London Equations from HVBK
Derivation: London Equations from HVBK
This derivation addresses open derivation #1: starting from the HVBK two-fluid mutual friction equations (Two Fluids → Quantum Potential), derive the two London equations with the Cooper pair condensate as the superfluid component. The derivation proceeds in three phases: the textbook target (Phase A), the substrate derivation (Phase B), and the mass consistency check (Phase C).
Target equations:
\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m_e}\,\mathbf{E} \qquad\text{(1st London equation)}
\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m_e}\,\mathbf{B} \qquad\text{(2nd London equation)}
Success criterion: Both equations emerge from the HVBK mutual friction equations with the substrate identifications, the B \to 0 mechanism is physically motivated by the pair vortex topology, and the London penetration depth \lambda_L uses m_e (not m_\text{eff}) consistently with the mass relation m_\text{eff} \cdot \alpha_\text{mf} = m_e (Constraint C2).
Epistemological status: This is primarily a consistency demonstration — showing that the HVBK framework reproduces the London equations with physically motivated substrate identifications. The genuinely new substrate content is the topological mechanism for B = 0 and the connection of the Meissner effect to the reactive mutual friction channel. One potential distinguishing prediction emerges in Phase C.
Phase A: The Standard London Derivation (Target)
The standard derivation establishes the target and identifies exactly where “no dissipation” enters, so Phase B knows what to provide.
A1. Equation of motion for a charged superfluid. The superfluid component of a charged fluid obeys:
m_e \frac{\partial \mathbf{v}_s}{\partial t} = e\,\mathbf{E} + e\,\mathbf{v}_s \times \mathbf{B} - \nabla \mu_s
where \mathbf{v}_s is the superfluid velocity and \mu_s is the chemical potential. The key assumption: no friction term. The superfluid accelerates freely under applied fields.
A2. Supercurrent identification. The supercurrent density is \mathbf{J}_s = -n_s e\,\mathbf{v}_s, where n_s is the superfluid number density (superconducting electrons per volume).
A3. First London equation. Substituting \mathbf{v}_s = -\mathbf{J}_s/(n_s e) into A1, neglecting \mathbf{v}_s \times \mathbf{B} (small for non-relativistic electrons) and setting \nabla\mu_s = 0 (spatially uniform condensate):
\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m_e}\,\mathbf{E}
This says the pair condensate accelerates frictionlessly under an applied field. What Phase B must provide: a physical mechanism for why the friction term vanishes for paired electrons but not for unpaired ones.
A4. Second London equation. The canonical momentum of a Cooper pair (charge q = 2e, mass 2m_e) is:
2m_e\,\mathbf{v}_s = \hbar\nabla\theta - \frac{2e}{c}\,\mathbf{A}
The condensate is irrotational away from vortex cores: \nabla \times (\hbar\nabla\theta) = 0. Taking the curl:
\nabla \times \mathbf{v}_s = -\frac{e}{m_e c}\,\mathbf{B}
Substituting \mathbf{J}_s = -n_s e\,\mathbf{v}_s:
\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m_e}\,\mathbf{B}
Combined with Ampère’s law, this gives exponential field decay \mathbf{B} \propto e^{-x/\lambda_L} with the London penetration depth:
\lambda_L = \sqrt{\frac{m_e}{\mu_0 n_s e^2}}
What Phase B must provide: (i) the irrotationality of the condensate from the HVBK framework, and (ii) the canonical momentum coupling to \mathbf{A} from the substrate’s electromagnetic identification.
A5. Assumptions requiring substrate justification:
| Assumption | Standard justification | What Phase B must provide |
|---|---|---|
| No friction on condensate | BCS energy gap prevents scattering | Pair vortex topology suppresses B term |
| \mathbf{v}_s couples to \mathbf{A} | Minimal coupling postulate | EM field = co-rotating substrate flow |
| Condensate irrotational | Macroscopic quantum coherence | Pair condensate has no internal vorticity |
| Mass = m_e in \lambda_L | Inertial mass of electron | EM sector sees m_e, not m_\text{eff} |
Phase B: HVBK → London (The Substrate Derivation)
B1. Starting point: HVBK two-fluid equations. The HVBK equations for two interpenetrating fluids with mutual friction (Two Fluids → Quantum Potential):
Superfluid component: \rho_s \left[\frac{\partial \mathbf{v}_s}{\partial t} + (\mathbf{v}_s \cdot \nabla)\mathbf{v}_s\right] = -\rho_s\,\nabla\mu_s + \mathbf{F}_{ns}
Normal component: \rho_n \left[\frac{\partial \mathbf{v}_n}{\partial t} + (\mathbf{v}_n \cdot \nabla)\mathbf{v}_n\right] = -\nabla P + \eta\nabla^2\mathbf{v}_n - \mathbf{F}_{ns}
Mutual friction force (Weinberg Angle): \mathbf{F}_{ns} = \frac{B\,\rho_n\,\rho_s}{2\rho}\;\hat{\mathbf{s}} \times [\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s)] + \frac{B'\,\rho_n\,\rho_s}{2\rho}\;\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s)
where B and B' are the dimensionless dissipative and reactive mutual friction coefficients, \hat{\mathbf{s}} is the unit vector along the vortex line, and the vortex line velocity \mathbf{v}_L is dropped for clarity (restorable for full treatment).
B2. Substrate identifications for the superconductor. The HVBK quantities map to the superconducting state:
| HVBK quantity | Substrate identification | Notes |
|---|---|---|
| Superfluid (\rho_s, \mathbf{v}_s) | Cooper pair condensate | Co-rotating channels carrying zero-resistance current |
| Normal fluid (\rho_n, \mathbf{v}_n) | Unpaired electrons + phonons | Dissipative flow |
| B (dissipative) | Energy transfer across pair boundary | Maps to g' (U(1)_Y) in electroweak sector |
| B' (reactive/Hall) | Chirality rotation, no energy transfer | Maps to g (SU(2)_L) in electroweak sector |
| \hat{\mathbf{s}} | Axis of each pair vortex | Direction along the shared counter-rotating seam |
| \rho_s / \rho | Superfluid fraction = n_s / n | Fraction of electrons in condensate |
The B \leftrightarrow g' and B' \leftrightarrow g identifications are inherited directly from the electroweak mapping (Weinberg Angle): the dissipative HVBK channel transfers energy across the fermion boundary (hypercharge coupling), while the reactive channel redirects flow without energy transfer (isospin coupling). The same decomposition applies at the Cooper pair boundary because the pair vortex is the same class of counter-rotating structure as the single-electron boundary — same mechanism, different energy scale.
B3. The key step: B \to 0 for the pair condensate. This is where the substrate framework contributes content beyond the standard treatment.
Standard BCS argument (for reference): At T \ll T_c, all electrons near the Fermi surface are paired. Scattering a paired electron requires energy \geq 2\Delta to break the pair. Below T_c, no thermal excitations have this energy, so the dissipative channel is suppressed exponentially: B \propto \exp(-\Delta/k_B T).
Substrate argument: The Cooper pair’s shared counter-rotating vortex is a topologically protected structure — the same class of boundary protection that makes nucleon binding robust (The Metal Ion Core). The Compton breathing mechanics (Electron) make this concrete:
The anti-phase energy budget. Each electron oscillates between a contracted phase (effective quantum at r_\text{eff} \approx 150 fm, peak kinetic energy \frac{1}{2}m_\text{eff}v_\text{rot,inner}^2 = m_e c^2) and an expanded phase (coherence envelope at \xi \approx 100\;\mum, peak boundary energy = m_e c^2), shuttling its full rest energy between these reservoirs every Compton period T_c = 8.1 \times 10^{-21} s. In the Cooper pair, the two electrons are \pi out of phase: when electron A is contracted (intense rotation at 0.776c, pulling substrate inward), electron B is expanded (diffuse boundary ripple, pushing substrate outward). The flow gradient between “tight 150 fm” and “diffuse 100 μm” — spanning 9 orders of magnitude in spatial scale — is what creates the shear zone that self-organizes into the shared counter-rotating vortex. This is the promenading-pair mechanism from pilot-wave hydrodynamics (Bush & Oza), now with explicit scales.
Neutral flow system. At every instant, one electron pulls substrate inward while the other pushes outward. The net disturbance to the surrounding medium averages to zero — the pair is invisible to the lattice’s scattering centers. This is not a cancellation of charges (both carry -e) but a cancellation of substrate flow perturbations. A single electron’s Compton breathing creates ripples that scatter off channel defects; the anti-phase pair’s breathing creates no net ripple for defects to scatter off.
Why B = 0 — topological protection via boundary parity. Dissipative mutual friction (B term) requires energy transfer between the co-rotating and counter-rotating layers at the pair boundary. But the pair vortex is the boundary — it is self-consistently maintained by the anti-phase relationship. Disrupting it requires breaking the anti-phase lock, which costs energy 2\Delta. The boundary parity argument (Spin-Statistics) provides the deeper topological underpinning: the Cooper pair has even boundary parity (two odd-parity fermions merge via anti-phase breathing into an even-parity boson). Even-parity composites have no net chirality mismatch with the background substrate. A dissipative interaction (B term) requires a chirality mismatch to transfer energy; even-parity objects present no mismatch. This is why the pair is structurally transparent to the dissipative channel.
Why B' \neq 0 — reactive channel remains active. Reactive mutual friction (B' term) redirects flow without energy transfer — it operates via the Magnus-like transverse force on the vortex, which does not require breaking the pair’s chirality balance. This channel remains active and is what generates the screening response.
The mathematical consequence. With B = 0, the mutual friction force reduces to the purely reactive term:
\mathbf{F}_{ns}\big|_{B=0} = \frac{B'\,\rho_n\,\rho_s}{2\rho}\;\hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s)
This is a Hall-type force. It does no work on the superfluid: \mathbf{F}_{ns} \cdot \mathbf{v}_s = 0 (since \hat{\mathbf{s}} \times (\mathbf{v}_n - \mathbf{v}_s) is perpendicular to (\mathbf{v}_n - \mathbf{v}_s), and the force’s projection onto \mathbf{v}_s vanishes for any flow geometry where \hat{\mathbf{s}} is along the vortex axis). The superfluid equation of motion becomes:
\rho_s \frac{\partial \mathbf{v}_s}{\partial t} = -\rho_s\,\nabla\mu_s + [\text{reactive (Hall) force, does no work}]
The reactive force redirects flow but does not resist acceleration — the condensate accelerates frictionlessly.
B4. First London equation from HVBK. The substrate identifies the electromagnetic field with co-rotating substrate flow (The Photon as Modon). For a charged superfluid, the Euler equation gains the Lorentz force:
\rho_s \frac{\partial \mathbf{v}_s}{\partial t} = n_s e\,\mathbf{E} + n_s e\,\mathbf{v}_s \times \mathbf{B}_\text{field} - \rho_s\,\nabla\mu_s + \mathbf{F}_{ns}^{(B')}
With B = 0 (Phase B3), neglecting the non-relativistic \mathbf{v}_s \times \mathbf{B}_\text{field} and assuming spatially uniform chemical potential (\nabla\mu_s = 0), dividing through by n_s (number density) and noting \rho_s/n_s = m_e (mass per superconducting electron):
m_e \frac{\partial \mathbf{v}_s}{\partial t} = e\,\mathbf{E}
Using \mathbf{J}_s = -n_s e\,\mathbf{v}_s:
\boxed{\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m_e}\,\mathbf{E}} \qquad\checkmark\;\text{1st London equation}
The HVBK machinery delivers this through a specific physical chain: the pair vortex topology suppresses B, removing dissipative friction; the remaining reactive force B' does no work; therefore the condensate accelerates freely under the applied field, exactly as the first London equation requires.
B5. Second London equation from HVBK. The condensate’s macroscopic wavefunction \psi = \sqrt{n_s}\,e^{i\theta} has velocity field:
\mathbf{v}_s = \frac{\hbar}{2m_e}\nabla\theta
(factor of 2 for Cooper pair mass 2m_e). Away from vortex cores, \nabla \times \nabla\theta = 0, so:
\nabla \times \mathbf{v}_s = 0
In the substrate picture, this irrotationality has a physical origin that connects to the universal boundary-matching pattern (Hydrogen Atom). The same Bessel matching — oscillatory interior, decaying exterior, discrete eigenvalues from joining at the boundary — operates at three scales: modon speeds (l = 0, The Photon as Modon), hydrogen orbitals ($l = $ integer), and spin (l = 1/2, Spin-Statistics). For the Cooper pair condensate, the standing wave condition 2\pi r = n\lambda_B that quantizes hydrogen orbitals becomes the macroscopic coherence condition: the pilot wave phases of all Cooper pairs must be mutually consistent across the entire material, locking \nabla\theta into a single-valued irrotational field. A vorticity \nabla \times \mathbf{v}_s \neq 0 away from quantized vortex cores would create a multi-valued \theta — a boundary-matching inconsistency that the substrate cannot sustain, just as a fractional standing wave cannot exist in the hydrogen raceway.
The phase \theta is the literal oscillation phase of synchronized counter-rotating vortex eddies spanning the material (One Mechanism, Five Scales). In the Compton breathing picture (Electron), \theta encodes the collective anti-phase breathing clock of the condensate: each pair’s contracted/expanded cycle is phase-locked to every other pair’s, and \nabla\theta encodes the spatial gradient of that collective clock. The pilot wave self-reinforcement mechanism — each electron’s Compton ripples carve the raceway that guides it back to itself — now operates cooperatively across \sim 10^{22} pairs.
The canonical momentum relation 2m_e \mathbf{v}_s = \hbar\nabla\theta - (2e/c)\mathbf{A} gives, on taking the curl:
2m_e (\nabla \times \mathbf{v}_s) = -\frac{2e}{c}\,\mathbf{B}_\text{field}
\nabla \times \mathbf{v}_s = -\frac{e}{m_e c}\,\mathbf{B}_\text{field}
Substituting \mathbf{J}_s = -n_s e\,\mathbf{v}_s:
\boxed{\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m_e}\,\mathbf{B}_\text{field}} \qquad\checkmark\;\text{2nd London equation}
(In SI units, the factor is n_s e^2/m_e with \mathbf{B} = \nabla \times \mathbf{A}; the Gaussian 1/c factors are absorbed into the SI definitions.)
The HVBK framework contributes to this step through the canonical momentum coupling: the substrate identifies \mathbf{A} with the co-rotating substrate flow pattern. The minimal coupling \mathbf{p} \to \mathbf{p} - (e/c)\mathbf{A} is not a postulate — it follows from the identification of charge with co-rotating substrate flow and the electromagnetic field with perturbations of that flow (The Photon as Modon).
B6. What the HVBK framework adds. The standard derivation (Phase A) assumes frictionless flow. The HVBK framework provides the mechanism:
B = 0 has a physical cause. The topological protection of the pair vortex (even boundary parity of the Cooper pair boson) suppresses the dissipative channel. This is not just the empirical consequence of an energy gap — it is a structural feature of the pair’s boundary topology that explains why the energy gap prevents dissipation. The gap is the energy cost of breaking even boundary parity, and even parity means no chirality mismatch for the B channel to act on.
B' generates the Meissner effect. The reactive channel is the counter-rotating response to co-rotating perturbation. An external magnetic field (co-rotating substrate flow) is met by the condensate’s collective reactive response (screening currents), which is exactly the B' Hall-type mutual friction acting across the boundary of every pair vortex. The screening currents curl proportional to local \mathbf{B}, producing exponential field decay with penetration depth \lambda_L.
The B/B' decomposition connects to the electroweak sector. In the Weinberg angle derivation (Weinberg Angle, Constraint C8), B \to g' (U(1)_Y) and B' \to g (SU(2)_L). The superconductor’s Meissner effect (screening of EM field by reactive mutual friction) is therefore structurally parallel to the SU(2)_L gauge interaction:
| Context | B (dissipative) | B' (reactive) | Physical consequence |
|---|---|---|---|
| Electroweak (Weinberg Angle) | g' (U(1)_Y) | g (SU(2)_L) | Weinberg angle: \tan^2\theta_W = B/B' = \alpha_\text{mf} |
| Superconductor (this derivation) | Pair-breaking scattering → suppressed | Screening currents (Meissner) → active | London equations: frictionless flow + field expulsion |
At the electroweak scale, both channels are active with ratio \alpha_\text{mf} = 0.3008 (30% dissipative, 70% reactive). In the superconductor at T \ll T_c, the topological protection of the pair condensate suppresses the dissipative channel entirely, leaving 100% reactive — the Meissner effect is the limiting case of the same B/B' physics that gives the Weinberg angle, with B driven to zero by the pair vortex topology.
Phase C: The \alpha_\text{mf} Consistency Check
C1. Why \lambda_L uses m_e, not m_\text{eff}. The London penetration depth:
\lambda_L = \sqrt{\frac{m_e}{\mu_0\,n_s\,e^2}}
uses the bare electron mass m_e = 0.511 MeV/c^2, not the effective quantum mass m_\text{eff} = m_e/\alpha_\text{mf} \approx 1.70 MeV/c^2. This is a consistency requirement, not an anomaly.
The mass relation m_\text{eff} \cdot \alpha_\text{mf} = m_e (Constraint C2, Two Fluids → Quantum Potential) bridges two sectors:
| Sector | Relevant mass | Coupling | Quantum |
|---|---|---|---|
| Electromagnetic | m_e (full orbital system) | Charge e | \Phi_0 = h/(2e) |
| Superfluid | m_\text{eff} (effective quantum) | \alpha_\text{mf} | \kappa_q = h/m_\text{eff} |
\lambda_L derives from the electromagnetic response — how the condensate couples to co-rotating substrate flow (the EM field) via charge e. The Compton energy identity (Electron) makes the mass assignment transparent:
\tfrac{1}{2}m_\text{eff}\,v_\text{rot,inner}^2 = \tfrac{1}{2}\frac{m_e}{\alpha_\text{mf}} \cdot 2\alpha_\text{mf}\,c^2 = m_e c^2
The peak kinetic energy of the effective quantum at contraction equals the full electron rest energy. The EM field couples to the time-averaged total energy of the Compton oscillation — the electron’s rest mass m_e c^2 — not to the internal substructure. At any instant the energy is split between kinetic (contracted, r_\text{eff}) and boundary (expanded, \xi), but the EM field sees the sum, which is m_e. This is why \lambda_L naturally uses m_e: the EM sector couples to the full orbital system complex, and the full orbital system complex has mass m_e.
The effective quantum mass m_\text{eff} is an internal substrate property — the mass of the condensed dc1 cluster that carries the electron’s spin and angular momentum. It enters the circulation quantum \kappa_q = h/m_\text{eff} (superfluid sector), not the flux quantum \Phi_0 = h/(2e) (electromagnetic sector).
Both sectors use the same Planck’s constant h because quantization comes from the same boundary-matching physics at the soliton scale \xi — the Bessel matching at the counter-rotating interface that gives discrete states for hydrogen (Hydrogen Atom), modon speeds (The Photon as Modon), and spin (Spin-Statistics).
C2. Temperature dependence — a potential distinguishing prediction. The standard Gorter-Casimir two-fluid model gives:
\lambda_L(T) = \frac{\lambda_L(0)}{\sqrt{1 - (T/T_c)^4}}
In the HVBK framework, as T \to T_c:
- The superfluid fraction \rho_s/\rho \to 0 (pairs break as thermal energy exceeds 2\Delta)
- The dissipative coefficient B grows from zero (topological protection weakens as the pair vortex becomes thermally fragile)
- \lambda_L \to \infty (screening fails, the Meissner effect is lost)
The HVBK framework predicts B(T) \propto \exp(-\Delta(T)/k_BT) through the same Boltzmann suppression that gives the BCS quasiparticle density. The question is whether the HVBK mutual friction equations, with the substrate’s specific B(T) form and the reactive channel’s T-dependence, reproduce the Gorter-Casimir (T/T_c)^4 dependence or predict corrections.
The Gorter-Casimir form is phenomenological — it assumes n_s(T)/n = 1 - (T/T_c)^4 without microscopic justification. BCS theory gives a different temperature dependence near T_c: n_s \propto (1 - T/T_c) (mean-field, linear vanishing). The HVBK framework with the B/B' decomposition might predict a form that interpolates between these — with the specific interpolation determined by how the reactive channel (B') responds to the gradual weakening of topological protection near T_c.
If the substrate predicts corrections to the Gorter-Casimir form near T_c that differ from standard Ginzburg-Landau theory, this would be a forward prediction — measurable in penetration depth measurements on clean superconductors near the critical temperature. This is the most exciting potential outcome of the derivation, but the quantitative calculation remains open: it requires solving the HVBK equations with temperature-dependent B(T) and B'(T) and computing the resulting \lambda_L(T).
C3. The \Phi_0 / \kappa_q ratio. The superconducting flux quantum \Phi_0 = h/(2e) and the circulation quantum \kappa_q = h/m_\text{eff} encode different physics. Their ratio:
\frac{\Phi_0}{\kappa_q} = \frac{m_\text{eff}}{2e} = \frac{m_e}{2e\,\alpha_\text{mf}}
has dimensions [kg/C] = [T·s/m]. Whether this combination has independent physical meaning — for instance as a substrate “impedance” connecting the electromagnetic and superfluid sectors — is worth investigating but not required for the London derivation. The ratio involves \alpha_\text{mf} explicitly, so if the flux quantum and circulation quantum could be measured independently in the same system, their ratio would provide a direct measurement of the mutual friction parameter at the electron scale.
Summary: What Is New vs. What Is Textbook
| Element | Content | Epistemic status |
|---|---|---|
| Phase A (standard London) | Textbook target equations | Well-established |
| B1–B2 (HVBK setup + identifications) | Textbook HVBK + substrate identifications | Interpretive mapping — identifications are framework-specific |
| B3 (B \to 0 mechanism) | Topological protection via even boundary parity | New substrate content — pair vortex topology suppresses dissipation |
| B4–B5 (derive London equations) | Standard math with HVBK starting point; irrotationality from universal boundary-matching pattern | Follows from B3; boundary matching connects to hydrogen, modon, spin |
| B6 (B' = Meissner, electroweak connection) | Reactive channel generates screening | New substrate interpretation — connects Meissner to SU(2)_L |
| C1 (m_e vs m_\text{eff} in \lambda_L) | Two sectors bridge via m_\text{eff} \cdot \alpha_\text{mf} = m_e | Framework consistency check — required by mass relation |
| C2 (\lambda_L(T) near T_c) | HVBK B(T)/B'(T) temperature dependence | Potential forward prediction — quantitative calculation open |
| C3 (\Phi_0/\kappa_q ratio) | Connects EM and superfluid sectors via \alpha_\text{mf} | Exploratory — interesting if measurable |
The derivation closes open derivation #1: the London equations follow from the HVBK mutual friction equations with the substrate identifications, the B = 0 mechanism is physically motivated, and \lambda_L uses m_e consistently with the mass relation. The derivation opens a new potential prediction (\lambda_L(T) near T_c) and a new connection (Meissner effect as the limiting case of the B/B' decomposition that gives the Weinberg angle).