The Fine Structure Constant in the DRUMS Framework
1. Coherent Superfluid Field
In DRUMS, the universe is modeled as a coherent superfluid with a phase field:
\[
\Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)}
\]
The phase gradient defines local velocity:
\[
\mathbf{v} = \frac{\hbar}{m} \nabla \theta
\]
2. Electron Orbital Dynamics
Electrons are treated as excitations of the superfluid, orbiting a proton at radius \(r\). The centripetal acceleration is balanced by Coulomb-like interaction emerging from phase gradients:
\[
m_e v^2 / r = e^2 / (4\pi \varepsilon_0 r^2)
\]
Velocity from phase quantization:
\[
\oint \mathbf{p} \cdot d\mathbf{l} = \oint m_e \mathbf{v} \cdot d\mathbf{l} = n h
\]
3. Bohr Radius Relation
From quantization:
\[
v = \frac{n \hbar}{m_e r}
\]
Substitute into centripetal balance:
\[
\frac{m_e}{r} \left( \frac{n \hbar}{m_e r} \right)^2 = \frac{e^2}{4\pi \varepsilon_0 r^2} \Rightarrow r = \frac{4\pi \varepsilon_0 n^2 \hbar^2}{m_e e^2}
\]
This recovers the Bohr radius \(a_0\) for \(n=1\).
4. Emergence of the Fine Structure Constant
Define fine structure constant:
\[
\alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}
\]
Express orbital velocity in terms of \(c\):
\[
v = \alpha c
\]
Within DRUMS, \(\alpha\) arises naturally from:
Ratio of phase-induced interaction energy \(e^2 / 4\pi \varepsilon_0\)
Planck’s constant \(\hbar\) setting quantum of circulation
Speed of light \(c\) as superfluid phase propagation speed
5. DRUMS Interpretation
The fine structure constant is not arbitrary but a dimensionless ratio of fundamental superfluid parameters:
\[
\alpha = \frac{\text{Phase Interaction Energy}}{\text{Quantum of Circulation} \times \text{Phase Propagation Speed}}
\]
It determines the strength of coupling between charged excitations and the coherent background field.
6. Energy Level Splitting
Fine structure splitting arises from relativistic and spin effects in the DRUMS superfluid, expressed as:
\[
\Delta E_{fs} \sim \alpha^2 m_e c^2 \left( \frac{1}{n^3} \right)
\]
Consistent with observed atomic spectra.
7. Final Interpretation
Within the DRUMS framework, the fine structure constant is a natural consequence of:
Superfluid phase coherence
Quantized circulation of electron excitations
Speed of phase propagation (light speed)
Phase-induced interaction strength (effective charge coupling)
It emerges directly from the dynamics of the coherent medium rather than being an independent empirical parameter.