A Microscopic Model for the Dependence of Vacuum Energy on Matter Density: Toward a Quantum-Field-Theoretic Interpretation

This paper develops a quantum-field-theoretic framework in which the effective vacuum energy density becomes a function of the local matter density, ρ_vac = ρ_vac(ρ_m), arising naturally from quantum fields propagating in curved spacetime in the presence of matter sources.

Using the Schwinger–DeWitt heat-kernel expansion at one-loop order, we show that matter-induced curvature modifies the spectrum of vacuum fluctuations and thereby the renormalized vacuum energy density. Two physically distinct channels are identified and carefully distinguished:

• Channel (a) — rigorously derived from the curvature-dependent effective action via the a₁ heat-kernel coefficient — yields a coupling |α| ~ 10⁻³⁴ (negligible).

• Channel (b) — a physically motivated conjecture involving direct vacuum polarization at gauge/Yukawa coupling strength — suggests |α| ~ 10⁻² (potentially observable).

We prove that the leading matter-dependent correction is linear in ρ_m under explicit assumptions (weak curvature, slowly varying matter distributions, one-loop order, self-consistency), with error terms from quadratic curvature invariants and gradient contributions estimated separately. The sign of the coupling is not uniquely determined and both possibilities (vacuum suppression and vacuum enhancement by matter) are explored.

The paper establishes structural connections to the running vacuum model of Solà Peracaula and collaborators, in which ρ_vac evolves as a function of the Hubble rate, and identifies a structural analogy — clearly marked as the present author's interpretive observation — with Brown's Quantum-Kinetic Dark Energy (QKDE) framework (Int. J. Mod. Phys. D 35, 2650006, 2026), in which a curvature-motivated kinetic normalization produces H²-dependent corrections to the dark-energy sector.

The manuscript has undergone peer review. The reviewer's principal criticism — that the observationally relevant estimate (|α| ~ 10⁻²) rests on a conjectured channel not derived from the heat-kernel formalism — has been addressed by restructuring the paper to clearly distinguish rigorous results from conjectures. The full reviewer report and point-by-point responses are included in the manuscript.

This is Paper #3 in the research program "What If the Vacuum Gravitates Locally? Separating Cosmic Expansion from Quantum Vacuum Energy." It provides the microscopic QFT foundation for the phenomenological matter–vacuum coupling introduced in Paper #4 and connects to the separation thesis established in Paper #1a.

Recommended for submission to Foundations of Physics.

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Keywords: vacuum energy, matter density, cosmological constant problem, effective action, curved spacetime, vacuum polarization, running vacuum model, Schwinger–DeWitt expansion, dark energy, effective field theory, QKDE, heat-kernel expansion

License: CC BY 4.0

Related identifiers:
• Paper #1a: "Theoretical Separation of Quantum Vacuum Energy from the Cosmological Constant" (companion review paper)
• D. Brown, "Quantum-Kinetic Dark Energy (QKDE)," Int. J. Mod. Phys. D 35(4), 2650006 (2026), DOI: 10.1142/S0218271826500069

Boris Kriger
Information Physics Institute, Gosport · Institute of Integrative and Interdisciplinary Research, Toronto
ORCID: 0009-0001-0034-2903