Quantum Entanglement in the DRUMS Framework
1. Global Superfluid Field
In DRUMS, all physical systems are excitations of a single coherent field:
\[
\Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)}
\]
This field is globally continuous, implying that spatially separated systems remain embedded in the same phase structure.
2. Multi-Particle State as Shared Phase Structure
Two particles are not independent objects but coupled excitations of the same field. A joint state is represented as:
\[
\Psi_{AB} = \Psi(\mathbf{x}_A, \mathbf{x}_B, t)
\]
Non-factorizability corresponds to entanglement:
\[
\Psi_{AB} \neq \Psi_A(\mathbf{x}_A) \Psi_B(\mathbf{x}_B)
\]
3. Phase Correlation Constraint
Entanglement arises from a shared phase constraint:
\[
\theta(\mathbf{x}_A, t) - \theta(\mathbf{x}_B, t) = \Delta \theta_{AB} = \text{constant}
\]
This enforces correlated outcomes regardless of spatial separation.
4. Measurement as Local Phase Projection
A measurement corresponds to a local constraint on the field:
\[
\theta(\mathbf{x}_A,t) \rightarrow \theta_A^{(meas)}
\]
Because the phase field is continuous, this imposes a global adjustment:
\[
\theta(\mathbf{x},t) \rightarrow \theta(\mathbf{x},t) + \delta \theta(\mathbf{x})
\]
5. Correlation Emergence
The shared constraint yields correlated observables:
\[
\langle A B \rangle = \int d\lambda \, \rho(\lambda) A(\lambda) B(\lambda)
\]
In DRUMS, \(\lambda\) corresponds to hidden phase configuration variables.
6. Bell-Type Correlations
Measurement outcomes depend on local phase projections:
\[
A = \text{sign}[\cos(\theta_A - \alpha)]
\]
\[
B = \text{sign}[\cos(\theta_B - \beta)]
\]
With shared phase constraint, correlation becomes:
\[
E(\alpha,\beta) = -\cos(\alpha - \beta)
\]
This reproduces quantum mechanical predictions.
7. No Signal Propagation
No superluminal signaling occurs because:
- The phase field is pre-existing and global
- Measurement reveals correlations rather than transmitting information
Mathematically:
\[
\frac{\partial \rho_B}{\partial t} \bigg|_{A} = 0
\]
8. Decoherence
Interaction with environment randomizes phase:
\[
\theta \rightarrow \theta + \delta \theta_{env}
\]
This destroys coherent phase relationships:
\[
\langle e^{i(\theta_A - \theta_B)} \rangle \rightarrow 0
\]
9. Physical Interpretation
Entanglement is not a mysterious connection but a consequence of:
- Global phase continuity
- Shared origin in a single field
- Constraints imposed by topology and coherence
10. Final Interpretation
Within the DRUMS framework, entanglement arises naturally as:
- A manifestation of a single coherent superfluid field
- A constraint on relative phase between regions
- A global property rather than a transmitted interaction
Thus, quantum correlations reflect underlying field coherence rather than nonlocal signaling.