Beltrami flows and holography = holomorphy vision

Beltrami flows (see this) appear in several contexts. Google AI informs that Beltrami flow is a force-free flow field at 3-sphere. The simplest Hopf fibration (see this) is from 3-sphere to 2-sphere and fibers correspond to circles and there are numerous generations of Hopf fibration: the fibration S⁵→ CP₂ is of special interest in TGD.

Some background

Some background about Beltrami flows is in order.

1. For the Beltrami flow (see this) velocity field satisfies curl(v) = Λ×v so that curl(v) is parallel to v. In fluid dynamics Beltrami flow corresponds to a flow for which vorticity ω= ∇× v and velocity v are parallel. ω× v=0 gives ω =∇× v= α (x,t)v. Beltrami flows in S³ satisfy this condition and are exact solutions to Euler equations.
2. In magnetohydrodynamics one can replace velocity field with magnetic field B and of the current j satisfies j= ∇× B= α B implying the vanishing of the Lorentz force j× B. The current flows along field lines and in TGD the flow of particles along monopole flux tubes is the counterpart for this flow. These Beltrami flows involve the linking and knotting of magnetic field lines. Similar situation prevails in hydrodynamics.

Jenny Lorraine Nielsen has proposed that the Hopf fibration S¹→ S⁹→ CP₄ could provide a theory of everything and that Beltrami flows (see this) associated with this kind of fibrations play a key role in physics. The scalar Λ, which depends on position, appearing in the definition of Beltrami flow has dimensions of 1/length. Mass has dimension of ℏ/length so that 1/Λ should be identified as an analog of Compton length. These flows are topologically very interesting and involve linking and knotting of the flow lines.

The claim of Jenny Nielsen is that it is possible to understand particle massivation in terms of Beltrami flows. Higgs expectation defining the mass spectrum in the standard model is identified as hbarΛ for the eigenvalue Λ of the lowest eigenmode of Beltrami flow. It would seem that Λ is assumed to be constant: this is not necessary. It must be possible to relate Λ to the radius of S³ and one chooses it suitably to get Higgs vacuum expectation. To get masses of fermions one must put them in by hand as couplings of fermions to Higgs so that one does not really predict fermion masses: the situation remains the same as in the standard model.

Beltrami flows in TGD

The generalization of Beltrami flows to 4-D context is one of the key ideas of TGD (see for instance this and this) but I have not discussed them explicitly in the recent framework based on holography = holomorphy vision (H-H) ) (see for instance this and this and this).

1. The motivation is that TGD is formally hydrodynamics in the sense that field equations express local conservation of isometry charges of M⁴× CP₂. There is actually infinite-dimensional algebra of conserved charges. The proposal is that in TGD, the Beltrami flows become genuinely 4-dimensional and correspond to classical field configuration for which the 4-D Lorentz force involving electric components vanishes.
2. The definition of the Beltrami flow is however different since one cannot regard the magnetic field as a vector field in 4 dimensions. For field equations Kähler current typically vanishes but can be also light-like. The counterpart of Beltrami flow states that Kähler current is proportional to the corresponding axial current:

j^=D_νJμν = α × ε^{μν αβ}A_νJ_αβ.

The divergence of j^μ vanishes and this must be true also for the instanton current unless α=0 holds true This is the case if the CP₂ projection of the space-time surface is at most 3-dimensional. If it is 4-D the parameter α must vanish since the divergence of the axial current gives instanton density ε^μναβ J_μνJ_αβ, which is non-vanishing for CP₂ and by self-duality proportional to J^μν J_μν. Hence the only option is α=0 for D=4.

If these Beltrami flows are integrable, they can give a physical realization of some, perhaps all, space-time coordinates as coordinates varying along the flow lines of some isometry current. The time component of 4-force has interpretation as dissipation power and also vanishes. These non-dissipative configurations play a key role in TGD and are natural when space-time surfaces are identified as quantum coherence regions.

The key idea sharpening dramatically the notion of Beltrami flow supported by H-H vision is that complex analytic maps f: z→ f(z) allow us to construct integrable flows. What matters physically would be singularities: poles and zeros. Without them these maps would be mere general coordinate transformations.

In TGD, this generalizes to 4 dimensions by the introduction of generalized complex structure in H=M⁴ × CP₂. The presence of hypercomplex coordinates in M⁴ motivates the term “generalized”. In 4-D context, poles and zeros as singularities of a flow correspond to string world sheets and partonic 2-surfaces. The second key idea is that fermions at the flow lines serve as markers and provide information about the flow. In the cognitive sector they realize Boolean logic. Flows in the complex plane

Flows in the plane are usually regarded as interesting Beltrami flows since in this case the condition ∇ × v = α v cannot be satisfied unless vorticity and eigenvalue α vanish. It is however possible to consider the situation α=0 also in the D=2 case. What is remarkable is that this brings in the notion of holomorphy. Moreover, the integrability is the key notion in TGD and means that flow lines have interpretation as coordinate lines.

One can start from flows in plane, in particular integrable flows.

1. Integrability means that the flow lines of the flow define coordinate lines. This requires that the velocity v for the flow line is a gradient grad(phi) of the coordinate in question. This condition is very strong. Without integrability, the flow would be a random motion analogous to the motion of gas particles. Integrability brings in smoothness and the flow looks like a fluid flow.

Most importantly, exotic smooth structures possible in TGD would correspond to flows for which smoothness fails at singularities to make possible fermionic interactions, although fermions are free in TGD. But this is possible only for 4-D space-time.

One can go even further and require that there are two integrable coordinates. They could be assigned with velocity v and vorticity curl(v). Both would define gradient flows apart from singularities. These conditions give Cauchy-Riemann conditions expressing complex analyticity. The flow can be expressed as an analytic map z→ f(z) of the complex plane and the flow lines correspond to coordinate lines of the new coordinates defined by f. Locally the conditions state that the flow is locally incompressible and irrotational apart from singularities. Globally this need not be the case.

These maps however have poles and zeros as singularities. Poles act as sources and sinks at which the flow fails to be incompressible. Zeros correspond to vortex cores at which the flow velocity must approach zero.

One can also allow cuts. They appear if a complex analytic map is many-valued, such as fractional power and it is made discontinuous by taking only a single branch. Second option is to allow a covering in which case the complex plane becomes many-sheeted. In TGD, this picture is generalized to a 4-D situation. The 2-dimensional flows induced by the simplest Hopf fibration S³→ S²

Consider next the Hopf fibration S³→ S². One can consider Betrami flows ∇ v=α v in S³ but also the flows of S² induced by Hopf fibration are very interesting concerning the generalization to the 4-D case in the TGD framework. Since S² has complex – and Kähler structures, also these integrable flows should be reducible to analytic maps f: z→ f(z) of S² to itself.

1. At the fermionic the presence of the S¹ as fiber of S³ brings in a coupling of S² spinors to a covariantly constant Kähler form of S², which corresponds to a U(1) symmetry assignable to S¹. In the case of S², the coupling is not necessary but in the case of CP₂ the Hopf fibration S⁵→ CP₂ allows Spin_c structure and leads to the standard model couplings and symmetries in TGD.
2. S³ with Kähler structure can be visualized for the standard embedding S² → E³ as a covariantly constant magnetic field B orthogonal to S². Another way to describe B is as a covariantly constant antisymmetric 2-tensor in S².
3. At the hydrodynamical level, one can consider hydrodynamics in which geodesic free motion couples to the magnetic field defined by the Kähler form via Lorentz force. The magnetic force causes a twisting so that the motion is not anymore along a big circle. The flow lines tend to turn towards the North Pole or South Pole and approach/or leave the poles from South or North. Chiral symmetry is clearly violated.
4. Also now one can ask when the flow is integrable rather than a random motion of gas molecules. One needs two coordinates defined by the flow and there are several candidates, acceleration defined by the Lorentz force, velocity and vorticity. Complex and Kähler structures make sense also for S². The conclusion is that analytic maps z→ f(z) of a complex coordinate of S² define an integrable flow. The real and imaginary parts of f(z) define the velocity field v.
5. There are two kinds of singularities at which the analyticity fails: zeros correspond to vortices and poles to sources and sinks. Everywhere else the flow is locally incompressible and irrotational so that both the divergence and rotor of the velocity field vanish. If the flow has no singularities it can be regarded as a mere coordinate change. Singularities contain the physics.

Hopf fibration S⁵→ CP₂

In TGD the projection S⁵→ CP₂ is the crucial Hopf fibration since it makes it possible to provide CP₂ with a respectable spinor structure. The Kähler coupling gives rise to the standard model couplings and symmetries and H=M⁴× CP₂ is physically unique: weak interactions are color interactions in CP₂ spin degrees of freedom (charge and weak isospin). What is essential is the coupling of the Kähler gauge potential to spinors. This in turn leads to a Dirac equation in H=M⁴× CP₂ and the induced Dirac equation at the space-time surface X⁴.

1. At the hydrodynamical level one has geodesic flow coupled to the self-dual Kähler form of CP₂ . One has Euclidian analogs of constant electric and magnetic fields, which are of the same magnitude. They would be orthogonal in E⁴ but in CP₂ their inner product gives constant instanton density.
2. Also now complex analytic maps f: CP₂→ CP₂ define i×ntegrable flows with singularities. There are two complex coordinates and one can have poles with respect to both of them. Both poles and zeros are replaced with 2-D surfaces and also the analogs of cuts appearing if many-valued maps f are allowed.

CP₂ type extremals

At the next level one can consider CP₂ type extremals, which are deformations of the canonical embedding of CP₂ as an Euclidean 4-surface of H=M⁴× CP₂ for which M⁴ coordinates are constant. They can be said to define basic building bricks of particles in TGD. The CP₂ type extremal has locally the same induced metric and Kähler structure as CP₂ but its M⁴ projection is a light-like curve, light-like geodesic in the simplest situation. It also ends, that is holes realized as 3-surfaces.

1. The above situation for which time is time parameter as 5:th coordinate is replaced with M⁴ time coordinate u varying along the light-like curve. Also now the complex analytic functions f: CP₂ → CP₂ define integrable flows. Time coordinate labels 3-D sections of the flow.
2. Now these flows would carry real physics. Induced Dirac equation effectively reduces to 1-D Dirac equation for fermion lines identified and holomorphy solves it, very much like in string models.

The physical interpretation is very concrete. The addition of fermions to fermion lines serves as an addition of a marker making the flow visible. Fermions as markers allow to get information about the underlying geometric flow making itself visible via the time evolution of the many-fermion state.

In TGD, fermions also realize Boolean logic at quantum level and the time evolutions between fermionic states can be seen as logical implication A→ B. Spinor structure as square root of metric structure fuses logic and geometry to a larger structure. Integrable flows at Minkowskian space-time surfaces X⁴ ⊂ H

In holography = holomorphy vision space-time surfaces are roots for a pair f=(f₁,f₂): H→ C² of two generalized analytic functions f_i of one real hypercomplex coordinate u of M⁴, and the remaining 3 complex coordinates of H. Let us denote one of the complex coordinates by w. It can be either an M⁴ or CP₂ coordinate.

1. The roots give space-time surfaces as minimal surfaces solving the field equations for any classical action as long as it is general coordinate invariant and constructible in terms of induced geometry. The extremely nonlinear field equations reduce to local algebraic conditions and Riemannian geometry to algebraic geometry.
2. X⁴ shares one hypercomplex coordinate and one complex coordinate with H and both X⁴ and H have generalized complex structure. X⁴ has hypercomplex coordinate u (u=t-z of M² in the simplest situation) and complex coordinate w (coordinate of complex plane E² in the simplest situation). This defines the Hamilton-Jacobi structure of X⁴.
3. Complex analytic maps of X⁴ are of the form by (u→ f(u), w→ g(u,w)). Integrable flows are induced by these maps. If there are no singularities they correspond to general coordinate transformations. The map by f having singularities generates a new Hamilton-Jacobi structure.
4. Poles and zeros in the w-plane correspond to 2-D string world sheets. The counterparts of zeros and poles for hypercomplex plane, parameterized by a discrete set of values of the real hypercomplex coordinate u correspond to singular partonic 2-surfaces with complex coordinate w at the light-like orbit of a partonic 2-surface. These singular partonic 2-surfaces can be identified as TGD counterparts analogs of vertices at which fermionic lines can change their direction. At these surfaces the trace H of the second fundamental form vanishing everywhere else by minimal surface property has a delta function like singular. Its CP₂ part has an interpretation as analog of Higgs vacuum expectation value. The claim of Jenny Nielsen is analogous to this result. In TGD also the M⁴ part of H is non-vanishing and corresponds to a local acceleration concentrated at the singularity. An analog of Brownian motion is in question.

One could very loosely say that the parameter α for Beltrami flow vanishes everywhere except at singularities where it has interpretation as value of the analog of Higgs expectation as the trace of the second fundamental form.

String world sheets in turn mediate interactions since they connect to each other the light-like orbits of partonic 2-surfaces. This view conforms with the basic physical picture of TGD. To sum up, the new elements brought by the holography = holomorphy principle are as follows.

1. The flows in CP₁ and CP₂ are more important than flows in S³ and S⁵. The spheres provide the needed Kähler form guaranteeing the twisting of the flow and making possible arbitrarily complex flow topologies as knotting, braiding, and linking. Also 2-knots are possible in 4-D context.
2. The realization of the role of analytic maps with singularities as a way to generate new Hamilton-Jacobi structures and at the same time build string world sheets and singular partonic 2-surfaces as interaction vertices. This is like the action of spectrum generating algebra. Also the functional composition of maps f= (f₁,f₂): C²→ C² acts as a spectrum generating algebra.
3. The realization that fermion lines very concretely serve as markers of a hydrodynamic flow.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/11/beltrami-flows-and-holography.html