Emergent Gravity from Quantum Information Flow: A Modified Dirac- Equation Perspective

We present a unified thermodynamic framework in which the flow of quantum information -- described covariantly by the emergent entropy current vector s^μ-- couples to fermionic matter and suggests a possible effective origin for spacetime geometry within a low-energy effective field theory. The Dirac equation is modified as (iℏγ^μ ∂_μ-mc+λγ^μ s_μ)ψ=0, where dimensional analysis identifies λ as a coupling constant of mass dimension [λ]=-2, naturally placing the theory as a low-energy effective field theory. The modified dispersion relation E=-λs⁰ c±c√(p²+m² c² ) shows that uniform entropy flow produces only an overall energy shift, while non-uniform flows induce a complete gauge-like structure in the non-relativistic limit: the spatial entropy current s acts as a vector potential, and its curl generates an effective Zeeman-type field B_eff=(λ/m)∇×s, driving spin precession via ω_entropy=(λ/m)∇×s. Extending the Proca-type dynamics of s^μ to curved spacetime with non-minimal curvature coupling, we quantize the entropy flow field and integrate out its quantum fluctuations. In this framework, the Einstein-Hilbert action emerges as the low-energy effective action, with Newton's constant G determined by the entropy field's vacuum expectation value. The linearized theory yields a massive spin-2 excitation -- the emergent graviton -- with dispersion E²=p²+m_s², where the entropy field mass m_s is constrained by LIGO/Virgo gravitational-wave observations to m_s≲10^-20eV. Using STAR data on Λ hyperon global spin polarization in heavy-ion collisions, we estimate λ∼10^-5-10^-3 GeV^-2, corresponding to an entropy-flow energy scale Λ_EFT∼10^1.5-10^2.5GeV. This framework naturally connects to Jacobson's thermodynamic gravity, Verlinde's entropic gravity, and the generalized second law ∇_μ s^μ≥0, suggesting a possible quantum-information theoretic interpretation of spacetime geometry and offering a concrete, testable realization of gravity as an effective entropic phenomenon within the EFT framework. This framework naturally connects to Jacobson's thermodynamic gravity, Verlinde's entropic gravity, Bianconi's Gravity from Entropy (GfE) theory, and the generalized second law ∇_μs^μ≥0, providing a quantum-information theoretic foundation for the emergence of spacetime and a concrete, testable realization of gravity as an entropic phenomenon.