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From the Bronshtein cube of limits to the degrees of freedom of relativistic quantum gravity

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It is argued that the quadruple gravitational constant 4G can be seen as a fundamental limit of nature. The limit holds across all gravitational systems and distinguishes bound from unbound systems. Including the maximum force c^4/4G allows extending the Bronshtein cube of physical theories to a cube of limits. Every theory of physics refining Galilean physics -- universal gravitation, special relativity, general relativity, quantum theory and quantum field theory -- is defined by one fundamental limit. As a result, also relativistic quantum gravity is defined by a limit: the minimum length in nature. The minimum length is used to deduce the Planck-scale structure of space. Numerous options are eliminated. Then, the minimum length is used to deduce the main properties of the common constituents that make up space and particles.
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From the Bronshtein cube of limits
to
the degrees of freedom of relativistic quantum gravity
Christoph SchilleraID
2023
Abstract
It is argued that the quadruple gravitational constant 4Gcan be seen as a fundamental limit
of nature. The limit holds across all gravitational systems and distinguishes bound from
unbound systems. This allows extending the Bronshtein cube of physical theories to a cube
of limits. Every theory of physics refining Galilean physics universal gravitation, special
relativity, general relativity, quantum theory and quantum field theory is defined by one
fundamental limit. As a result, also relativistic quantum gravity is defined by a limit: the
minimum length in nature. The minimum length is used to deduce the Planck-scale structure
of space. Numerous options are eliminated. Then, the minimum length is used to deduce
the main properties of the common constituents that make up space and particles.
aMotion Mountain Research, 81827 Munich, Germany, cs@motionmountain.net, ORCID 0000-0002-8188-6282.
2
I The Bronshtein cube
It is common to attribute to Matvei Petrovich Bronshtein, born in 1906 in Vinnytsia in Ukraine,
the physics cube that is based on the speed of light c, the quantum of action ~and the gravitational
constant G[13]. The physics cube has regularly been used to illustrate the relation between the
main physical theories [47].
In the following, it is first argued that Bronshtein’s usual physics cube can be extended to a
cube of limits, illustrated in Figure 1. Since their inception, special relativity and quantum theory
are based on the invariant maximum speed cand the invariant minimum action ~. In the past
decades, it became clear that general relativity can be based on the invariant maximum force
c4/4G, which is realized on gravitational horizons [812]. Below, the approach is completed
by showing that also classical gravity can be based on an invariant limit, namely the quadruple
gravitational constant 4G, which is realized in parabolic gravitational motion. In this way, each
part of fundamental physics is defined using an invariant limit principle. In particular, it follows
that relativistic quantum gravity is based on the minimum length or on an equivalent limit.
It is then argued that the minimum length at the top of Bronshtein’s limit cube implies a unique
and common description for space and particles that is based on fluctuating filiform constituents.
The Bronshtein cube of limits thus appears to determine the theory of relativistic quantum gravity.
II Maximum force
In 1973, Elizabeth Rauscher, followed by many others, discovered that general relativity implies a
local maximum force c4/G [839]. In 2002, Gibbons, and the present author in 2003, included the
factor 1/4and showed that in 3+1 dimensions the value of the force the change of momentum
of a system with time produced by black holes on any system at a point is never larger than the
maximum value c4/4G[8,9].
Black holes are systems that realize several limits. The ratio of mass and diameter for a
Schwarzschild black hole is c2/4G, independently of the size and mass of the black hole. The
ratio is not exceeded by any other physical system. For a Schwarzschild black hole, the ratio
between its energy Mc2and its diameter D= 4GM/c2is given by the maximum force value
Fmax =c4/4G3.0·1043 N. This (local) maximum force value is not exceeded in any system
in nature. Also the force on a test mass that is lowered with a rope towards a gravitational horizon
whether charged, rotating or both never exceeds the force limit, as long as the minimum size
of the test mass is taken into account. All apparent counter-arguments and counter-examples to
maximum force disappear when explored in detail [11,3033].
The maximum force c4/4Gcharacterizes general relativity. Indeed, maximum force c4/4G
3
Galilean physics,
without limits Precision increases upwards.
Quantum theory,
Relativistic quantum gravity,
the complete description of motion,
de
ned by
upper limit
lower limit
and preceding limits
limits
tiny motion
General relativity,
de
ned by
upper limit
lower limit
and preceding limits
Classical gravity,
de
ned by
upper limit
lower limit
de
ned by
lower limit
upper limit
limits fast
motion
de
ned by
upper limit
lower limit
Quantum
eld theory,
de
ned by
upper limit
lower limit
and preceding limits
Quantum theory
with classical
gravity, def-
ined by
upper limit
lower limit
and preceding
limits
Special relativity,
limits
unbound motion
c
4G~
1/4G
4Gc
1/c 1/~
~
c4/4G
4G/c4
1/4G~
4G~
c/~
~/c
c3/4G~
4G~/c3
FIG. 1: The Bronshtein limit cube of physical theories is illustrated, in which each modern
theory of physics is described by a limit. Following the lines towards the top increases the
precision of the descriptions and their fascination.
implies Einstein’s field equations of general relativity [812]. This result, only valid in 3+1 di-
mensions, can be reached in two ways: it can be deduced from the elastic properties of space-time
implied by maximum force, and it can be deduced from c4/4Gwith the help of the first law of
black hole horizons. As a result of these deductions, the (local) maximum force limit can be seen
as the defining principle of general relativity. The situation resembles special relativity, for which
the (local) maximum speed limit ccan be seen as the defining principle.
The maximum force limit is not the only option to describe general relativity. Other limit
quantities that combine cand G, such as maximum power c5/4G[11,17,21,25,36,37,4043],
maximum mass flow rate c3/4G[11,34], or the maximum mass per length ratio c2/4G[4446]
4
can also be taken as principles of relativistic gravitation. All these limits values are achieved only
by black holes. A similar situation arises in special relativity, where, e.g., either cor c2can be
taken as defining limit. All these limits agree with all experiments.
In short, special relativity can be deduced from the principle of maximum speed c; general
relativity can be described by adding maximum force c4/4G. A question arises: can classical
inverse square gravity also be deduced from a limit?
III The quadruple gravitational constant as a limit distinguishing unbound from
bound particles
The gravitational constant Gis defined as the constant appearing in the gravitational acceleration
of a small mass, or particle, at a distance rto a large mass M:
a=GM
r2.(1)
This expression, independent of the value of the small mass, is due to the work of Hooke, Newton
and Cavendish. Because the expression unified sublunar and translunar observations, the inverse
square dependence of classical gravity is often called universal gravity. Rewriting the equation
using the diameter d= 2ryields
a=4GM
d2.(2)
The expression implies that a small mass or particle is unbound from a large mass Mif its kinetic
energy is larger than the gravitational potential energy. In other terms, a particle is unbound if the
double centre-to-centre distance d= 2rand the speed vobey
dv2
M4G2.7·1010 m3/kg s2.(3)
The constant 4Gthus describes the difference between unbound and bound particles near a mass
M. If a particle has a product dv2/M that is larger than 4G, it is unbound; otherwise, it is bound.
For example, a rocket ‘orbiting’ Earth with diameter d, flying faster than the escape velocity
v=p4GM/d, is unbound. In contrast, a stone on the ground is bound to Earth.
Equivalently, a particle is unbound from a large mass Mif its acceleration obeys ad2/M 4G.
Alternatively, a particle near a mass Mis unbound if it obeys
ρT 21
4G3.7·109kg s2/m3.(4)
In this expression, the ‘effective’ particle cycle time T=d/v around the central mass and the
‘effective’ density ρ=M/d3are used. In all other cases, the particle is gravitationally bound.
When a particle approaches the limit 4Gfrom below, its orbital time increases without limit,
until parabolic motion is reached. Approaching the limit 4Gfrom above leads from unbound
5
hyperbolic to parabolic motion, for which gravitational potential energy and kinetic energy are
exactly equal. Crossing the limit bounds the particle or test mass to the attracting mass M.
As shown in equation (4), the value 1/4Gis the largest possible value for effective density
times time squared that can arise in a gravitationally unbound system. This limit can also be
applied to the universe as a whole. The quantity 1/4Gshould limit the product T 2
Hof the matter
density in the universe and the (Hubble) time squared. Present data on the energy density and the
Hubble time [47] indeed confirm that, within measurement errors, the limit 1/4Gis realized by the
universe. The mass-energy in the universe, seen at a large scale, is generally at the limit between
being bound and unbound as expected. In other words, the limit 4Galso holds in cosmology.
In short, for a gravitationally unbound particle, 4Gis the limit value of any product of quantities
containing the nearby mass M. All experiments in the solar system confirm the result.
IV Testing the limit 4Gin rotating galaxies
At present, the validity of the limit 4Gis in discussion for one class of physical systems. The
rotation of outer stars orbiting galaxy centres is an intense topic of research. In almost all galaxies
the most distant stars are measured to rotate faster than predicted from inverse square gravity
with the estimated central mass values. Similar results are found for globular clusters and galaxy
groups. Different explanations have been proposed for these observations.
In the most common explanation, the deviation is explained with yet unobserved (cold) dark
matter [47]. The explanation postulates that stars (or galaxies) at the outer edge remain bound
because the actual mass of galaxies is larger than the luminous mass, because of additional dark
matter. This approach retains the limit 4G.
In contrast, some researchers explore modifications of the inverse square dependence of grav-
itation [48]. They postulate that at distances that would lead to accelerations smaller than an
observed constant a01.2·1010 m/s2, the actual acceleration due to gravitation is larger than
the one predicted by inverse square gravity, and thus outer stars (or galaxies) remain bound even
if their speeds are larger than the conventional escape velocity. This approach predicted the ob-
served baryonic Tully-Fisher relation between galaxy mass and asymptotic rotation velocity. Some
versions of modified gravity put into question the limit 4G.
It might be that the observational constant a0and the baryonic Tully-Fisher relation are due to
some quantum effect on cosmological scale [49]; it might also be that the Weyl metric explains the
rotation curves [50,51], or that measurements are not interpreted correctly. If any of these option
sis confirmed, the limit 4Gwould remain valid. Future research will show which explanation is
correct.
In short, so far, there is no definite observation contradicting the limit 4Gfor unbound motion.
6
V Properties of the limit 4G
The factor 4 in the limit for gravitationally bound motion arises because of the historical preference
to use the radius instead of the diameter. High-precision experiments confirm equations (3) and
(4), as well as the expression for the escape velocity. These expressions allow stating: the limit
4Gcontains and implies the inverse square law of universal gravitation. The limit 4Gthus implies
classical gravity in the same way that the limit cimplies special relativity or the limit c4/4G
implies general relativity.
The limit 4G like the fundamental limits ~,c,c2/4Gand c4/4G is invariant: it is inde-
pendent of the observer. Neither relativistic boosts nor other coordinate transformations of any
type change the limit value. Like the other limits, also the limit 4Gapplies only to real and free
particles. The limit 4G, like all other invariant limits of nature, is also needed to define units of
measurements.
In short, the expression 4Gis a fundamental, invariant limit principle of nature. This result
allows a striking formulation of Bronshtein’s physics cube.
VI The Bronshtein limit cube of physical theories
As illustrated in Figure 1, the above results for universal gravity and for general relativity allow
defining a limit value at every corner of the physics cube except for Galilean physics. The origin
of this possibility is the following summary of modern physics:
Universal gravity is equivalent to dv2/M 4Gfor unbound systems.
Special relativity is equivalent to vcfor physical systems.
General relativity is equivalent to Fc4/4Gor m/d c2/4Gfor physical systems.
Quantum theory is equivalent to W~for all physical systems.
Quantum field theory is equivalent to ml ~/c for real, i.e., physical particles.
Non-relativistic quantum gravity is equivalent to free particles with Av34G~.
Relativistic quantum gravity is equivalent to l24G~/c3for physical systems.
In these expressions, lis length, Wis action, and Ais area. Each of the seven limit expressions is
associated with one of the upper corners of Bronshtein’s physics cube, as illustrated in Figure 1.
The choice of the limit for every field of physics is not unique; at each corner of the cube, the
nonzero exponents of 4G,cand ~can be changed within certain boundaries. These choices allow
assigning several equivalent upper or lower limits to each corner of the cube. If desired, the
limit for entropy, defined by the Boltzmann constant kcan be added to the description of nature,
yielding a hypercube [4,7,52].
Several properties of the Bronshtein limit cube are worth pointing out. First, apart from the
7
correcting factor 4, all limits are Planck limits. In other terms, one can say: the (corrected) Planck
limits define modern physics. Each Planck limit is a principle for the corresponding theory.
Each Planck limit is deduced from experiments. No experiment ever performed contradicted
any of these limits. In other terms, there is no evidence for trans-Planckian physics from any
observation. The Bronshtein limit cube implies that there is no physics beyond general relativity,
even for the strongest gravitational fields. The cube implies that there is no physics beyond quan-
tum field theory, even for the smallest scales or the highest energies. The cube further implies that
there is no physics beyond quantum theory in classical gravity. This agrees with all experiments
ever performed.
Whenever a limit c,4Gor ~is added to a given description of motion, a more precise descrip-
tion is obtained. Because relativistic quantum gravity takes into account all limits, it is the most
precise description of fundamental physics. Thus, relativistic quantum gravity is the complete and
final theory of physics.
In particular, the combination of all limits implies a statement on length. Using the expression
for action W=F lt =F l2/v, and inserting the limits for force F, speed vand action W, one finds
that length values are limited by lp4G~/c3, or twice the Planck length. This result has been
derived in many ways in the past [5357]. The smallest length limits both the observable values
and the achievable measurement precision. Many other, equivalent limits such as the minimum
area, the minimum time, the minimum volume, the maximum acceleration or the maximum mass
density can also be deduced for the top corner of the physics cube. In fact, at the top corner,
there is a limit for every observable in contrast to the lower corners.
In short, each field of fundamental physics can be defined with a Planck limit principle. This
characterization is also useful for teaching: learning physics starts at the bottom of the Bronshtein
cube, where no limits are assumed, and proceeds towards the top, where all physical observables
are limited by the corrected Planck values. The agreement with experiments, the simplicity, and
the explanatory power of the fundamental limits at each corner of the physics cube are fascinating.
The fascination is especially intense at the top corner.
VII Relativistic quantum gravity and its predictions
In the Bronshtein cube of limits, any path towards the summit leads to the theory of relativistic
quantum gravity. This implies that the minimum length limit p4G~/c3 or any equivalent quan-
tum gravity limit alone by itself, completely describes and implies relativistic quantum gravity.
Every limit that can be used to define the top of the physics cube from minimum length
to minimum time, minimum area, minimum volume, maximum acceleration or maximum mass
density is inaccessible to experiments. Therefore, the Bronshtein limit cube suggests that rela-
8
tivistic quantum gravity is already known in all its observable effects: the only observable effects
of relativistic quantum gravity are predicted to be general relativity, quantum field theory with
the standard model, and quantum theory in classical gravity. These are the theories for the three
corners below the top of the cube.
In addition, minimum length, together with its various equivalent limits of nature at the upper-
most corner, leads to the following conclusions about relativistic quantum gravity:
At the Planck scale, minimum length and minimum time imply that space-time is not a man-
ifold and thus has no defined dimensionality. In contrast, at macroscopic scale i.e., at all scales
larger than the Planck scale space and time are effectively continuous. Space and time emerge
from the Planck limits as macroscopic approximations. At scales larger than the Planck scale,
space-time is effectively, i.e., approximately a 3+1 dimensional manifold. But because of min-
imum length, neither higher nor lower dimensions can be deduced or detected neither at the
Planck scale nor at larger scales.
Because of minimum length and minimum time, it is impossible to deduce or to establish the
existence of any kind of space-time manifold or of any kind of manifold with any additional struc-
ture, neither at the Planck scale nor at larger scales. Therefore, no space-time foam, no microscopic
wormholes, no diffeomorphism invariance, no fermionic coordinates, no non-commutative coor-
dinates, no twistors, no conformal symmetry, no supersymmetry, no supergravity, no asymptotic
safety, no causal fermion systems, no geometrodynamics, no T-duality, no exact UV-IR symmetry,
no exact holographic principle, no canonical quantum gravity, no continuous gauge-gravity dual-
ity, and no additional continuous or discrete space-time symmetries can be deduced, measured or
confirmed.
Minimum length and minimum time also imply that there are no measurable points in space
or instants in time, but that there is a Planck-scale non-locality in nature. In contrast, at larger
scales, points in space and instants in time are useful approximations.
Minimum length and minimum time imply that both at the Planck scale and at larger scales
space-time structures based on discrete points cannot be measured, detected or be proven to
exist. This includes space-time lattices, causal dynamical triangulations, spin networks, Turaev-
Viro models, and graphs. Likewise, because of minimum length, there are no singularities in
nature, of any type. The limit cube implies that they do not and cannot occur.
Minimum length and minimum time, and in particular the lack of continuous space and time
manifolds, imply that derivatives cannot be used at the Planck scale. In addition, the Planck
scale prevents following the motion of constituents derived in the next section. This implies that
evolution equations and Lagrangians cannot be used in the unified theory of relativistic quantum
gravity.
In short, the physics cube and the contained limit values, and in particular the minimum length
9
and the minimum time, eliminate all descriptions of space-time and of quantum gravity that use
trans-Planckian quantities, such as descriptions based on manifolds or on discrete points. The lack
of any trans-Planckian effect is also predicted. All these predictions agree with all experiments and
all observations ever performed. The lack of points implies that the unified theory of relativistic
quantum gravity cannot be described by evolution equations or Lagrangians. The conclusions are
almost dramatic and at first sight appear discouraging. Fortunately, this appearance is deceptive.
VIII From horizons to the common constituents of space and particles
In the 1970s, Bekenstein and Hawking showed that black holes have entropy. The Schwarzschild
black hole is the simplest case, with an entropy given by S/k =A/Amin, where kis the Boltzmann
constant. The entropy value is finite and depends on the minimum area Amin = 4G~/c3, the square
of the minimum length. The Bronshtein limit cube thus implies that the factor 4 in black hole
entropy is due to the factor 4 appearing in the limits of general relativity and of classical gravity.
The finite value of black hole entropy implies that gravitational horizons are made of many
discrete degrees of freedom with Planck-sized area. This connection has been explored in var-
ious approaches to quantum gravity [58]; for these explorations, the limit cube turns out to be
particularly helpful.
Black holes can be seen either as specific configurations of curved space or as highly con-
centrated matter systems. As a consequence, both space and particles are made of Planck-sized
degrees of freedom. These tiny degrees of freedom of nature are the common constituents of space
and particles. The known properties of space and particles, together with the limit cube, allow
deducing the basic properties of these constituents.
Because the common constituents make up empty space, which is extended, the constituents
must be extended as well. Because the common constituents realize the minimum length and
make up particles, which are almost point-like, the constituents must have Planck radius. The
common constituents cannot be of Planck size in all three dimensions. If that were the case,
physical observables that depend on the enclosed volume would exceed the density and entropy
limits set by black holes, such as Bekenstein’s entropy bound [59]. Only discrete constituents that
are extended comply with the limits of the physics cube.
The common constituents cannot be membranes extended in two dimensions. Such mem-
branes cannot form localized structures, such as particles. The common constituents cannot be
bands described by an additional width parameter. If that were the case, black hole entropy would
depend on that parameter, and not depend purely on the horizon area. As a consequence, the
common constituents must be filiform [52,60,61], with a Planck-scale cross-section.
The filiform constituents cannot be or form loops, chain rings or connecting structures of fixed
10
size or shape. For Planck-scale sizes or shapes, observables would again exceed the density and
entropy limits. For measurably large sizes and shapes, the common constituents cannot be fixed in
shape or size, because in that case space would be anisotropic or not Lorentz-invariant or both.
The common filiform constituents must therefore be fluctuating.
The common constituents cannot have ends, branches or crossings, and cannot disappear into
other dimensions. All these options disagree with the minimum length, with black hole entropy
and with the properties of empty space. The common constituents must be filiform and be, apart
from their Plank-scale cross-section, essentially one-dimensional.
To comply with minimum length, the common constituents cannot pass each other, but must
lead to tangles, weaves and similar configurations. Otherwise, minimum length could not be
ensured, macroscopic empty space would not be three-dimensional, and spatial curvature could not
be recovered. Inhomogeneous configurations of the filiform constituents lead to spatial curvature.
The filiform constituents thus act like a specific type of causal sets [62].
The common constituents cannot have (or ‘carry’) observable properties such as field inten-
sities, mass, energy, momentum, charges, spin or other quantum numbers because in that case,
the vacuum would have the same properties and not be empty, i.e., with vanishing observables and
quantum numbers.
Because the minimum length and the minimum time do limit the measurement precision of
every observable and thus lead to large measurement uncertainties at Planck scales, the filiform
constituents cannot be observed individually and cannot have an equation of motion [52]. The
constituents can only be described statistically. As a consequence, no new quantum gravity effects
are expected to be observable.
The common filiform constituents lead to physical observables only when they form certain
configurations that are linked or tangled. Particles, probability densities, horizons, curvature, and
all physical observables are all due to specific linked or tangled configurations of the filiform
constituents.
In black hole horizons, the nearly two-dimensional configurations of the filiform constituents
lead to entropy and mass. Because of the two-dimensional configurations, for an observer at spatial
infinity, mass is distributed over the horizon and not concentrated at its centre [63].
In flat space, specific, almost localized, fluctuating, linked or tangled configurations of the
filiform constituents lead to particles and wave functions.
None of these conclusions disagrees with observations. However, filiform constituents are inac-
cessible, unobservable, and thus impossible to check directly. Are they pure speculations? Clearly,
filiform constituents are only worth considering if they imply both general relativity and particle
physics. And it appears that they do.
In the approximation of localized mass-energy distributions, the configurations of fluctuating
11
filiform constituents of Planck radius lead to black hole entropy, black hole temperature, and
black hole mass. Using Jacobson’s argument, Einstein’s field equations arise [6468]. Filiform
constituents exclude any extension or modification to general relativity. This is as observed [69].
In flat space, tangled configurations of the filiform constituents lead to particles. Specifically,
rational tangles of constituents lead to elementary and composed particles, spin, wave functions
and their evolution equations. Classifying rational tangles leads to the three particle generations
and to all the elementary fermions and bosons of the standard model, including the Higgs and to
no additional ones [65,66,70,71]. Filiform constituents exclude elementary dark matter particles,
anyons or any other additional elementary particles. This is as observed [72].
In flat space, deformations of the filiform constituents lead to gauge interactions. Classifying
deformations with Reidemeister moves leads to the observed gauge groups U(1), broken SU(2),
and SU(3) [65,66,70,71]. The Lagrangian of the standard model arises. Filiform constituents
exclude unified gauge groups and additional symmetries and any modification of the standard
model. This is as observed [72].
In flat space, the spatial configurations of the filiform constituents with Planck radius lead
to unique elementary particle mass values, unique coupling constants and unique mixing angles;
their values are close to the observed values [6568,70,71]. So far, only rough estimates are
available. Precise numerical calculations still need to be performed. They are predicted to agree
with the data.
In short, Bronshtein’s limit cube allows deducing that nature is made of fluctuating filiform
constituents of Planck radius, so-called strands, that fluctuate in 3 dimensions. Rational tangles
and their Reidemeister moves yield both the Lagrangian of general relativity and the Lagrangian
of the standard model of elementary particle physics. As expected from the physics cube of limits,
strands predict the lack of deviations from known physics. No consequence deduced from tangles
of strands disagrees with experiments.
IX Conclusion and outlook
Using the quadruple gravitational constant 4Gand the maximum force c4/4G, Bronshtein’s
physics cube can be extended to exhibit an invariant limit at each of its upper corners. Each
limit defines a theory of modern physics and is given by a Planck value, with Gsubstituted by 4G
everywhere. The limit cube of physics predicts the lack of trans-Planckian physics, in agreement
with all experiments so far. The limit cube also implies that the minimum length p4G~/c3is
the single principle defining relativistic quantum gravity. As a consequence, space-time is only
effectively continuous and only effectively 3+1-dimensional. Therefore, no final equations of
motion exist.
12
The minimum length implies that space and particles are made of common constituents that
are filiform, of Planck radius, and fluctuating. In separate publications, the statistics of filiform
constituents, or strands, have been shown to explain the origin of general relativity and its field
equations, of elementary particles, wave functions, gauge interactions and the full Lagrangian of
the standard model, with unique particle masses, gauge coupling constants and mixing angles.
Strands predict the lack of new physics, in agreement with all experiments so far.
Bronshtein’s limit cube of physics thus predicts that relativistic quantum gravity, the complete
description of motion, is near completion. To achieve completion, the fundamental constants of
the standard model need to be calculated to high precision using strands.
Acknowledgments
The author thanks Chandra Sivaram, Arun Kenath, Peter Schiller, Thomas Racey, Uwe Hohm, and
Isabella Borgogelli Avveduti for fruitful discussions. This work was partly supported by a grant
from the Klaus Tschira Foundation. The author declares that he has no conflict of interest and no
competing interests. There are no additional data available for this manuscript.
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