## Monday, July 19, 2021

### Beltrami flow as space-time correlate for non-dissipative flow

In the standard model of superconductivity SC is characterized by a complex order parameter for which the Berry phase would serves as an analog in BPM. Berry phase is a consequence of adiabaticity and characterizes collective phase. One can assign to the Berry phase effective U(1) gauge field which reduces to magnetic field in a static situation. What are the TGD counterparts of these notions?

TGD provides the geometrization of classical physics in terms of space-time surfaces carrying gravitational and standard model fields as induced fields so that both the supra current and the phase should have geometric intepretation. This serves as a powerful constraint on the model.

1. Supra current must correspond to a flow. The flow must be integrable in the sense that the coordinate defined along flow lines defines a global coordinate at flux tubes. One can indeed argue that an operational defition of a coordinate system requires that coordinates correspond to coordinates varying along flow lines of some physical flow. The exponential of the coordinate would define the phase factor of the complex order parameter such that its gradient defines the direction of the supracurrent.

If the motion of particles is random one cannot talk of a hydrodynamic flow but something analogous to the motion of gas particles or Brownian motion. In the TGD framework this situation corresponds to disjoint space-time sheets as a representation of particle orbits. The flow property could however hold true inside the "pieces" of space-time. The coherence scales of flow would become short.

2. One must make it clear that here an approximation is made. Elementary particles have as building bricks wormhole contacts defining light-like partonic orbits to which one can assign light-like curves as M4 projections. For a vanishing value \Lambda=0 of cosmological constant (real analytic functions at M8 level), these curves are light-like (light-likeness condition reduces to Virasoro conditions) whereas for \Lambda>0 (real polynomials) at M8 level the projections consist of pieces which are light-like geodesics somewhat like in the twistor diagrams \cite{minimal}. Smooth curve is replaced with its approximation.

For massive particles, this orbit would be analogous to zitterbewegung orbit and the motion in the long scales would occur with velocity v<c: this provides a geometric description of particle massiation. The supracurrent would not actually correspond to the flow as such but to CP2 type extremals along the flow lines.

3. The 4-D generalization of so called Beltrami flow \cite{Beltrami,Beltramia,Beltramib,Beltramic}, which defines an integrable flow in terms of flow lines of magnetic field, could be central in TGD. Superfluid flows and supra currents could be along flux lines of Beltrami flows defined by the Kähler magnetic field \cite{class,prext}.

If the Beltrami property is universal, one must ask whether even the ordinary hydrodynamics flow could represent Beltrami flow with flow lines interpreted in terms of flow lines Kähler magnetic field appearing as a a part of classical Z0 field. Could hydrodynamical flow be stabilized by a superfluid made of neutrino Cooper pairs. heff hierarchy of dark matters in turn inspires the question whether weak length scale could be scaled up to say cellular length scales (neutrino mass corresponds to a length scale of a large neuron).

4. The integrability condition

j∧ dj=0

of the Beltrami flow states that the flow is of form

j= Ψ dΦ ,

where Φ and Ψ are scalar functions, which means that Ψ defines a global coordinate varying along the flow lines.

5. Beltrami property means that the classical dissipation characterized by the contraction of the Kähler current

jα=DβJαβ

with Kähler form Jαβ is absent:

jβJαβ=0 .

In absence of Kähler electric field (stationary situation), this condition states the 3-D current is parallel with the magnetic field that it creates.

In 4-D case, the orthogonality condition guarantees the vanishing of the covariant divergence of the energy momentum tensor associated with the Kähler form. This condition is automatically true for the volume part of the energy momentum tensor but not for the Kähler part, which is essentially energy momentum tensor for Maxwell's field in the induced metric. As far as energetics is considered, the system would be similar to Maxwell's equations.

The vanishing of the divergence of the energy momentum tensor would support Einstein's equations expected at QFT limit of TGD when many-sheeted space-time is approximated with a slightly curved region of M4 and gauge and gravitational fields are defined as the sums of correspond induced fields (experienced by test particles touching all space-time sheets).

6. An interesting question is whether Beltrami condition holds true for all preferred extremals \cite{prext} \cite{minimal}, which have been conjectured to be minimal surfaces analogous to soap films outside the dynamically generated analogs of frames at which the minimal surface property fails but the divergences of isometry currents for volume term and Kähler action have delta function divergences cancelling each other. The Beltrami conditions would be satisfied for the minimal surfaces.

If the preferred extremals are minimal surfaces and simultaneous extremals of both the volume term and the Kähler action, one expects that they possess a 4-D analog of complex structure \cite{minimal}: the identification of this structure would be as Hamilton-Jacobi structure \cite{prext} to be discussed below.

7. Earlier I have also proposed that preferred extremals involving light-like local direction as direction of the Kähler current and orthogonal local polarization direction. This conforms with the fact that Kähler action is a non-linear generalization of Maxwell action and minimal surface equations generalize massless field equations. Locally the solutions would look like photon like entities.

This inspires the question whether all preferred extremals except CP2 type extremals defining basic building bricks of space-time surfaces in H have a 2-D or 3-D CP2 projection and allow interpretation as thickening of flux tubes? CP2 type extremals have 4-D CP2 projection and light-like M4 projection and an induced metric with an Euclidean signature.

See the article Comparing the Berry phase model of superconductivity with the TGD based model or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.