Effects of Quantum Spin-Connection Foam in the Solar System, Galaxies, and the Universe

We argue that effects of the quantum spin-connection foam, which describes quantum gravity according to the precanonical quantization of General Relativity, may already be observed in the form of the small cosmological constant and a modification of Newtonian dynamics at small accelerations, manifested in the flat rotation curves of galaxies. We obtain a modification of the Newtonian potential that takes into account the existence of a fundamental small acceleration scale, a∗ = 8πGħκ, where κ is a parameter with the dimensions of inverse spatial volume that appears on dimensional grounds. The connection between κ and the hadronic scale of the mass gap in the pure Yang–Mills sector of the Standard Model leads to an estimated value of a∗ compatible with the Milgromian acceleration scale in MOND. The connection between a2∗ and the cosmological constant leads to a realistic value of the latter. Milgromian MOND, together with a theoretically distinct interpolating function, is derived under the assumption that classical dynamics is modified by the mean-field acceleration calculated from the simplest solution of precanonical quantum gravity in the nonrelativistic approximation. We also indicate that the effects of Newtonian dynamics modified by the spin-connection foam may be observable in the Solar System and even in laboratory experiments.

The paper uses the framework of precanonical quantization of gravity, which is based on the De Donder-Weyl (DDW) Hamiltonian formulation that treats all spacetime dimensions on an equal footing and requires no space+time decomposition. The quantization of fields based on the generalized Poisson-Gerstenhaber algebra of differential forms is called precanonical quantization. It results in a hypercomplex generalization of quantum mechanics to field theory where quantum fields are described in terms of a Clifford-algebra-valued precanonical wave function on the bundle of field variables over spacetime. The standard functional Schrödinger representation in quantum field theory emerges from the precanonical quantization in the case of infinitesimal 1/κ, where κ is an ultraviolet parameter of the dimension of the inverse spatial volume introduced when the dual basis of differential forms υµ is represented by dimensionless elements of Clifford algebra. The study of the spectrum of the DDW Hamiltonian operator of the SU(2) quantum Yang-Mills theory suggests that the scale of physical κ is related to the hadronic scale of the mass gap in the nonabelian gauge theories of the Standard Model.

The precanonical quantization of pure tetrad gravity starts from the Palatini Lagrangian density for general relativity in tetrad variables. The analysis of the primary constraints in the DDW Hamiltonian formulation and the calculation of the corresponding generalized Dirac brackets of forms leads to the representation of the operators of the tetrad components and the operator of the metric tensor. The precanonical analog of the Schrödinger equation reads (iħκ ̸∇ − Ĥ)Ψ = 0, where Ĥ is the operator of the DDW Hamiltonian function and ̸∇ is the Dirac operator in which the curved space-time Dirac matrices render into differential operators because the tetrad components are differential operators. The physical picture of quantum gravity according to the precanonical quantization is what we call the spin connection foam, characterized by the Clifford-algebra-valued 1-point amplitudes Ψ(ω, x) and the 2-point amplitudes that describe the probability distribution and correlations of quantum fluctuations of spin connection at different spacetime points.

For the quantum Minkowski spacetime, the precanonical Schrödinger equation with Λ = 0 assumes a simple form, and the square of this equation decouples different components of Ψ so that it can be assumed to be a scalar. Combining the square of the equation with the requirement of the correspondence of the average of the metric operator with the Minkowski metric on average, the modes of the precanonical wave function on the spin connection bundle satisfy wave equations that imply the precanonical wave function of the Minkowski spacetime has light-like modes propagating on the base of the spin connection bundle and massive modes in the fibers. The range of the massive modes in the space of spin connection coefficients defines an invariant scale of accelerations a∗ = 8πGħκ, at which the classical notion of inertial frames is violated by quantum fluctuations in the spin connection foam. The relation between the acceleration scale a∗ and the square root of the cosmological constant resembles the relation between the Milgromian acceleration scale a0 and the square root of the cosmological constant in MOND.

For the quantum gravitational modification of the Newtonian dynamics (qMOND), the paper restricts to the motion of non-relativistic test particles. The geodesic equation in the non-relativistic approximation reads ẍi + Γi00 = 0, where the components of the Christoffel symbol Γi00 are equal to the components of spin connection ω0i0 to be denoted by ω i. For the gravitational field of a point mass M at the center of coordinates, ω i = GM xi /r3. Assuming this system is immersed into a non-relativistic limit of the quantum Minkowski space, the only fluctuating components of spin connection are ω̃ i with ⟨ω̃ i⟩ = 0 and their statistics is described by the solutions of the 00-component of the wave equations. The ground state solution of this modified Helmholtz equation has the form of the Yukawa potential Ψ(ω̃ i) = √(1/(16π2Għκ ω)) e−ω/(8πGħκ). The relevant expectation values yield ⟨ω̃ i⟩ = 0, ⟨ω̃ i ω̃ i⟩ = (1/2)(8πGħκ)2 = a2∗, and ⟨√(ω̃ i ω̃ i)⟩ = 4πGħκ = a∗.

The movement of a test particle in the gravitational field of the point mass M immersed in the nonrelativistic approximation of the spin connection foam can be described by the equation ẍi + GM xi/r3 + ω̃ i = 0. Since ⟨ω̃ i⟩ = 0, the average of this equation just reproduces the classical equation of motion. However, the nonvanishing variance ⟨ω̃ 2⟩ ≠ 0 leads to non-trivial consequences of the average of the square of the equation. Assuming that the averaging is over quantum fluctuations of spin connections, it is a direct consequence of ⟨ω i⟩ = 0 that ⟨xi ω̃ i/r3⟩ = 0. Therefore, by denoting a = ẍ = √(ẍ2) and ā = √(⟨ω̃ 2⟩), the following modified Newton’s law of universal gravitation for a point mass is obtained, which is called quantum-gravitationally modified Newtonian dynamics or qMOND: a = √(G2M2/r4 + ā2). The right hand side can be considered as the radial component of the negative gradient of a qMOND generalization of the Newtonian gravitational potential of a point mass which takes into account non-relativistic quantum fluctuations of spin connection: Φ(r) = −(GM/r) 2F1(1, 3/4; 5/4; −ā2r4/(G2M2)), where 2F1 is the standard Gauss hypergeometric function. For small r, the Newton’s law a = GM/r2 for the absolute values is reproduced, and for large r, a = ā, which corresponds to the linear potential Φ(r) = ār that emerges here as the “antiscreening” effect of fluctuations of spin connection. This asymptotic behavior also appears in the Cornell potential postulated in the context of the quark confinement problem, in Grumiller’s model of “gravity at large distances” with Λ = 0 and ā identified with the Rindler acceleration, and in spherically symmetric solutions of conformal Weyl gravity.

When classical gravitation dominates over quantum fluctuations of spin connection, i.e. GM/r2 ≫ ā, the correction is a ≈ GM/r2 + ā2r2/(2GM). When quantum fluctuations of spin connection dominate over the classical gravitation, i.e. ā ≫ GM/r2, the result is a ≈ ā + G2M2/(2ār4) − G4M4/(8ā3r8), where the last two terms are similar to the force that can be derived from the “7-3 Lennard-Jones potential”. In both cases, the geometric mean of the gravitational radius of the mass M, rS = 2GM, and the cosmological radius rΛ ∼ ā−1, namely, r∗ = √(rS rΛ), is the border between two different physical regimes.

Due to the omnipresent quantum fluctuations of spin connection, classical accelerations are being measured with respect to the “mean field” acceleration ā. In this reference system, the standard Newton’s law reads a − ā = GM/r2. Taking this equation as a redefinition of the right hand side in terms of the left hand side, which is based on non-relativistic experiments at normal accelerations a ≫ ā, and then extrapolating its validity to arbitrary accelerations, equation (39) without the last O(r−8) term can be rewritten in terms of the acceleration with respect to the mean field background g = a − ā as follows: g2/(2ā) = GM/r2. Therefore, in the regime of small accelerations g, the equation of Milgrom’s Modified Newtonian Dynamics (MOND) in the deep-MOND regime is reproduced: g2/g0 = GM/r2, where the Milgromian acceleration g0 = 2ā. Moreover, applying the same procedure to the qMOND law in equation (35) yields GM/r2 = √(g2 + ā2) − ā. According to MOND, the modified dynamics for arbitrary accelerations has the form GM/r2 = µ(g/g0)g, where µ(x) is an interpolating function such that µ(x → 0) → x and µ(x → ∞) → 1. By comparing with the qMOND law, the theoretical interpolating function is derived from the first principles of precanonical quantum gravity: µ(x) = (1/(2x))(√(4x2 + 1) − 1), with x = g/g0 and g0 = 2ā. This interpolating function satisfies the required asymptotics for x → 0 and x → ∞, and very closely matches the “simple interpolating function” µsimple(x) = x/(1 + x) with the maximal deviation of around 12% near x ≈ 1.3 and less than 2% in the deep-MOND regime, thus making it phenomenologically viable.

The acceleration a∗ and the cosmological constant Λ depend both on the Planck scale of Għ and the unknown scale of the parameter κ. The study of the spectrum of the DDW Hamiltonian of quantum Yang-Mills theory has shown that the order of magnitude of the mass gap ∆m is related to the scale of κ as follows: κ ∼ (∆m)3/(ħ4gs2), where gs is the gauge coupling constant in the classical YM Lagrangian. Then the invariant scale of acceleration a∗ is given by a∗ ∼ 8πG(∆m)3/(ħ3gs2). Identifying the classical coupling constant gs with the QCD running constant at the momentum transfer Q = 0, where the non-perturbative consideration and the experimental data indicate that the infra-red limit of gs2(Q) = 4παs(Q) stabilizes to the value gs2 ≈ 4π2 at Q → 0, and assuming that the mass gap corresponds to the mass of the lowest observable excitation in QCD at ∆m ∼ 10−1 GeV, the numerical value a∗ ∼ 10−27 m−1 is obtained. This is consistent with the value of the Milgromian acceleration g0 = 2a∗ = 0.12 nm/s2 or ≈ 10−27 m−1 in geometrized units, although the error of this estimation is several orders of magnitude. The same approximate consistency up to several orders of magnitude with the observed cosmological constant Λ ≈ 10−52 m−2 follows from the relation between a∗ and Λ with λ = 3 for the Weyl ordered spin connection operator.

In the Solar system with GM⊙ ≈ 1.5 km (in geometrized units), the correction in (38) will be 1% of the gravitational acceleration from the Sun at the distance rM = (0.01 × 2G2M⊙2/ā2)1/4 ∼ 3 × 103 au, i.e. the deviation from the Newtonian dynamics may be detectable for objects from the inner edge of the Öpik-Oort cloud. The deviation at the current location of Voyager 1 spacecraft (at 166 au from the Sun) is about 10−4%. The effects of the spin connection foam in the vicinity of the Earth orbit lead to the corrections of the order of 10−9% to the duration of the Earth year T and the locations of the Lagrange points, based on the following correction to Kepler’s third law: (2π/T)2 = G(M1 + M2)/R3 + ā2R(M1 + M2)/(2GM1M2) + O(ā4), where R is the semi-major axis of the Earth orbit, and M1 and M2 are the masses of the Sun and Earth, respectively. Moreover, for the mass M = 1 kg with GM ∼ 10−27 m the condition GM/r2 ≫ ā is satisfied for r ≪ 1 m. At the distance r = 10 cm from the mass M the correction to the Newtonian acceleration in (38) is ∼ 10−2g0 ≈ 10−12 m/s2. When acting on the test mass of 1 mg it leads to the force of 10−18 N. As the sub-attonewton sensitivity of force sensors is already reachable, and the masses and distances in the estimation here are compatible with the experimental setup which has already achieved the attonewton sensitivity, the correction in (38) can already be experimentally tested.

For the movement around point masses of the order of the mass M of galaxies, the formula for the orbital velocities v such that v2/r = a is derived: v(r) = (G2M2/r2 + ā2r2)1/4. This function has the minimum at rm = √(GM/ā), where the velocity vm:= v(rm) = (2āGM)1/4. Near this point the rotation curve v(r) is approximated by v(r) = (2āGM)1/4 + ā2/(3vm)(r − rm)2 + O((r − rm)3). This is a very flat parabola around rm since the scale of ā is cosmological and the scale of GM for most galaxies is below 1 ly. The first term corresponds to the velocity of flat rotation curves according to MOND and the phenomenological Baryonic Tully-Fisher relation. For example, for a galaxy of the baryonic mass M ∼ 1011M⊙, GM ≈ 1.5 × 10−2 ly, rm ≈ 5 × 104 ly, and vm ≈ 0.65 × 10−3 (i.e. 195 km/s). With the error margin of ±10%, the rotation velocity v(r) in (52) can be approximated by a flat rotation curve v(r) ≈ 210 km/s in the region between 30 kly and 90 kly, consistent with the observed flat rotation curves of such galaxies (like M31 or Milky Way) considering that the behavior of the rotation curves at r < 30 kly is predominantly determined by the mass distribution in the galactic disk, which the approximation of the central point mass completely ignores. At larger distances r > 100 kly (31 kpc), equation (52) predicts the linear growth of the rotation velocity at the scale of hundreds of kly and then, at even larger distances, the √(ār) growth, as dictated by the asymptotic linear behavior of the potential Φ(r). The hierarchy of scales of different regimes which follows from the formula for the rotation curve v(r) around an isolated point mass M and its potential Φ(r) suggests that it can play a key role in the dynamics of galactic clusters, superclusters, voids, and larger scale structures formation. Because the potential Φ(r) ensures deeper and longer range potential wells, it can account for even more of the missing mass than MOND at those scales, as a MOND-like description of galaxy clusters requires, and it can also explain a faster clumping of matter than expected in the standard ΛCDM and even MOND due to it allowing for the earlier mass fluctuations to attract matter from the more distant regions of space due to the asymptotic linear grown of the potential Φ(r). Moreover, it may be interesting to check via numerical simulations if the one-dimensional nature of the linear potential up to a Gly scale, where the effects of the cosmological background will take over, could actually promote the formation of quasi one-dimensional large scale structures like the galaxy filaments in the cosmic web. It should be noted that the linear and the square root growth of the rotation velocity at the distances above 100 kly from the gravitating center of 1011 solar masses, which follows from equation (52), seems to contradict observations of flat rotation curves at the Mly scales, while those are described by MOND by design. However, the Milgromian MOND is obtained by accounting for the non-inertial effects of the mean field of fluctuating accelerations in the quantum spin connection foam. This is why the observable effects of the interactions of very distant objects via the potential in (36) can be described by MOND using the accelerations g, g0 and an interpolating function like the one derived in (46). In more realistic situations, however, the interpolating function should also take into account both the multi-body interactions (mass distributions) and correlations between macroscopic massive bodies immersed in the quantum spin connection foam (as a field of fluctuating accelerations), in accordance with the equivalence principle and the two-point solutions of the precanonical Schrödinger equation.

Based on the paper, here are specific improvements for AI systems:

1. Modified gravitational dynamics engine: Implement the qMOND law a = √(G2M2/r4 + ā2) and the derived interpolating function μ(x) = (1/(2x))(√(4x2+1) − 1) as a drop-in replacement for Newtonian gravity in N-body simulations, enabling accurate modeling of galactic rotation curves, galaxy cluster dynamics, and large-scale structure formation without dark matter assumptions.

2. Multi-scale physics integrator: Build an AI system that automatically switches between regimes—Newtonian (GM/r2 ≫ ā), qMOND transition (r* = √(r S r Λ)), and deep-MOND (g2/g0 = GM/r2)—based on the acceleration scale ā ≈ 10−27 m−1, allowing seamless simulation from laboratory scales (1 kg masses, 10 cm distances) to galactic scales (1011 M, 100 kly).

3. Predictive anomaly detector for solar system tests: Train a model to predict deviations from Keplerian motion (e.g., 1% correction at 3×103 AU from the Sun, 10−4% at Voyager 1's 166 AU, 10−9% at Earth's orbit) and flag observations from the Oort cloud, Kuiper belt objects, or spacecraft telemetry that match the predicted correction term ā2r2/(2GM).

4. Laboratory experiment optimizer: Use the paper's force prediction (10−18 N on a 1 mg test mass at 10 cm from a 1 kg source) to design optimal torsion balance or cantilever experiments, recommending mass configurations, distances, and measurement durations to maximize signal-to-noise for detecting the spin-connection foam effect.

5. Rotation curve synthesizer: Generate synthetic galactic rotation curves using v(r) = (G2M2/r2 + ā2r2)(1/4), including the flat region (v ≈ 210 km/s for M ≈ 1011 M between 30–90 kly), the parabolic minimum at r m = √(GM/ā), and the linear/√(ā r) growth at >100 kly, for training AI models to classify observed galaxies or test dark matter vs. modified gravity hypotheses.

6. Cosmological constant calibrator: Implement the relation a*2 = 3Λ (with λ=3 for Weyl ordering) to allow AI systems to cross-validate independent measurements of the cosmological constant (Λ ≈ 10−52 m−2) against local acceleration-scale observations, improving consistency checks across cosmological and galactic datasets.

7. Quantum-gravity-informed field theory simulator: Incorporate the precanonical quantization framework (Clifford-algebra-valued wave functions, DDW Hamiltonian, spin-connection foam statistics with Yukawa-type ground state) into AI-driven simulations of quantum gravity effects, enabling predictions for spin-connection fluctuations in curved spacetime that could be tested against future high-precision experiments.

8. Hierarchical structure formation predictor: Use the deeper/longer-range potential wells from the modified potential Φ(r) = −(GM/r)2F1(1, 3/4; 5/4; −ā2r4/(G2M2)) to train AI models that predict enhanced clumping of matter, earlier structure formation, and the promotion of quasi-1D filamentary cosmic web structures, which can be validated against galaxy survey data (e.g., SDSS, DESI).