Experimental Constraints on Deformed Constraint Algebra via Gravity-Mediated Entanglement
Canonical general relativity is governed by first-class constraints:
H(x) ≈ 0, Hᵢ(x) ≈ 0,
with hypersurface deformation algebra
{H(N), H(M)} = D[q^{ij}(N∂ⱼM − M∂ⱼN)].
Quantum consistency requires anomaly freedom:
[Ĥ(N), Ĥ(M)] = iℏ Ḋ[q^{ij}(N∂ⱼM − M∂ⱼN)].
Effective quantum gravity models may preserve closure but deform the structure function:
{H(N), H(M)} = D[Ω q^{ij}(N∂ⱼM − M∂ⱼN)],
with Ω = 1 ε, |ε| ≪ 1. Anomaly corresponds to closure failure; deformation corresponds to ε ≠ 0 but closed algebra.
Gravity-mediated entanglement provides an operational probe.
Two masses m in spatial superpositions acquire branch-dependent phases:
φ_{ab} = −(G m² t)/(ℏ r_{ab}).
Relative phase:
Δφ ≈ (G m² t / ℏ) · (Δr / r²).
Entanglement is generated if Δφ ≠ 0 and non-factorizable.
Under deformation:
Δφ → Δφ(1 ε).
Thus precision measurement of Δφ constrains ε.
For Δφ ≈ 1, required interaction time:
t* = (ℏ / (G m²)) · (r² / Δr).
Representative regimes (Δr = 10⁻⁶ m):
m = 10⁻¹⁴ kg, r = 10⁻⁶ m → t* ≈ 1.6×10⁻² s
m = 10⁻¹⁴ kg, r = 10⁻⁵ m → t* ≈ 1.6 s
m = 10⁻¹³ kg, r = 10⁻⁵ m → t* ≈ 1.6×10⁻² s
Second-scale coherence under cryogenic ultra-high vacuum approaches these regimes.
Deformation affects causal structure: effective propagation speed satisfies
c_eff² ∝ Ω.
Thus:
Ω > 0 → Lorentzian structure preserved
Ω = 0 → ultralocal limit
Ω < 0 → signature change
Experimental bounds on ε therefore constrain both algebraic consistency and causal stability.
Irreversibility must arise via decoherence or boundary conditions if constraint closure holds.
Fundamental collapse or anomaly would imply violation of linear constraint structure.
BMV-type experiments therefore test whether irreversibility is emergent (closure preserved) or fundamental (closure broken).
In Loop Quantum Gravity, holonomy corrections generate Ω-deformation; Δφ measurements bound residual ε at low energy.
In AdS/CFT, geometry emerges from entanglement; gravity-mediated entanglement tests the reciprocal direction (gravity → entanglement), linking information–geometry duality to measurable correlations.
Thus gravity-mediated entanglement provides a single observable (Δφ and entanglement witness) capable of constraining:
• Collapse models
• Deformation parameter ε
• Constraint-algebra anomaly
• Effective causal structure
• Emergence of irreversibility
The central question becomes:
Does macroscopic gravitational interaction preserve linear constraint closure (ε = 0), or produce measurable deformation/anomaly?
Experimental resolution informs whether spacetime geometry is fundamental or emergent from amplitude over geometry.
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