\rm
sin}\chi_{1})^{D-3}Y_{l,m,n,\ldots}}\,\partial_{\chi_{1}}\bigg{[}({\rm
sin}\chi_{1})^{D-3}\,\partial_{\chi_{1}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}$
$\displaystyle+\frac{1}{({\rm sin}^{2}\chi_{1})}\,\frac{1}{({\rm
sin}\chi_{2})^{D-4}}\frac{1}{Y_{l,m,n,\ldots}}\,\partial_{\chi_{2}}\bigg{[}({\rm
sin}\chi_{2})^{D-4}\,\partial_{\chi_{2}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}+\ldots\bigg{]}~{}.$
(74)

In Eq. (74), we equate both sides with $\hbar^{2}\,l(l+1)$ and, therefore, the
RHS which is the angular part becomes

$\displaystyle\frac{\hbar^{2}}{({\rm
sin}\chi_{1})^{D-3}Y_{l,m,n,\ldots}}\,\partial_{\chi_{1}}\bigg{[}({\rm
sin}\chi_{1})^{D-3}\,\partial_{\chi_{1}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}$
$\displaystyle+\frac{\hbar^{2}}{({\rm sin}^{2}\chi_{1})}\,\frac{1}{({\rm
sin}\chi_{2})^{D-4}}\frac{1}{Y_{l,m,n,\ldots}}\,\partial_{\chi_{2}}\bigg{[}({\rm
sin}\chi_{2})^{D-4}\,\partial_{\chi_{2}}Y_{l,m,n,\ldots}(\chi_{1},\chi_{2},\chi_{3},\ldots,\chi_{D-2})\bigg{]}+\ldots=\,-\,\hbar^{2}\,l(l+1)$
(75)

and, consequently, the LHS part of Eq. (74) will be

$\displaystyle-\frac{\hbar^{2}r^{2}}{c^{2}f(r)\tau(t)}\partial_{t}^{2}[\tau(t)]+\frac{\hbar^{2}r^{2}}{r^{D-2}}\partial_{r}\bigg{[}r^{D-2}f(r)\partial_{r}\psi(r)\bigg{]}\frac{1}{\psi(r)}-m^{2}\,r^{2}\,c^{2}=\hbar^{2}\,l(l+1)~{}.$
(76)

In $D=4$ spacetime dimensions, the angular momentum operator can be identified
as follows

$\displaystyle
L^{2}=\,-\hbar^{2}\bigg{[}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\bigg{(}\sin\theta\frac{\partial}{\partial\theta}\bigg{)}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\bigg{]}$
(77) $\displaystyle
L^{2}\,Y_{l,m}(\theta,\phi)\,=\,\hbar^{2}\,l(l+1)Y_{l,m}(\theta,\phi)~{}.$
(78)

By adopting the analysis followed before, Eq. (76) for $D=4$ reads

$\displaystyle-\bigg{(}\frac{\hbar^{2}}{c^{2}}\bigg{)}\frac{1}{\tau(t)}\partial_{t}^{2}[\tau(t)]=-\frac{\hbar^{2}f(r)}{r^{2}}\partial_{r}\bigg{[}r^{2}f(r)\partial_{r}\psi(r)\bigg{]}\frac{1}{\psi(r)}+m^{2}\,c^{2}\,f(r)+\frac{\hbar^{2}\,l(l+1)}{r^{2}}f(r)~{}.$
(79)

Now we equate both sides of Eq. (79) with the quantity $\frac{E^{2}}{c^{2}}$
and obtain

$\displaystyle-\frac{1}{\tau(t)}\partial_{t}^{2}[\tau(t)]=\frac{E^{2}}{\hbar^{2}}$
(80) $\displaystyle\implies\tau(t)\sim Exp\,[\pm\,iEt/\hbar]$ (81)

while the radial part reduces to

$\displaystyle-\frac{\hbar^{2}}{r^{2}}\frac{d}{dr}\bigg{[}r^{2}f(r)\frac{d\psi(r)}{dr}\bigg{]}+\bigg{[}\frac{\hbar^{2}\,l(l+1)}{r^{2}}-\,\frac{E^{2}}{c^{2}\,f(r)}+m^{2}c^{2}\bigg{]}\psi(r)=0$
(82)

where

$\displaystyle-\frac{\hbar^{2}}{r^{2}}\frac{d}{dr}\bigg{[}r^{2}f(r)\frac{d\psi(r)}{dr}\bigg{]}=-\frac{f(r)\,\hbar^{2}}{r^{2}}\frac{d}{dr}\bigg{[}r^{2}\frac{d\psi(r)}{dr}\bigg{]}-\hbar^{2}\,f^{\prime}(r)\frac{d\psi}{dr}$
$\displaystyle=$
$\displaystyle-f(r)\,\hbar^{2}\nabla^{2}\,\psi(r)-\hbar^{2}\,f^{\prime}(r)\frac{d\psi}{dr}$
(83) $\displaystyle=$ $\displaystyle
f(r)\,p_{0}^{2}\psi(r)-\hbar^{2}\,f^{\prime}(r)\frac{d\psi}{dr}~{}.$

##  Appendix B Boundary condition

In order to be precise, we impose the boundary conditions of the form

$\psi(r=\epsilon)\,=\,0\,\,\,\,\,\,\,\,\psi(r=L)=0$ (84)

where $\epsilon$ is a radial distance close to BH singularity, i.e., $r=0$.
The above boundary conditions imply

$\displaystyle\psi(r=\epsilon)=C_{1}\,{\rm
exp}\bigg{[}\frac{i\epsilon}{8\alpha\hbar}\bigg{\\{}1-\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}+C_{2}\,{\rm
exp}\bigg{[}\frac{i\epsilon}{8\alpha\hbar}\bigg{\\{}1+\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}$
(85) $\displaystyle\psi(r=L)=C_{1}\,{\rm
exp}\bigg{[}\frac{iL}{8\alpha\hbar}\bigg{\\{}1-\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}+C_{2}\,{\rm
exp}\bigg{[}\frac{iL}{8\alpha\hbar}\bigg{\\{}1+\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}~{}.$
(86)

Solving the above equations, we obtain

$\displaystyle C_{2}\bigg{[}{\rm
exp}\bigg{[}\frac{i\epsilon}{8\alpha\hbar}\bigg{\\{}1+\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}\,{\rm
exp}\bigg{[}\frac{iL}{8\alpha\hbar}\bigg{\\{}1-\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}$
$\displaystyle-\,{\rm
exp}\bigg{[}\frac{i\epsilon}{8\alpha\hbar}\bigg{\\{}1-\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}\,{\rm
exp}\bigg{[}\frac{iL}{8\alpha\hbar}\bigg{\\{}1+\bigg{(}1-\frac{8ic^{2}\,\alpha\hbar\,l(1+l)}{GM}\bigg{)}^{1/2}\bigg{\\}}\bigg{]}\bigg{]}\,=\,0~{}.$
(87)

Then, we simplify Eq. (87) and we get

$\displaystyle{\rm
exp}\bigg{[}\frac{\epsilon+L}{8\alpha\hbar}-\frac{\pi}{2}\bigg{]}\,{\rm
sin}\bigg{[}\frac{L-\epsilon}{8\alpha\hbar}\,(1-i\alpha\beta)^{1/2}\bigg{]}=0~{}.$
(88)

Finally, the above equation reduces to

$\displaystyle\frac{L+\epsilon}{8\alpha\hbar}\,=\,(n+1)\pi,\,\,\,\,\,\,\,\,\,n\in\mathbb{N}$
(89)
$\displaystyle\frac{L-\epsilon}{8\alpha\hbar}\,\approx\,p\pi,\,\,\,\,\,\,\,\,\,p\in\mathbb{N}~{}.$
(90)

It can be noted that for $\epsilon\to\,0$ limit the above equation will give
rise to the quantization conditions as obtained in section(II).

## References

  * (1) G. Amelino-Camelia, Living Rev. Rel. 16, 5 (2013) [arXiv:0806.0339 [gr-qc]].
  * (2) S. Hossenfelder, Living Rev. Rel. 16, 2 (2013) [arXiv:1203.6191 [gr-qc]].
  * (3) D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 41-47 (1989)
  * (4) L. J. Garay, Int. J. Mod. Phys. A 10, 145-166 (1995) [arXiv:gr-qc/9403008 [gr-qc]].
  * (5) A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52, 1108-1118 (1995) [arXiv:hep-th/9412167 [hep-th]].
  * (6) M. Maggiore, Phys. Lett. B 304, 65-69 (1993) [arXiv:hep-th/9301067 [hep-th]].
  * (7) M. Maggiore, Phys. Rev. D 49, 5182-5187 (1994) [arXiv:hep-th/9305163 [hep-th]].
  * (8) M. Maggiore, Phys. Lett. B 319, 83-86 (1993) [arXiv:hep-th/9309034 [hep-th]].
  * (9) F. Scardigli, Phys. Lett. B 452, 39-44 (1999) [arXiv:hep-th/9904025 [hep-th]].
  * (10) S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008) [arXiv:0810.5333 [hep-th]].
  * (11) S. Das and E. C. Vagenas, Phys. Rev. Lett. 104, 119002 (2010) [arXiv:1003.3208 [hep-th]].
  * (12) S. Das and E. C. Vagenas, Can. J. Phys. 87, 233-240 (2009) [arXiv:0901.1768 [hep-th]].
  * (13) A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B 678, 497-499 (2009) [arXiv:0906.5396 [hep-th]].
  * (14) A. F. Ali, S. Das and E. C. Vagenas, in “On recent developments in theoretical and experimental general relativity, astrophysics and relativistic field theories”. Proceedings, 12th Marcel Grossmann Meeting on General Relativity, Paris, France, July 12-18, 2009. Vol. 1-3, pp 2407–2409 (2010) [arXiv:1001.2642 [hep-th]].
  * (15) F. Scardigli, G. Lambiase and E. Vagenas, Phys. Lett. B 767, 242-246 (2017) [arXiv:1611.01469 [hep-th]].
  * (16) E. C. Vagenas, S. M. Alsaleh and A. Farag, EPL 120, no.4, 40001 (2017) [arXiv:1801.03670 [hep-th]].
  * (17) A. Kempf, J. Phys. A 30, 2093-2102 (1997) [arXiv:hep-th/9604045 [hep-th]].
  * (18) F. Brau, J. Phys. A 32, 7691-7696 (1999) [arXiv:quant-ph/9905033 [quant-ph]].
  * (19) S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer and H. Stoecker, Phys. Lett. B 575, 85-99 (2003) [arXiv:hep-th/0305262 [hep-th]].
  * (20) R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14, 1371 (1999) [arXiv:gr-qc/9904026 [gr-qc]].
  * (21) F. Scardigli and R. Casadio, Class. Quant. Grav. 20, 3915-3926 (2003) [arXiv:hep-th/0307174 [hep-th]].
  * (22) P. Jizba, H. Kleinert and F. Scardigli, Phys. Rev. D 81, 084030 (2010) [arXiv:0912.2253 [hep-th]].
  * (23) A. F. Ali, S. Das and E. C. Vagenas, Phys. Rev. D 84, 044013 (2011) [arXiv:1107.3164 [hep-th]].
  * (24) S. Basilakos, S. Das and E. C. Vagenas, JCAP 09, 027 (2010) [arXiv:1009.0365 [hep-th]].
  * (25) F. Scardigli and R. Casadio, Eur. Phys. J. C 75, no.9, 425 (2015) [arXiv:1407.0113 [hep-th]].
  * (26) J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002) [arXiv:hep-th/0112090 [hep-th]].
  * (27) G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35-60 (2002) [arXiv:gr-qc/0012051 [gr-qc]].
  * (28) G. Amelino-Camelia, Phys. Lett. B 510, 255-263 (2001) [arXiv:hep-th/0012238 [hep-th]].
  * (29) J. Magueijo and L. Smolin, Phys. Rev. D 67, 044017 (2003) [arXiv:gr-qc/0207085 [gr-qc]].
  * (30) J. Cortes and J. Gamboa, Phys. Rev. D 71, 065015 (2005) [arXiv:hep-th/0405285 [hep-th]].
  * (31) S. Das, E. C. Vagenas and A. F. Ali, Phys. Lett. B 690, 407-412 (2010) [Erratum: Phys.Lett.B 692, 342–342 (2010)] [arXiv:1005.3368 [hep-th]].
  * (32) S. Deb, S. Das and E. C. Vagenas, Phys. Lett. B 755, 17-23 (2016) [arXiv:1601.07893 [gr-qc]].
  * (33) T. Thiemann, J. Math. Phys. 39, 3372-3392 (1998) [arXiv:gr-qc/9606092 [gr-qc]].
  * (34) C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593-622 (1995) [arXiv:gr-qc/9411005 [gr-qc]].
  * (35) H. Nicolai, K. Peeters and M. Zamaklar, Class. Quant. Grav. 22, R193 (2005) [arXiv:hep-th/0501114 [hep-th]].
  * (36) R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986)
  * (37) P. Bosso and S. Das, Annals Phys. 383, 416-438 (2017) [arXiv:1607.01083 [gr-qc]].
  * (38) D. Baumann, [arXiv:0907.5424 [hep-th]].
  * (39) S. Hossenfelder, Phys. Rev. D 73, 105013 (2006) [arXiv:hep-th/0603032 [hep-th]].
  * (40) M. Kober, Phys. Rev. D 82, 085017 (2010) [arXiv:1008.0154 [physics.gen-ph]].
  * (41) S. Capozziello, G. Lambiase and G. Scarpetta, Int. J. Theor. Phys. 39, 15-22 (2000) [arXiv:gr-qc/9910017 [gr-qc]].
  * (42) C. Quesne and V. Tkachuk, Czech. J. Phys. 56, 1269-1274 (2006) [arXiv:quant-ph/0612093 [quant-ph]].
  * (43) V. Todorinov, P. Bosso and S. Das, Annals Phys. 405, 92-100 (2019) [arXiv:1810.11761 [gr-qc]].
  * (44) P. Bosso, S. Das and V. Todorinov, [arXiv:2005.03771 [gr-qc]].
  * (45) P. Bosso, S. Das and V. Todorinov, [arXiv:2005.03772 [gr-qc]].
  * (46) P. Bosso and S. Das, Int. J. Mod. Phys. D 28, no.04, 1950068 (2019) [arXiv:1812.05595 [gr-qc]].


[30] J. Koncki, Comparison of motivic Chern classes and stable envelopes for cotangent bundles, J. Topology 15 (2022), 168–203.

[31]
J. Koncki and A. Weber, Twisted motivic Chern class and stable envelopes, Advances in Mathematics, 404, Part A, 2022

[33]
B. Kostant and S. Kumar, T-equivariant K-theory of generalized flag varieties, J. Diff. Geom. 32 (1990), 549–603.

[34]
T. Lam,  Ribbon Schur operators,  Europ. J. Combin. 29 (2008), 343–359.

[35]
T. Lam and P. Pylyavskyy, Combinatorial Hopf algebras and K-homology of Grassmanians,  Int.
Math. Res. Not. IMRN 9 (2007), rnm125.

[36]
A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Math. Acad. Sci. Paris, Sér. I Math.  294 (1982), 447–450.

[37]
A. Lascoux and M.-P. Schützenberger,  Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982),   629–633.

[38]
C. Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Combin. 4 (2000), 67–82.

[40]
C. Lenart,  C. Su, K. Zainoulline and C. Zhong, Geometric properties of the Kazhdan–Lusztig Schubert basis, Algebra & Number Theory  17 (2) (2023), 435–464.

[41]
M. Levine and F. Morel, Algebraic cobordism, Springer Science & Business Media, 2007.

[42]
G. Lusztig, Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Amer. Math. Soc.  277 (1983),  623–653.

[43]
G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (3) (1989), 599–635.

[44]
I. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford Math.
Mon., 1995.

[45]
R. MacPherson,  Chern classes for singular algebraic varieties, Ann. Math.  100 (1974), 423–432.

[46]
L. Mihalcea, H. Naruse and C. Su,
Left Demazure–Lusztig operators on equivariant (quantum) cohomology and K-theory, Int. Math. Res. Not. IMRN 16 (2022), 12096–12147.

[47]
L. Mihalcea, H. Naruse and C. Su,
Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes, 	arXiv:2312.17200.

[48]
L. Mihalcea,  C. Su and D. Anderson, Whittaker functions from motivic Chern classes, Transform. Groups 27 (3) (2022), 1045–1067.

[49]
T. Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math.
Proc. Camb. Philos. Soc. 140 (2006), 115–134.

[50]
A. Okounkov,  Lectures on K-theoretic computations in enumerative geometry, Geometry of moduli spaces and representation theory, 251-V380, IAS/Park City Math. Ser., 24, AMS, 2017.

[52]
R. Rimányi,  $\hbar$-Deformed Schubert Calculus in equivariant cohomology, K-theory, and elliptic cohomology, Singularities and their interaction with geometry and low dimensional topology: In honor of András Némethi (2021), 73–96.

[53]
M. Schwartz, Classes caractéristiques définies par une stratification
d’une variété analytique complexe, I, C. R. Acad. Sci. Paris 260 (1965), 3262–3264.

[54]
M. Schwartz, Classes caractéristiques définies par une stratification
d’une variété analytique complexe, II, C. R. Acad. Sci. Paris, 260 (1965), 3535–3537.

[55]
W. Soergel, Kazhdan–Lusztig polynomials and a combinatoric for tilting modules, Represent. Theory  1 (6) (1997), 83–114.

[57]
R. Stanley, Enumerative Combinatorics, Vol. 2,  Cambridge University Press, Cambridge, 1999.

[60]
S. Yokura, A singular Riemann–Roch theorem for Hirzebruch characteristics,  Banach Center Publ. 44 (1998), 257–268.