How to create monopole-flux loops?

Yesterday evening we had a very fruitful discussion with Marko Manninen about wormhole contacts and monopole flux loops. The question was following:

We pondered this a lot, but the final outcome remained vague and bothered me and I continued to ponder the problem during the night. Finally, I realized that the problem was structurally the same as another problem: how is the creation of fermion pairs possible in TGD even though spinor fields in H=M⁴×CP₂ are free fields? The solution was also the same and possible only for 4-D spacetime surfaces.

I will summarize the basic facts first and then the results.

1. A wormhole contact is a Euclidean spacetime region that connects two Minkowski spacetime sheets. They appear as two different types.
   1. One that arises when the sheets touch each other. This region is not stable because there is no net magnetic flux flowing from one sheet to the other.
   2. One with a monopole flux. One can talk about a very short piece of flux tube. Since the flux is conserved, it cannot arise when the sheets touch. On the other hand, the conservation of flux stabilizes the tube so that it cannot be split.

How could monopole wormhole contacts arise? This is the basic question we pondered. During late-night reflections it became clear that the attempts that first came to my mind did not work.

More facts.

1. The boundary conditions of the field equations of TGD do not allow open flux tubes, i.e. cylinders with ends from which flux would escape into vacuum. The flux losses at the ends would compensate each other, but this is not enough. Local conservation is required and is not possible. Flux tubes must be closed loops in which monopole flux flows.
2. This implies that wormhole contacts carrying monopole flux must be appear in pairs. They can be visualized as two wormhole contacts, i.e. magnetic flux flows from sink A1 to sink B₁, from there to sink B₂ on the lower leaf and from there to sink A₂ and further to sink A₁ on the upper leaf. A closed loop, then. At least massive elementary particles would correspond to such. Point-like fermions would inhabit the sinks.
3. Magnetic flux corresponds to K hler magnetic charge. It must be conserved as it flows along the tube. The total magnetic charge would be zero if open flux tubes were allowed. Here the hydrodynamic analogy to incompressible flow, which is mathematically very accurate, helps.
4. But what happens in the case of closed flux tubes? Does the sum of the charges conserved for closed flux tubes? This would generalize the conservation of flux inside the tube. The flux for a closed flux tube would be analogous to an electric charge.

Would it be concluded that if a monopole flux tube breaks up into two flux tubes with fluxes n₁ and n₂, then the flux remains n=n₁+n₂?

In any case, even the creation of a single closed monopole flux tube seems impossible. So the alternative would be that they are created in pairs so that the fluxes are opposite. How would this happen? Here emerges a connection to a similar problem regarding the fermion number.

1. In TGD, the spinor fields are free. That is the only option. If a quartic term were added to the action, the result would be a non-renormalizable theory and thus a catastrophe.

But is the creation of fermion pairs from a vacuum or a classical induced field then possible at all? In QED, it is when the sm field is quantized but also when it is classical.

An intuitive picture of what happens has been clear since Dirac’s time. The fermion line turns back in time when a pair is created. This picture generalizes to TGD. The fermion line has an edge with an infinite acceleration.

Now comes the connection with zero energy ontology. In TGD, the particle is geometrically replaced by a Bohr orbit of the 3-surface and the infinite acceleration at the edge corresponds to the bracing of the holomorphism on the 3-surface. These singularities are the fundamental prediction and correspond to the mild non-determinism of holography and the poles of analytic functions. These surfaces correspond also to interaction vertices.

At the edge, the minimal surface property braces and the trace of the second fundamental form, which is the generalization of the acceleration form 1-D to a 4-D object, is infinite. Its CP₂ part transforms in symmetries like the Higgs. The Higgs would be non-zero only at the vertices. The M⁴ part corresponds to the ordinary acceleration and the particle’s earthly pilgrimage would be geometrically analogous to 8-D Brownian motion such that the vertices are special moments in its life since they correspond to conscious choices.

A couple of years ago I realized that the edge singularity of a fermion line corresponds to the phenomenon of “exotic smooth structure”. It is a smooth structure, which reduces to the standard smooth structure except for defects that are 3-surfaces: they are “edges” and correspond geometrically to particle vertices in TGD. Differentiability is broken.

The exotic smooth structures only occur in dimension D=4. Pair creation and non-trivial fermionic dynamics are only possible in 4-D spacetime! A really bad problem of general relativity turns into a triumph in TGD. Now we are ver close to solving the original problem. So: how do monopole flux loops arise?

1. The emergence of a pair formed by closed monopole flux tubes corresponds to a situation when a closed monopole flux tube coming from the geometric future turns back to the geometric future! More general reversals can occur and the emission of virtual particles at vertices corresponds to this.

Therefore, a particle pair in geometric sense can be created in the classical induced field in a geometric sense. What happens to fermion lines also happens to 3-surfaces.

This view generalizes: a closed monopole flux tube can break up, thus creating, for example, two monopole flux tubes and the total flux is preserved. See for instance the article Does M⁸−H duality reduce to local G₂ symmetry? or the a HREF= “https://tgdtheory.fi/pdfpool/M8Hstill.pdf”>chapter with the same title.

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/how-to-create-monopole-flux-loops.html