Step 6: Physical Consequences - Decoherence and Clock Correlations

The redundancy functional makes concrete, testable predictions about how quantum systems behave in gravitational fields:

1. Decoherence Rate

For a quantum system with frequency \(\omega\) and mass \(M\), the redundancy functional predicts a specific decoherence rate:

\[\Gamma = \alpha \omega^2 M\]

where \(\alpha = \kappa^2/\xi^3\) is determined by the redundancy parameters.

Key features:

• Scales as \(\omega^2\) (not \(\omega\)) - higher frequency clocks decohere faster
• Linear in mass \(M\) (not \(M^2\) as in collapse models)
• The coefficient \(\alpha\) can be measured experimentally

2. Spatial Correlations

The CTP relation \(C[\Phi] = \exp(-|\nabla\Phi|^2/\xi^2)\) connects lapse gradients to coherence quality. Through the constraint \(\Xi = C[\Phi]\), this creates correlations between nearby clocks. Two clocks separated by distance \(L\) experience correlated noise with:

\[C(L) \propto \exp(-L/\xi_{\text{eff}})\]

where the effective correlation length is:

\[\xi_{\text{eff}} \sim \frac{c}{\sqrt{8\pi G \rho_{\text{ambient}}}}\]

3. Observable Effects

• Visibility loss in interferometers: \(V = \exp\left[-\frac{1}{2}\omega^2 \int\int C_\Phi(t,t') dt dt'\right]\)
• Clock network correlations: Nearby optical clocks show correlated frequency noise
• Direct measurement of G: Cross-correlations reveal \(G\) without measuring forces

The profound implication:

These effects show that gravity isn't just about forces and curved spacetime - it's fundamentally about the capacity to maintain temporal coherence. Massive objects create lapse gradients that limit how well nearby quantum systems can keep track of time.

Mini-Glossary

Symbol	Name	Meaning	Value / Units	Metaphor
\(\Gamma\)	decoherence rate	rate of quantum coherence loss	1/time	"Quantum memory decay rate"
\(\alpha\)	decoherence coefficient	\(\kappa^2/\xi^3\)	mass⁻¹ time⁻³	"Quantum fragility factor"
\(M\)	mass	mass of quantum system	mass	"Clock weight"
\(C(L)\)	correlation function	correlation between separated clocks	dimensionless	"Clock synchronization strength"
\(\xi_{\text{eff}}\)	effective correlation length	\(c/\sqrt{8\pi G \rho}\)	length	"Correlation reach"
\(V\)	visibility	interferometer fringe visibility	0 to 1	"Quantum pattern clarity"
\(C_\Phi(t,t')\)	lapse correlator	temporal field correlations	dimensionless	"Time noise memory"