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Absence of equations of motion and of a Lagrangian
for the unified theory of relativistic quantum gravity
Christoph SchilleraID
October 2023
Abstract
The possibility of describing the yet unknown theory of relativistic quantum gravity using
equations of motion or a Lagrangian is evaluated. First, it is found that any such description
would contradict either the speed limit, the quantum of action, or the black hole limits, be-
cause any such description contradicts the minimum length in nature. Second, it is shown
that the minimum measurement errors in nature exclude any unified evolution equation di-
rectly. Third, it is found that the impossibility of actually reaching the Planck scale prevents
any description of the yet unknown constituents of space and particles with equations of
motion or a Lagrangian. Fourth, it is shown that any Lagrangian and any equation of motion
is logically incompatible with unification. All four arguments are independent of the yet
unknown unified theory. The arguments also show that space is not continuous, not discrete,
and not fundamental. This explains the failure of several attempts at achieving unification.
Given the absence of fundamental space, of equations of motion, and of Lagrangians, the
unified theory will purely describe the statistics of the yet unknown constituents of space
and particles.
Keywords: unified equations of motion; unified Lagrangian; quantum gravity.
aMotion Mountain Research, Munich, Germany, cs@motionmountain.net, ORCID 0000-0002-8188-6282.
2
I The challenge of unification
At present, both the theory of general relativity and the standard model of elementary particle
physics – with massive mixing Dirac neutrinos, as implied in the following – are in complete
agreement with the experimental results [1–4]. Some doubts, e.g, about dark matter, still remain
but have no influence on the rest of this article. Both theories are based on a Lagrangian. How-
ever, their mathematical descriptions differ markedly and are not compatible. In addition, the two
theories do not explain the origins of the fields, of space, and of the fundamental constants in the
standard model and in general relativity: elementary particle masses, mixing angles, CP violating
phases, coupling constants, number of dimensions, and the cosmological constant appear arbitrary
[5–7].
To complete the description of nature, the mathematical descriptions of general relativity and of
the standard model need to be combined into a common, unified description of relativistic quantum
gravity that describes nature with full precision and complete explanations. The achievement of the
explanation of the fundamental constants distinguishes the unified theory of relativistic quantum
gravity from low-energy quantum gravity [8,9] or from the Wheeler-DeWitt equation [10].
The search for this unified theory of relativistic quantum gravity is assumed to be, sometimes
explicitly, often tacitly, the search for a unified Lagrangian or for a set of unified, fundamental evo-
lution equations [11–18]. To date, no experimental or theoretical research approach has succeeded
in this search [1–4,19–21].
The present article argues that the unified theory of relativistic quantum gravity cannot have
a Lagrangian nor evolution equations. Four reasons for this impossibility are presented, and the
consequences are evaluated. The reasons leave open one class of approaches towards the unified
theory. Their general characteristics are outlined.
II Past success as a reason to search for a unified Lagrangian
The search for a unified theory of relativistic quantum gravity is often assumed to be the search
for a unified Lagrangian. This assumption has historical reasons.
Special relativity is characterized by the invariant speed limit c≈3.0·108m/s, as clarified by
Einstein [22]. The speed limit cis closely approached by elementary particles of low mass, such
as neutrinos or highly accelerated electrons, and is realized by photons. Every elementary particle
3
moves according to a Lagrangian that is relativistically invariant and involves the speed limit.
Quantum theory is characterized by the invariant action limit ℏ≈1.1·10−34 Js [23–27], dis-
covered by Planck. The action limit ℏis realized by quantum systems such as photons in laser
light or electrons in metals or semiconductors. The action limit also applies to the motion or
spin flip of a single electron in vacuum. The quantum of action implies the existence of wave
functions. When combined with the maximum speed cand Maxwell’s equations, the quantum
of action yields the Dirac equation, the quantum theory of light, and quantum electrodynamics.
Every quantum particle that constitutes matter moves according to a Lagrangian that involves the
quantum of action.
General relativity is characterized by the invariant mass to length limit c2/4G≈3.4·1026 kg/m
of black holes, or by a physically equivalent limit, such as the maximum force c4/4G≈3.0·1043 N
or the maximum power c5/4G≈9.1·1051 W[28–31]. All of these limits are realized by black
hole horizons. In general, the limits apply to the motion and properties of a horizon and space.
Every gravitational horizon moves according to the laws of black hole mechanics; therefore every
volume of space curves and moves according to the field equations that involve the mass to length
limit or the force limit [30,31].
Similar to the three limits for speed, for action, and for the mass-to-length ratio or force, also
the motion in the domain of relativistic quantum gravity is expected to derive from a Lagrangian
that is based on the combination of these limits.
III The Planck limits of quantum gravity
In the domain of relativistic quantum gravity, the three experimental limits of nature on speed
v≤c, on action W≥ℏ, and the black hole limit on mass per length m/l ≤c2/4Gplay a role at
the same time. The equivalent black hole limit on force on F≤c4/4Gcan also be used instead.
The three limits can be combined.
A straightforward expression for action is
W=Et =mc2t= (m/l)c l2,(1)
where Eis energy and mis mass. This expression allows inserting the limits for action and for
mass per length. An alternative expression for action is
W=F l t =F l2/v . (2)
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This alternative expression allows inserting the limits for action, for force and for speed. In both
cases, solving for length lyields
l≥p4Gℏ/c3≈3.2·10−35 m.(3)
Nature limits length. The experimental limits of special relativity, quantum theory and general
relativity imply a minimum length in nature, given by twice the Planck length. The existence
of a minimum length, which also implies a minimum precision for length measurements, has
been known for a long time [32–40]. Relativistic quantum gravity describes nature whenever this
minimum length is approached.
Equivalently, relativistic quantum gravity is characterized by the minimum area 4Gℏ/c3≈1.0·
10−69 m2or by the minimum time p4Gℏ/c5≈1.1·10−43 s. Any other limit that contains all three
constants c,ℏand 4Gcan also be used to characterize relativistic quantum gravity. (The limits
given by the Planck energy, Planck momentum, and Planck mass only apply to single elementary
particles, as their derivation shows.)
In other terms, nature has no trans-Planckian effect of any kind. This statement follows from
all observations ever made about speed, quanta and general relativity. The impossibility of exceed-
ing Planck limits was already understood by Matvei Bronstein, who explained in the 1930s that
observing areas smaller than the limit implies either crossing a (microscopic) black hole horizon
or exceeding the indeterminacy relation [41–46]. Both cases are impossible.
IV Lagrangians and equations of motion are always approximations
Quantum gravity, the combination of special relativity, general relativity and quantum theory,
implies a minimum length. Conversely, continuity, and in particular the continuity of space and
time, results from neglecting quantum gravity. This contradiction between continuity and quantum
gravity is essential in the following.
Every equation of motion and every equation of motion makes use of continuous pace and time.
Because of the use of continuity, no Lagrangian formulation can yield a minimum length. The
opposite is possible: a minimum length can yields a Lagrangian or an equations, as will become
clear below. In other words, every Lagrangian and every evolution equation is an approximation
that arises from what is happening at the Planck scale. As all experiments show, the Lagrangians
of general relativity and of the standard model are extremely good approximations. These two
Lagrangians are also required approximations, and there is no way to avoid them. The same
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applies to equations of motion: they are approximations as well, based on the approximation of
continuous space and time. However, approximate concepts cannot be the basic concepts in a
unified theory.
Lagrangians and equations of motion arise only once continuous space and time are introduced.
Therefore, a continuous description that takes into account all three limits of nature – the maximum
speed, the quantum of action, and any of the black hole limits – is intrinsically impossible, because
the three limits, taken together, contradict continuity. Any proposed equation of motion and any
proposed Lagrangian for quantum gravity would imply continuous space and time and thus would
contradict at least one of the three experimentally established limits. This is the first argument
against a unified Lagrangian and against unified evolution equations.
V Errors prevent the description of evolution with equation
Feynman gave a humorous argument claiming that all the laws of physics can be contained in
one single equation [14]. He started by defining U1= (F−ma)2for mechanics, then U2=
(∇ · E−ϱ/ε0)2for electrostatics, and imagined to continue for all other equations of physics. He
claimed that continuing in this way, the equation for the unworldliness
U=U1+U2+... = 0 (4)
contains all of physics. However, quantum gravity shows that this humorous argument is wrong.
In the usual domains of physics, the squared differences UNbetween the right- and left-hand
sides of an evolution equation indeed vanish. Why can such a statement be made? Outside quan-
tum gravity, measurement errors can – in a suitable limit – always be reduced as much as desired.
In general relativity or in quantum theory, length values, time values and the values of all ob-
servables can always be measured – in a suitable limit – with errors as small as desired. Sloppily
speaking, in quantum theory and in general relativity, measurement errors are due to limited exper-
imental effort. In other terms, in these domains of nature, measurement errors are due to practical
limitations, but can be made negligibly small with sufficiently well-designed experiments. Mea-
surement errors in these domain are not intrinsic, but human artefacts. Therefore, in quantum
theory and in general relativity, one can speak about equations as objective concepts, even though
they are somewhat hidden behind human limitations.
In quantum gravity, in contrast to general relativity and to quantum theory, in addition to errors
due to limited effort, certain measurement errors are intrinsic to nature. In particular, the minimum
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length and the minimum time imply minimum measurement errors for length and time. Because
of error propagation, quantum gravity also implies minimum measurement errors for all other
observables. The Planck limits always provide intrinsic measurement errors. These minimum
measurement errors prevent from proving that any expression involving differences of observables,
such as the unworldliness U, vanishes identically. Because of nature’s intrinsic measurement
errors, all expressions obey
UN>0(5)
and Feynman’s unworldliness Uis not exactly zero – not even for general relativity or the standard
model.
In fact, the intrinsic measurement errors due to quantum gravity prevent the confirmation of
any proposed evolution equation, both in quantum gravity and in all other domains. Due to the
intrinsic measurement errors, in quantum gravity and in physics in general, evolution equations
are always approximate. There is no way to confirm any evolution equation. In quantum gravity,
no limit leads to situations with vanishing measurement errors. When quantum gravity is taken
into account, evolution equations lose their objective reality.
The intrinsic errors of nature imply that no equation is exactly valid. Equations do not exist in
nature. The second argument thus states that no evolution equation and no principle of least action
that uses a Lagrangian can be exact, or fully precise. But full precision is a requirement for any
unified theory. In other words, minimum measurement errors due to quantum gravity prevent the
existence of evolution equations and Lagrangians for the unified theory.
VI Black holes as a reason to search for a unified Lagrangian
A further reason to look for unified equations of motion starts with the thermodynamics of every-
day systems. In all thermodynamic systems studied in physics – including liquids, solids, electron
gases, and stars – the thermodynamic properties are averages of the motion of the underlying
constituents, such as molecules, atoms, nuclei, nuclei or electrons.
Also black holes are thermodynamic systems. Black holes can either be seen as large num-
bers of particles compressed to the limit or as space curved to the limit by the contained energy
value. Thus, black hole horizons are systems at the border between matter and radiation on the one
hand, and between space on the one hand. Because black holes have entropy and temperature, all
horizons, all particles, and space itself must be composed of some common constituents. There-
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fore, the future unified theory of relativistic quantum gravity is a theory of common constituents
that describes both particles and space. These common constituents are the fundamental concepts
beyond the standard model. At present, the details of the common constituents are subject of re-
search. The common constituents could be superstrings, loops, bands, strands, graphs, tetrahedra,
or other something else. In the following arguments, these details are unimportant, except for two
general properties.
All black hole horizons have in common that their entropy is finite. So far, the Bekenstein-
Hawking expression [47,48] relating entropy Sand horizon area A
S=kc3
4GℏA , (6)
where kis the Boltzmann constant, has never been confirmed in direct experiments. Nevertheless,
there is little doubt that an entropy value of a Schwarzschild black hole horizon exists, that it is
finite, and that the Bekenstein-Hawking expression gives at least the correct order of magnitude.
The finite entropy of horizons implies that space, particles and horizons must consist of countable,
discrete constituents. But the constituents must also be extended, in order to form space.
The value of the black hole entropy implies that each constituent occupies – within a factor of
order O(1) – the minimum area of 4Gℏ/c3≈1.0·10−69 m2. Thus, in two spatial dimensions, the
common constituents are of Planck size; in the third dimension, they must be extended.
In short, the thermodynamics of black holes is due to the disordered motion of their extended
and discrete, Planck-sized constituents – whatever they may be. This conclusion demands to
search for the Lagrangian and the evolution equations for single common constituents of space
and particles.
VII The impossibility to observe single constituents
The minimum length implies that the quantum gravity limit 4Gℏ/c3specifies the minimum area in
nature [33–35]. However, these and all other quantum gravity limits differ in an important aspect
from any limit that arises from the combination of only one or two quantities c,ℏand 4G: there is
no known physical system that realizes the area limit. Observation of a single Planck-scale system
would require a probe with a wavelength as small as the Planck length. However, such probes have
never been developed or observed. Therefore, no observation that realizes, contains, or confirms
the area limit has ever been conducted.
8
In apparent contrast, the area limit appears in the black hole entropy and in the temperature of
black holes. However, no experimental verification of these expressions has yet been achieved,
despite intense efforts in Unruh radiation [49] and in systems analogous to black holes [50]. In
other words, the area limit cannot be experimentally verified using black holes.
As a consequence, the area limit cannot be achieved. Why is this the case? A general way to
explain the result is the following. Every measurement is a comparison with a unit. To measure a
system with Planck size, the unit has to have Planck size, and so must be the comparing device.
But a system with Planck size also has Planck energy and Planck density. Such a system is a
microscopic black hole. And these have never been produced or even observed. This will be true
also in the future.
The impossibility of testing or approaching quantum gravity limits is general. For example,
it also applies to the time limit p4Gℏ/c5, to the volume limit, the density limit c5/(16G2ℏ), the
acceleration limit pc7/4Gℏ, the energy limit pℏc5/4Gfor single elementary particles, and to the
momentum limit pℏc3/4Gfor single elementary particles. At present, the quantum gravity limit
that has been most closely approached is the minimum length. The measurement of the electron
dipole moment performed in 2023 is still more than three orders of magnitude away from the
Planck length times the electron charge [51]. All other quantum gravity limits are many orders of
magnitude away from the best available experimental results. Simply stated, no quantum gravity
limit that contains all three quantities c,ℏand 4Ghas been realized or observed, can be realized
or observed, or can even be approached closely. Above all, the quantum gravity limits will never
be achieved.
In other words, even though relativistic quantum gravity is due to the motion of discrete and
extended Planck-sized degrees of freedom, or common constituents, single degrees of freedom
or single common constituents – both expressions are used interchangeably here – cannot be ob-
served. Because of the impossibility of reaching the Planck limits, it is even impossible to imagine
their observation. For example, there is no proposal in the literature that would allow counting
common constituents. The impossibility of observing any quantum gravity limit also implies the
impossibility of defining any physical variable, such as the position or velocity, that might describe
single common constituents.
It should be noted that the impossibility of achieving quantum gravity limits does not imply the
impossibility of obsering quantum gravity effects. The latter is a separate topic that is not treated
here.
9
Describing the motion of any fundamental constituent – with an equation or Lagrangian – re-
quires the ability to follow the (possibly approximate) position of the constituent over time. How-
ever, for a single common constituent of space and matter, this cannot be achieved: the observed
speed limit, observed black hole limits, and observed quantum of action, taken together, imply that
a single constituent cannot be measured and followed. If a single constituent cannot be followed
in its motion, its motion cannot be described with an equation. This surprising result must and can
be checked in more detail.
VIII Attempts to observe gravitons and common constituents
In a way that is still subject of research, gravitons are composed of the constituents of space. Some
time ago, various researchers stated that single gravitons cannot be observed [52,53]. In fact, not
everyone agrees [54,55] and theoretical efforts to find a way are still ongoing. To date, single
gravitons have never been observed. In contrast, large numbers of gravitons can be observed.
Examples include the first detection of gravitational waves in 2015 [56] and ongoing measurement
projects [57].
The impossibility of measuring single gravitons is corroborated by the impossibility of describ-
ing the motion of single gravitons. Even using the latest research results on quantum gravity [8],
single gravitons cannot be described by equations with complete precision; their scattering from
matter cannot be calculated with complete confidence.
The constituents of space could, in principle, be measured by observing the granularity of
space or by detecting space noise [58,59]. However, no such effect has been observed. Likewise,
attempts to observe Planck-scale constituents of matter, such as preons, or constituents of wave
functions [4,60], have all failed. One day, possibly, quantum gravity effects will be observed. But
no observation will detect single gravitons or single constituents of space.
The impossibility to reach the Planck scale prevents the observation of single common con-
stituents, and therefore prevents the existence of unified evolution equations and of a unified La-
grangian. This is the third argument making the point.
IX Logical impossibility of a unified Lagrangian or unified equations of motion
In his 2017 Faraday lecture, David Tong presented a partition function that combined general
relativity with the standard model in a single expression [61]. This symbolic expression avoids
10
mentioning common constituents and does not explain the origin of the fields or of the fundamental
constants. Thus, despite its appeal, the description is not unified. In fact, the attempt highlights a
fourth reason that speaks against a unified Lagrangian, or a unified partition function, or unified
equations of motion.
Any theory that is described with a Lagrangian density does not explain the origin of that
Lagrangian density. The same is valid for any equation of motion. A Lagrangian or equation of
motion does not explain the origin of the specific fields appearing in it, the origin of the specific
symmetries, and the origin of continuous space. A Lagrangian does not explain the origin of the
specific constants in it. Indeed, these are the open issues of the standard model and of general
relativity, and these issues are the reason that neither theory is unified.
The unified theory of relativistic quantum gravity must explain how the two Lagrangians of
general relativity and of the standard model arise. It must explain the origin of fields, symmetries
and fundamental constants. All of these concepts are based on continuous observables. The Planck
limits of quantum gravity imply that continuous observables result from approximations. The
unified theory must explain the details of this approximation process. This cannot be achieved with
a unified Lagrangian or with unified equations of motion, which would raise the same questions.
In addition, the principle of least action must itself be explained by a unified theory. Again, this
cannot be achieved with a unified Lagrangian, which assumes the principle from the start.
Finally, any Lagrangian and any equation of motion is based on space and time. A unified
theory must explain the origins of continuous space and time. Once again, this cannot be achieved
with a unified Lagrangian or a unified equation of motion.
In other words, any theory based on a Lagrangian cannot be unified for a logical reason: Uni-
fication and Lagrangians are mutually exclusive.This is the fourth argument against a unified
Lagrangian and against unified evolution equations. This result has important consequences.
X Continuous and discrete descriptions
The length measurement limit forbids describing space as a set of points. Therefore, in a unified
theory, a region of space cannot be described by a manifold, by a lattice of points, or by any set of
discrete elements. Likewise, the length limit bans the use of Riemannian space, scale invariance,
conformal invariance, space-time foam, holography, twistor space, categories, non-commutative or
fractal space from being part of any unified theory. The length measurement limit also prevents the
11
observation and the existence of higher or lower dimensions. Similarly, the length measurement
limit forbids the observation and existence of any additional spatial structure or symmetry at the
Planck scale that is added to a manifold or that is based on discrete points or elements. In simple
words, the minimum length implies that space is neither continuous nor discrete.
All physical observables are limited by Planck limits. The Planck limits, the lack of points,
and the lack of manifolds rule out continuous symmetries and continuous fields. Gauge fields,
wave functions, gauge symmetries, and space-time symmetries cannot be fundamental. But they
cannot be discrete either. In other words, the disregard for the Planck limits of quantum gravity
has prevented unification in the past.
Space, time, particles, and all physical observables must somehow be due to the common con-
stituents of space and particles. Due to the intrinsic measurement errors and the ensuing lack of
discrete values, nature cannot form discrete sets. Due to the intrinsic measurement errors and the
ensuing lack of continuous values, nature cannot form continuous sets. Nature is neither continu-
ous nor discrete.
Given that a unified theory of relativistic quantum gravity cannot contain Lagrangians, sym-
metries, fields or equations, the only mathematical statement using continuous observables that
describes the unified theory of relativistic quantum gravity is the mentioned inequality
l≥p4Gℏ/c3≈3.2·10−35 m.(7)
Equivalently, the unified theory is described by any other Planck limit from quantum gravity that
combines the three quantities c,4Gand ℏ, such as the area limit or the acceleration limit.
Fortunately, the exclusion of Lagrangians and evolution equations does not exclude all math-
ematics from the unified theory. The arguments only exclude mathematical analysis, because it
is based on continuous variables and spaces. Instead, tools and concepts of discrete mathemat-
ics [62,63], must and will be included in a unified theory. This is as expected from common
constituents of space and particles that are partially discrete and partially continuous.
XI Outlook
In summary, all equations of motion and all Lagrangians are approximations that average out the
effects of quantum gravity. Equivalently, all equations of motion and all Lagrangians are effects
of the common, Planck-sized constituents of space and particles. Quantum gravity Lagrangians
or equations of motion cannot exist because they contradict observational limits, because they
12
contradict the intrinsic measurement errors of nature, and because they cannot be unified. Above
all, no domain of physics – including black hole physics, low-energy quantum gravity, or particle
physics – provides a method to observe single constituents. In short, a Lagrangian for the unified
theory is impossible. This result prevents numerous approaches to unification from succeeding.
Nevertheless, observations show that evolution equations and Lagrangians for many common
constituents exist and describe nature without any measurable deviations [1–4], although they are
not unified. Therefore, the unified theory of relativistic quantum gravity is a theory of the statistical
and collective behaviour of many constituents, whatever they may be. The known Lagrangians,
all physical observables, and space-time itself emerge through averaging.
In the case of general relativity, large numbers of common constituents somehow lead to the
emergence of space, curvature, the Hilbert Lagrangian, Einstein’s field equations and the cosmo-
logical constant. Curvature, gravitational waves, and continuous flat space somehow emerge from
a large number of constituents in ways that resemble the way in which the constituents lead to
black hole horizons and their thermodynamic properties. The details of the “somehow” – i.e.,
the precise averaging procedure that includes discrete mathematics and leads to curved space –
will depend on the details of the constituents. Examples for this approach are provided by causal
sets [64], loop quantum gravity [65], and other approaches to emergent quantum gravity [66–71].
However, most of these approaches do not explain the origin of gauge interactions and fundamen-
tal constants – elementary particle masses, coupling constants, and mixing angles.
In the case of quantum field theory, large numbers of common constituents somehow lead to
the emergence of wave functions, the Dirac equation, the gauge field equations with their gauge
groups, elementary particles with their quantum numbers, the standard model Lagrangian, and
to the fundamental constants. Again, the details of the “somehow” – i.e., the precise averaging
procedure that uses discrete mathematics – will depend on the details of the constituents. Attempts
to deduce the standard model Lagrangian with its fundamental constants are still rare [72,73].
In physics, all testable expressions for observables are based on large numbers of the yet un-
known common Planck-scale constituents of space and particles. The challenge of unification is to
deduce the correct constituents of nature, despite the impossibility of observing them one by one.
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Acknowledgements and declarations
The author thanks Uwe Hohm, Michael Good, C. Sivaram, Arun Kenath and Isabella Borgogelli
for fruitful discussions.
Funding: part of this work was supported by a grant from the Klaus Tschira Foundation.
Competing interests, additional data and material, ethical concerns: none exist for this work.
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