Dark Matter and Dark Energy as Quantum Spin Connection Foam Manifestations

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Journal of Physics: Conference Series 3017 (2025) 012031
doi:10.1088/1742-6596/3017/1/012031
Dark matter and dark energy as manifestations of
quantum spin connection foam
I V Kanatchikov1,2 and V A Kholodnyi3,4
1
National Quantum Information Centre in Gdansk (KCIK), 80-309 Gdańsk, Poland
IAS Archimedes Project, 83700 Saint Raphaël, France
3
Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
4
Unyxon, Woodforest, TX, USA
2
E-mail: [email protected]
Abstract. This paper demonstrates how the simplest form of dark energy, represented by
the cosmological constant, and an alternative to dark matter, in the form of MOND, emerge
from the quantum spin connection foam picture of quantum gravity. We also discuss possible
experimental tests of the latter.
1. Introduction
Modern cosmological observations reveal that the mass-energy content of the Universe is
dominated by dark components, comprising approximately 68% dark energy and 27% dark
matter, with only 5% attributed to known forms of matter, such as baryons, leptons, and
photons [1]. The concept of dark matter [2] is invoked to explain the observed non-Keplerian
rotation curves of galaxies at large radial distances from their centers, as well as the dynamical
effects attributed to a “missing mass” within galaxy clusters. Dark energy, on the other hand,
is the leading explanation for the current accelerated expansion of the Universe. Within the
standard ΛCDM model of cosmology, dark energy is identified with the cosmological constant,
and dark matter is postulated to be a substance of unknown nature. In spite of a variety of
proposals of different candidates for the role of dark matter during the last decades, none of
dark matter particles has been detected so far.
A distinct approach modifies General Relativity and seeks to inherently embed the phenomena
attributed to dark matter and dark energy within a framework employing geometries and
Lagrangian densities more general than those of Riemannian geometry and the Hilbert-Einstein
action. As a result, a plethora of modifications to Einstein’s gravity have been proposed, wherein
classical solutions are constructed to replicate observational data or emulate the effects of diverse
dark matter and dark energy candidates.
A promising and phenomenologically viable hypothesis proposing a modification of
Newtonian dynamics at small accelerations was introduced by Milgrom in 1983 [3–6]. However,
subsequent efforts to formulate this hypothesis as a consequence of a generalization of General
Relativity have resulted in rather cumbersome expressions for the proposed Lagrangian densities
and the introduction of new fields (see [7] and [8] for a review). The key postulate of Milgrom’s
Modified Newtonian Dynamics (MOND) is the existence of an acceleration scale a0 ≈ 1.2×10−10
ms−2 at which the dynamics is modified, such that the new (“deep-MOND”) dynamics law at
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small accelerations
doi:10.1088/1742-6596/3017/1/012031
GM
g2
= 2
a0
r
(1)
automatically ensures, due to the relation g = v 2 /r between the acceleration g and the rotation
velocity v, that
v 4 = GM a0
(2)
is constant, in agreement with the phenomenological Tully-Fisher relation [3,8,9]. For arbitrary
accelerations, the MOND law
Å ã
GM
g
g= 2
(3)
µ
a0
r
interpolates between the Newtonian regime at large accelerations, where µ(u) → 1 at
u >> 1, and the deep-MOND regime at small accelerations, where µ(u) ≈ u at u << 1.
The drawback of this hypothesis is that the choice of the interpolating function is purely
phenomenological. Furthermore, the theory of MOND does not provide an explanation for the
observed phenomenological relation between the acceleration scale a0 , which is posited to play
a crucial role√in the dynamics of galaxies and galaxy clusters, and the fundamental cosmological
parameters Λ and H0 .
In [10], an invariant acceleration scale, related to the square root of the cosmological constant,
was found in the simplest solution of precanonical quantum gravity [11–15], corresponding to the
wave function describing a quantum analog of Minkowski spacetime in Cartesian coordinates.
This result poses the question of whether and how a modification of Newtonian mechanics could
emerge from the precanonical quantization approach to quantum gravity in the non-relativistic
limit, and whether this modification is equivalent to Milgrom’s MOND. This question was first
addressed in [16], where a non-Milgromian modification of Newtonian dynamics was obtained.
Subsequently, in two recent preprints [17, 18], we show that the true Milgromian MOND, with
a theoretically derived interpolating function, emerges as an effective description accounting for
non-inertial effects in the mean field of spin connection quantum fluctuations.
In this paper, we elucidate the emergence of MOND from precanonical quantum gravity and
discuss its implications for research on dark matter and dark energy, as well as the potential
for experimental detection of modified dynamics in a laboratory setting. Our analysis is
grounded in the framework of precanonical quantization of gravity within the vielbein Palatini
formulation [11–15] which uses methods and ideas from precanonical quantization of fields.
Precanonical quantization of fields [19–23] has been proposed as an alternative framework to
canonical quantization. Departing from the latter, which relies on the Hamiltonian formalism
and a distinguished time variable, precanonical quantization utilizes a spacetime-symmetric
generalization of the canonical Hamiltonian formalism, known in the calculus of variations
as the De Donder-Weyl (DDW) Hamiltonian formulation [24, 25]. The existence of the De
Donder-Weyl version of the Hamilton-Jacobi (HJ) theory [24–29] raises the question about
a formulation of quantum theory of fields which could reproduce the DDW-HJ theory in
the classical limit [20]. Precanonical quantization is predicated on the Dirac quantization
of a Heisenberg-like subalgebra of Poisson-Gerstenhaber brackets of differential forms, which
represent dynamical variables. These brackets were identified within the DDW Hamiltonian
formulation in [22, 30–34]. Quantizing brackets defined on differential forms naturally leads to a
hypercomplex generalization of quantum theory, where operators and wave functions are valued
in the complexified spacetime Clifford algebra [20–23]. This representation of differential forms
using Clifford algebra elements demands a dimensional factor. Specifically, as demonstrated
in [19], an ultraviolet parameter κ with the dimension of inverse spatial volume is required,
mapping the basis of (n−1)−forms, ϖµ , to the elements of the Clifford-Dirac algebra γµ /κ. The
parameter κ also appears in the expressions of physical quantities operators. The corresponding
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precanonical wave function, Ψ(ϕa , xµ ), defined on the finite-dimensional bundle of field variables
ϕa over spacetime variables xµ , satisfies a Dirac-like, spacetime-symmetric generalization of the
Schrödinger equation (in flat spacetime):
iℏκγ µ ∂µ Ψ = ĤΨ.
(4)
Here, Ĥ is a partial differential operator corresponding to the DDW analogue of the Hamiltonian
function (see [20, 21, 23] for further details).
In prior publications, the application of precanonical quantization has been extended to
various formulations of general relativity, including metric variables [38–41], the teleparallel
equivalent [42, 43], and vielbein variables [11–15].
The study of the relationship between the precanonical wave function Ψ(ϕ, x) and the
Schrödinger wave functional Ψ([ϕ(x)], t) in scalar field theory [45, 46], in curved spacetime
[47–49], and in pure Yang-Mills theory [35, 37] leads to the following conclusions: (i) The
functional Ψ can be interpreted as a product integral over all spatial points x of a precanonical
wave function Ψ restricted to a configuration ϕ = ϕ(x). (ii) The product integral is defined
after the Clifford algebra elements in Ψ are “dequantized” using the map γ 0 /κ 7→ dx. (iii)
The construction of the product integral implies an infinitesimal value of 1/κ or the infinite κ.
(iv) In this sense, the standard QFT appears to be a deregularized limit of the theory derived
from precanonical quantization. The parameter κ has been linked to the scale of the mass gap
in quantum non-Abelian gauge theories [36], and this connection is crucial for the conclusions
drawn in this paper regarding the emergence of MOND and the observed small value of the
cosmological constant representing the simplest form of what is called “dark energy”.
2. From precanonical quantum gravity to MOND
A concise introduction to the precanonical quantization of vielbein gravity [11–15] can be found
in [10, 16, 17]. To avoid redundancy, our presentation here will treat these earlier findings as
established.
Let us begin with the precanonical Schrödinger equation for quantum gravity:
Ç
å
1
∂
IJ ∂
KL ↔
KM
L
∂µ + ωµKL γ
∨ − KL ωµ
ωβM
Ψ(ω, x) + λΨ(ω, x) = 0,
(5)
γ
∂ωµIJ
4
∂ωβ
↔
↔
where ∨ denotes the commutator Clifford product γIJ ∨ Ψ = 21 γIJ , Ψ and λ = Λ/(8πGℏκ)2
is a dimensionless combination of the fundamental constants of the theory. One- and two-point
solutions of (5) describe quantum geometry of spacetime, termed spin connection foam (SCF),
in terms of probability amplitudes and transition amplitudes of spin connection components ωµIJ
at different spacetime points xµ .
In this formulation, the spacetime metric emerges as an expectation value
Z
µν
‘
⟨ĝ ⟩(x) = Tr Ψ(ω, x)ĝ µν [dω]Ψ(ω,
x) ,
(6)
of the operator of the metric tensor given by:
ĝ µν = −(8πGℏκ)2 η IK η JL
∂
∂
,
∂ωµIJ ∂ωνKL
(7)
where the invariant integration measure on the space of spin connection components can be
written in the form
Y
‘ = i (det
◊
[dω]
g µν )3
dωσIJ ,
(8)
σ,I,J
which can be related to the expressions from previous papers using e = det(eIµ ) = det(gµν )1/2 =
det(g µν )−1/2 and the operator of det(g µν ) derived from representation (7).
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2.1. SCF of quantum Minkowski spacetime
The simplest solution of (5) is obtained in the case of a quantum analog of Minkowski spacetime
in Cartesian coordinates when ωµIJ = 0. In this scenario, equation (5) with vanishing Λ = 0
simplifies as follows:
(9)
γIJ ∂ωµIJ ∂µ Ψ = 0.
Taking representation (7) into account, the square of (9) yields
ĝ µν ∂µ ∂ν Ψ = 0.
(10)
To describe the SCF corresponding to Minkowski spacetime, we have to require that ⟨ĝ µν ⟩ = η µν ,
which is satisfied by solutions of the following equations:
ĝ µν Ψ = η µν Ψ.
(11)
From (10) and (11), it follows that the modes of the precanonical wave function corresponding
to quantum Minkowski spacetime propagate on the base of the spin connection bundle according
to the wave equation
η µν ∂µ ∂ν Ψ = 0 ,
(12)
and in the fibres of the spin connection bundle according to the equations (11), which have the
explicit form
∂
1
∂
Ψ+
η µν Ψ = 0.
(13)
η IK η JL IJ
∂ωµ ∂ωνKL
(8πGℏκ)2
Therefore, the spin connection foam (SCF) corresponding to the quantum analogue of Minkowski
spacetime is described by equations (12) and (13), whose solutions represent massless modes
of the precanonical wave function propagating on the spacetime base and massive modes
propagating on the fibers, coordinatized by the spin connection components. The range of
these massive modes introduces an invariant scale of accelerations a∗ = 8πGhκ, which is related
to the cosmological constant Λ = λa2∗ through the dimensionless constant λ, determined by a
proper ordering of the second term in (5).
2.2. Test particle in the field of mass M in static SCF
Let us consider the motion of a point test particle in the field of a point mass M , both immersed
into a SCF with fluctuating spin connection coefficients. Given that we lack a solution of
precanonical Schrödinger equation or SCF corresponding to the Schwarzschild metric, we assume
that the motion is still described by the geodesic equation
α
β
dxµ
µ dx dx
+
Γ
αβ ds ds = 0 ,
ds2
(14)
in which the connection Γµαβ is a sum of the classical part and the quantum fluctuation. For the
non-relativistic test particle and a static approximation of the geometry we, therefore, can write
xi
ẍi + GM 3 + ω̃ i = 0 ,
r
(15)
where we use the expression of Γi00 = ω0i0 = GM xi /r3 + ω̃ i in the static approximation. The
distribution of quantum fluctuations of static SCF is given by the wave function of the ground
state, which satisfies the modified Helmholtz equation
δ ij ∂ω̃i ∂ω̃j Ψ −
1
Ψ = 0.
(8πℏGκ)2
4
(16)
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This is the only relevant component of (13) in the static approximation, where all the components
of the spin connection, except ω0i0 , are assumed to be frozen at the zero classical value.
The ground state of (16) is assumed to be spherically symmetric in ω-space and normalized
to unity:
Z ∞
dω̃ ω̃ 2 |Ψ0 |2 = 1 ,
(17)
⟨Ψ0 |Ψ0 ⟩ = 2π
0
where ω̃ = |ω̃ i |.
The ground state is readily found to be of the form of a Yukawa potential in
ω-space:
Ψ0 (ω̃) = ⟨ω̃|0⟩ = √
e−ω̃/8πGℏκ
.
ω̃
8π 2 ℏGκ
1
(18)
Consequently, a straightforward calculation yields:
⟨0|ω̃ i |0⟩ = 0,
1
⟨0|ω̃ 2 |0⟩ = a2∗ .
2
(19)
Then, the average of (15) reproduces classical Newton’s law, while the average of its square
leads to
G2 M 2
ẍ2 −
− ⟨0|ω̃ 2 |0⟩ = 0
(20)
r4
and therefore yields a qMOND law, a quantum-gravitational modification of Newtonian
dynamics:
G2 M 2 1 2
+ a∗ .
(21)
|ẍ| =
r4
2
This modified dynamics gives rise to an effective qMOND potential Φ(r), such that ẍ = −∇Φ,
where Φ(r) is given by
Φ(r) = −
Å
ã
GM
1 1 3
a2∗ r4
F
−
,
−
;
;
−
,
2 1
r
2 4 4 2G2 M 2
(22)
with 2 F1 (a, b; c; z) denoting the Gauss hypergeometric function. It is readily apparent that this
potential interpolates between the
for
√ small r ≪ r∗ and exhibits
√ Newtonian potential −GM/r
2
the linear potential Φ(r) ≈ a∗ r/ 2 for large r ≫ r∗ , where r∗ = 2GM/a∗ .
2.3. The origin of Milgromian MOND
The precanonical quantization of gravity leads us to the idea that all physical systems
are immersed in quantum spacetime characterized by fluctuating spin connections, the spin
connection foam. In the static approximation, the SCF exhibits a non-vanishing mean-field
acceleration in the ground state due to the non-vanishing covariance of the distribution of spin
connection coefficients derived from the wave function of the ground state: ⟨0|ω̃ 2 |0⟩ = 12 a2∗ . This
leads to the inherent non-inertiality of physical reference systems which should be accounted for
by adding a fictious force to the Newtonian gravitational force, i.e.
|ẍ| =
GM
1
+ √ a∗ .
2
r
2
(23)
In terms of the redefined force and the new acceleration
1
g = |ẍ| − √ a∗
2
5
(24)
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the qMOND law in equation (21) takes the form
…
Å ã
1 2
g
GM
1
2
= g + a∗ − √ a∗ = µ
g,
2
r
2
a0
2
where
a0 =
√
2a∗
(25)
(26)
and
ä
1 Äp 2
4u + 1 − 1
(27)
2u
satisfies the properties of interpolating functions in MOND: µ(u) → 1 for u ≫ 1 and µ(u) → u
for u ≪ 1.
The form of the dynamical law presented in (25) is recognized as MOND, a hypothesis
proposed by Milgrom in 1983 [2-4] to describe deviations from Keplerian behavior in the
dynamics of galaxies and galaxy clusters without the need for dark matter. This law interpolates
between Newtonian dynamics at accelerations g ≫ a0 and the “deep-MOND” dynamics at
g ≪ a0 , which describes the observed flat rotation curves of galaxies. Within MOND, the
interpolating functions µ(u) are selected based on observational data, and the theory itself offers
no theoretical foundation for preferring one over another. Our approach here, rooted in the first
principles of precanonical quantum gravity and applied to the approximation of non-relativistic
test particles moving within the field of a point-like central mass M fixed at the origin, both
immersed in the static approximation of SCF, not only derives MOND but also elucidates its
physical origin and theoretically determines the form of the interpolating function within the
approximations above.
Note that our consideration here explicitly neglects the back-reaction of the test particle on
the central mass M . However, in reality, this assumption can be violated in systems like the
Sun-Jupiter or wide binary systems, precisely because a0 is small but non-zero. Moreover, due
to the equivalence principle, the intrinsic correlations of spin connections at different spatial
points within the static SCF also contribute to the effective interaction between the test particle
and the central mass, even when the latter is significantly larger than the mass of the test
particle. Furthermore, our non-relativistic motion and static spin connection foam (SCF)
approximations disregard the non-instantaneous nature of gravitational interaction (and, in
our case, the propagation of the precanonical wave function), which may be crucial on galactic
scales [50,51]. Relaxing these assumptions will alter the interpolating function in (27), including
the relation (26) between the Milgromian a0 and our theoretical a∗ .
µ(u) =
3. Numerical values of a∗ and Λ
The numerical values of the characteristic acceleration a∗ = 8πℏGκ and the cosmological
constant Λ ∼ a2∗ depend on the Planck scale quantity ℏG ∼ L2Pl and the scale of the parameter
κ. The latter was related to the gap in the spectrum of the DDW Hamiltonian operator of the
quantum SU(2) Yang-Mills field that has allowed us to roughly relate it to the mass gap ∆m in
the pure Yang-Mills sector of the Standard Model as follows:
κ ∼ gs−2 ℏ−4 (∆m)3 ,
(28)
where gs represents the gauge coupling constant in the classical Yang-Mills Lagrangian, which
can be identified with the QCD running coupling constant at zero momentum transfer: gs2 ≈ 4π 2
[52, 53]. By identifying the mass gap with the scale of the lightest QCD meson excitations
∆m ∼ 10−1 GeV we obtain
a∗ ∼ 8πGgs−2 ℏ−3 (∆m)3 ∼ 10−27 m−1 ,
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(29)
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√
which is compatible with the observable value of the Milgromian acceleration a0 = 2a∗ =
1.2 × 10−10 m/s2 (approximately 10−27 m−1 in geometrized units). Consequently, the value of
the cosmological constant Λ = λa2∗ is close to the observed value, Λ ≈ 10−52 m−2 , for λ ∼ 1,
which is compatible with the Weyl ordering [16], with the discrepancies arising due to the
roughness of the spectral estimate underlying (28) and the uncertainty in the identification of
the mass gap in the pure Yang-Mills sector of the Standard Model.
In what follows, although the theory in its current stage reproduces the numerical values of a∗
and Λ to within several orders of magnitude, we will use the observable values in our subsequent
estimations.
4. Observational consequences
Let us consider a source of gravity with mass M = 1 kg and a test particles with mass m = 1 mg
in its gravitational field. One can assume that the gravitational field of Earth and surrounding
bodies in the vicinity of the test particle is compensated by magnetic
√ levitation, so it can be
ignored. Then, GM ∼ 10−27 m and the critical distance r∗ = ( 2GM/a∗ )1/2 ≈ 1 m. At
distances r ≪ r∗ , the qMOND equation in (21) is approximated by
|ẍ| ≈
GM
a2∗ r2
+
+ O(a4∗ ).
r2
4GM
(30)
Consequently, the correction to the Newtonian acceleration at a distance r = 0.1 m between the
source mass and the test mass is of the order of 10−2 a∗ ≈ 10−12 m/s2 , which corresponds to the
attonewton-level corrections to the force acting on the milligram test mass.
The sub-attonewton sensitivity of gravitational force sensors has already been achieved in a
number of experiments (see, e.g., [54–56]) and the masses and distances in our estimation are
close to the experimental setup in [56]. This opens a potential avenue for experimental testing
of the quantum gravitational correction to the Newtonian potential arising from the quantum
fluctuations of spin connections described by precanonical quantum gravity.
On the other hand, in the opposite limiting case r ≫ r∗ , we obtain
G4 M 4 G2 M 2
a∗
+O
.
|ẍ| ≈ √ + √
r8
2
2a∗ r4
(31)
It means that
r = 3 m the correction to the asymptotically constant accel√ at a distance of, say, −2
eration a∗ / 2 is of the order of 10 a∗ again. However, the surprising fact is that the theory
predicts a constant force of the order of 100 aN acting on a milligram test particle at distances
r ≫ 1 m from the M = 1 kg source of gravitational force. This prediction, of course, ignores the
feasibility of the controlled environment for a levitating test particle at several meters from the
source mass and even the influence of the involved classical gravitational and electromagnetic
field configurations on the vacuum state of the (static) SCF, whose anti-screening effect is
manifested in the first term in (31). To achieve more precise predictions, it is essential to know
the experimental setup’s configuration and to account for the correlations between fluctuating
spin connections within the SCF at different spatial points.
The effect of the first term in (31) could also be observed in outer space at distances r ≫ 100 m
from a spacecraft of mass 100 kg in the form of an anomalous constant decceleration of a small
body separated from the spacecraft
√ at a small constant velocity.
In the Solar System, r∗ = ( 2GM⊙ /a∗ )1/2 ∼ 0.1 ly, so that the effects of quantum SCF or
MOND affect the motion of objects from the Öpik-Oort cloud. Moreover, a qMOND correction
to the Kepler’s third law [17, 18] shortens the Earth year by
∆T ≈ −
πa2∗ R11/2
,
4GM⊕ (GM⊙ )3/2
7
(32)
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(in c = 1 units), where R is the semi-major axis of the Earth’s orbit. It results in ∆T ≈ −1/2 ms.
Using the same equation for Mars (R♂ ≈ 1.5 AU, M♂ ≈ 0.1M⊕ ), we obtain the correction to the
Martian year of ∆T♂ ≈ −50 ms. These deviations from Newtonian predictions should produce
observable effects in planetary ephemerides (cf. [57, 58]).
From (21) we obtain the orbital velocity around mass M at a distance r from it:
v(r) =
4
G2 M 2 1 2 2
+ a∗ r .
r2
2
(33)
√
The function v(r) has a minimum at r = r∗ , where v∗ = v(r∗ ) = ( 2a∗ GM )1/4 , and in the
vicinity of r∗ it is given by a flat parabola
v(r) ≈ v∗ +
a2∗
(r − r∗ )2 + O((r − r∗ )3 ) .
2v∗3
(34)
Due to the cosmological scale of a∗ , this flat parabola can be approximated as a constant
over a large range of distances around r∗ . Even for a central mass M ∼ 1011 M⊙ with
GM ≈ 0.5 × 10−2 pc, r∗ ≈ 15 kpc, and v∗ ≈ 200 km/s, the rotation curve (34) can be
approximated by a flat rotation curve v(r) ≈ 210 km/s for radial distances from 10 kpc to 30 kpc
with an error margin of 10%. Notwithstanding the point mass approximation of a galaxy, this
result exhibits approximate consistency with the behavior of galaxy rotation curves and their
typical measurement error, achieving this consistency without introducing dark matter.
5. Conclusion
This paper’s derivation of qMOND and MOND suggests that both the cosmological constant (or
its quantum gravitational contribution), which governs the universe’s accelerated expansion, and
Milgrom’s MOND, which describes motion in weak galactic and extra-galactic gravitational fields
without invoking dark matter halos (with a possible exception of galaxy clusters where MOND
still requires a fraction of dark matter compared to the ΛCDM [59, 60]), are fundamentally
connected as manifestations of quantum spin connection foam. This SCF arises from the
precanonical quantization of Einstein’s general relativity, providing a description of quantum
spacetime. Furthermore, our result, resembling Zeldovich’s hypothesis [61], elucidates the
small values of the cosmological constant and Milgromian acceleration by linking them to the
Planckian scale, determined by G and ℏ, and the hadronic scale of κ, a parameter introduced
by precanonical quantization and shown to relate to the mass gap in the pure Yang-Mills sector
of the Standard Model.
The emergence of MOND from quantum general relativity, as an effective non-relativistic
description accounting for non-inertial effects within the mean field of fluctuating spin connection
components (in the static approximation of SCF), suggests that attempts to construct a
“relativistic MOND” through classical general relativity modifications may be misdirected. Since
the original MOND is already derived from a general relativistic quantum theory of gravity, based
on a Dirac-like quantization of the covariant DDW Hamiltonian formalism, exploring dynamics
in SCF beyond non-relativistic and static approximations could be more fruitful.
Moreover, although the existence of unknown constituents in the Universe remains a
possibility, their introduction solely to account for galaxy rotation curves or accelerated
expansion also appears misdirected in light of this paper. Precanonical quantum gravity,
which synthesizes quantum theory and general relativity without additional hypotheses or exotic
matter, already provides a first-principles foundation for both MOND, as an alternative to dark
matter, and the cosmological constant, as the most straightforward dark energy, which already
accounts for a bulk of the observed phenomena (aside from the recent “tensions” and “crises”
in cosmology we hope to address in future work).
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