Hogan (2020) proposed that the observed cosmological constant emerges from the coherent quantum geometry of the Standard Model vacuum: gravitational 'drag' from fluctuating transverse momentum flows on causal-diamond surfaces produces rotational fluctuations of the inertial frame with variance ⟨ω2⟩∼G2c−4⟨pAB2/xAB4⟩CD\langle\omega^2\rangle\sim G^2c^{-4}\langle p_{AB}^2/x_{AB}^4\rangle_{CD}, and the hypothesis Λ=3⟨ω2⟩\Lambda=3\langle\omega^2\rangle - with the surviving contribution dominated by gluonic correlations at the quantum chromodynamics (QCD) scale - reproduces the observed cosmic acceleration within a rough scaling estimate (an effective mass mQ≈60m_Q\approx 60 MeV, close to the pion mass). The decisive correlator has never been computed. This problem asks for a controlled, nonperturbative evaluation of that causal-diamond QCD vacuum correlator (magnitude and sign), a derivation or refutation of the identification Λ=3⟨ω2⟩\Lambda=3\langle\omega^2\rangle from a coherent-geometry coarse-graining, and a quantitative error budget deciding whether the apparent QCD-scale numerical coincidence survives precise modeling. The only direct follow-up (Mackewicz and Hogan 2024) refines idealized classical gluonic bubble and string models and explicitly leaves the exact nonlinear computation of the QCD vacuum correlations open.

Background

The cosmological constant problem is the long-standing mismatch between quantum field theory and observed cosmic acceleration: summing the zero-point fluctuations of quantum fields up to a cutoff at the Planck scale predicts a vacuum energy density whose gravitational effect is a cosmological constant about 122 orders of magnitude larger than the value inferred from cosmology (Weinberg, Rev. Mod. Phys. 61, 1 (1989)). Hogan (2020) proposed a way out based on 'coherent quantum gravity': the hypothesis that quantum states of spacetime geometry, like quantum states of particles, remain coherent on light cones of any size, so the gravitational effect of virtual vacuum fluctuations must be averaged coherently over extended causal regions rather than summed mode by mode on a classical background. A causal diamond is the spacetime region between the past and future light cones of a segment of a timelike world line; its invariant spherical cross-sections are the surfaces over which the proposal averages. The classical building block is the Aichelburg-Sexl gravitational shock wave, the exact general-relativistic solution for a massless point particle of momentum pp, which produces a conical displacement δu=4Gp/c3\delta u = 4Gp/c^3 of the light cone - a physical 'drag' of causal structure by moving energy. In this picture, counterpropagating virtual particle pairs with transverse momentum difference pABp_{AB} passing at spacelike separation xABx_{AB} twist the local inertial frame with angular velocity ω∼GpAB/(c2xAB2)\omega\sim Gp_{AB}/(c^2x_{AB}^2); the mean twist vanishes, but the variance does not: ⟨ω2⟩∼G2c−4 ⟨pAB2/xAB4⟩CD,\langle\omega^2\rangle \sim G^2c^{-4}\,\langle p_{AB}^2/x_{AB}^4\rangle_{CD}, where the expectation value is taken over invariant spherical causal-diamond surfaces. The proposal then hypothesizes that these fluctuating twists act, in the time-averaged emergent geometry, like centrifugal acceleration, ⟨r¨/r⟩=⟨ω2⟩\langle\ddot r/r\rangle=\langle\omega^2\rangle, which is gravitationally equivalent to a cosmological constant Λ=3⟨ω2⟩\Lambda=3\langle\omega^2\rangle. Approximately pointlike virtual particles average away, so the surviving contribution comes from the scale of quantum chromodynamics (QCD, the theory of the strong interaction between quarks and gluons) - the energy scale, of order 0.1 GeV, at which the strong coupling becomes large and sets hadron masses - where vacuum fluctuations resemble spatially extended gluonic strings or condensates whose stress-energy is dominated by tension (negative pressure), and whose gravity is repulsive. The scaling estimate ⟨ω2⟩∼tP−2(mQ/mP)6\langle\omega^2\rangle\sim t_P^{-2}(m_Q/m_P)^6 (with tPt_P and mPm_P the Planck time and mass) reproduces the observed Λ/3≈1.0×10−61 tP−1\sqrt{\Lambda/3}\approx 1.0\times10^{-61}\,t_P^{-1} for an effective mass mQ∼60m_Q\sim 60 MeV, remarkably close to the pion mass mπ0=135m_\pi^0 = 135 MeV. The 2020 paper is, however, an explicit sketch: it introduces the identification with 'let us suppose', sets aside geometrical factors of order unity, and notes that the decisive quantity 'depends on nonlocal spacelike correlations of QCD fields, which can in principle be calculated on a lattice, as nucleon masses are'. No microscopic theory of the coherent quantum geometry - the quantum states of geometry, their coherent averaging rules, and how extended causal structure 'collapses' - is provided.

Problem Statement

Within Hogan's coherent-quantum-geometry framework, determine whether the spacetime-averaged gravitational effect of the Standard Model vacuum reproduces the observed cosmological constant. This requires, as jointly supporting parts of one objective: (1) a precise, computable definition of the causal-diamond correlator ⟨pAB2/xAB4⟩CD\langle p_{AB}^2/x_{AB}^4\rangle_{CD} - the variance of transverse momentum flows on invariant causal-diamond surfaces, with explicitly stated coherent-averaging rules - together with a controlled nonperturbative evaluation of it for the QCD vacuum (for example, a computation in lattice QCD - quantum chromodynamics formulated as a discretized path integral on a spacetime lattice and evaluated by numerical simulation - adapted to the required nonlocal spacelike correlations), establishing both the magnitude and the sign of the induced ⟨ω2⟩\langle\omega^2\rangle; (2) a justification or refutation of the identification Λ=3⟨ω2⟩\Lambda=3\langle\omega^2\rangle, i.e. a derivation (or disproof) from an explicit microscopic or coarse-graining model of coherent quantum geometry that time-averaged rotational-variance fluctuations of the inertial frame produce a mean outward radial acceleration ⟨r¨/r⟩=⟨ω2⟩\langle\ddot r/r\rangle=\langle\omega^2\rangle in the emergent geometry; and (3) a quantitative assessment of whether the apparent numerical coincidence with the QCD scale survives precise modeling - the geometrical numerical prefactors, the detailed structure of QCD vacuum excitations (string tension, correlation lengths), and the contributions of the other Standard-Model sectors. A complete answer decides, with a controlled error budget, whether the observed Λ\Lambda (with Λ/3≈1.0×10−61 tP−1\sqrt{\Lambda/3}\approx 1.0\times10^{-61}\,t_P^{-1}) is or is not accounted for by QCD vacuum correlations in a coherent quantum geometry.

References

  1. C. Hogan, 'Cosmological Constant in Coherent Quantum Gravity', Int. J. Mod. Phys. D 29, 2042004 (2020), DOI:10.1142/S0218271820420043, arXiv:2003.14255, https://arxiv.org/abs/2003.14255
  2. K. Mackewicz and C. Hogan, 'Gravity of gluonic fluctuations and the value of the cosmological constant', Class. Quantum Grav. 41, 015033 (2024), DOI:10.1088/1361-6382/ad13c2, arXiv:2305.04864, https://arxiv.org/abs/2305.04864
  3. K. Mackewicz and C. Hogan, 'Gravity of two photon decay and its quantum coherence', Class. Quantum Grav. 39, 075015 (2022), DOI:10.1088/1361-6382/ac5377, arXiv:2108.03264, https://arxiv.org/abs/2108.03264
  4. C. Hogan, 'Value of the Cosmological Constant in Emergent Quantum Gravity', arXiv:1804.00070 (2018), https://arxiv.org/abs/1804.00070
  5. J. W. Richardson, O. Kwon, H. R. Gustafson, C. Hogan, B. L. Kamai, L. P. McCuller, S. S. Meyer, C. Stoughton, R. Tomlin, and R. Weiss (Holometer Collaboration), 'Interferometric Constraints on Spacelike Coherent Rotational Fluctuations', Phys. Rev. Lett. 126, 241301 (2021), DOI:10.1103/PhysRevLett.126.241301, arXiv:2012.06939, https://arxiv.org/abs/2012.06939
  6. S. K. Kundu, A. Pradhan, and C. Rosenzweig, 'Does the vacuum gravitate on microscopic scales? Rydberg atoms indicate probably not', Phys. Rev. D 108, 043505 (2023), DOI:10.1103/PhysRevD.108.043505, arXiv:2208.14192, https://arxiv.org/abs/2208.14192
  7. C. Hogan, 'Cosmic shadows of causation', arXiv:2403.19870 (2024), https://arxiv.org/abs/2403.19870
  8. Kristian Mackewicz, 'Distributional Stress-Energy in General Relativity and Its Implications for Quantum Gravity', PhD thesis, University of Chicago (2024), https://inspirehep.net/literature/2877940
  9. S. Weinberg, 'The Cosmological Constant Problem', Rev. Mod. Phys. 61, 1-23 (1989), DOI:10.1103/RevModPhys.61.1, https://doi.org/10.1103/RevModPhys.61.1