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.

For massive particles, this orbit would be analogous to zitterbewegung orbit and the motion in the long scales would occur with velocity v2 type extremals along the flow lines.

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. h_eff 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).

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.

j^α=D_βJ^αβ

with Kähler form J_αβ is absent:

j^βJ_αβ=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).

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.

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.

Articles and other material related to TGD.