4-D generalization of the holomorphy allows conserved charges associated with the generalized holomorphies
Does the 4-D analogy of holomorphy as a realization of holography give rise to conserved quantities? Now the symmetries would not be isometries, nor some other symmetries of the action, but dynamic symmetries satisfied only by the Bohr orbits. A little calculation that one can do in your head shows that one obtains conserved currents: the reason is the same as in the case of field equations. The divergence of the Noether current is a contraction of tensors with no common index pairs for the generalization of complex coordinates.
Unlike those associated with the general coordinate invariance, these conserved quantities do not vanish. They correspond to the 4-D generalization of conformal transformations and give rise to a generalization of the Virasoro algebra and also of Super Virasoro algebra realized in terms of the modified Dirac action for the induced spinor fields obtained from the free second quantized spinor fields of H.
In the string model, these conformal charges are assumed to annihilate the physical states. In TGD, I have proposed that only a subalgebra that is isomorphic to the whole algebra, having conformal weights which are integer multiples of the entire algebra, does this. In TGD framework, the conformal weights are necessarily non-negative and ZEO allows this. One obtains a whole hierarchy of subalgebras and a sub-hierarchy of algebras for which conformal symmetry as gauge symmetry is "broken" to dynamical Lie symmetries for physical states having conformal weight below some maximum value. These hierarchies could correspond to the hierarchies of algebraic extensions for rationals defined by composite polynomials.
Are fermionic 2-vertices all that is needed in TGD?
In quantum field theories, already the interaction vertex for 3 particles leads to divergences. In a typical 3-vertex, fermion emits a boson or boson decays to a fermion-antifermion pair. In TGD, the situation changes.
- Fermions are the only fundamental particles in TGD. Since fundamental bosons are missing, there is no vertex representing emission of a fundamental boson emission from fermion or a vertex producing fermion antifermion pair from a fundamental boson. In TGD, bosons as elementary particles (distinguished from fundamental bosons) are fermion-antifermion pairs, and the emission of elementary bosons is possible. However, the problem is that the total fermion and antifermion numbers are separately conserved. Unless it is possible to create fermion pairs from classical fields!
- In the standard theory fermion-antifermion pairs can be indeed created in classical gauge fields. This creation is an experimental fact but it is thought that this description is only a convenient approximation. In TGD however, the classical fields associated with the Bohr orbits of 3-surfaces are an exact part of quantum theory. Could this description be accurate in TGD? In the classical induced fields associated with particles, pairs could arise. Approximation would become exact in TGD.
- I managed to identify the fermionic 2-vertex was specified towards the end of this year as I realized the connection to the problem of general relativity, which arises from the existence of GRT space-times for which the 4-D diffeo structure is non-standard. There are a lot of these. For an exotic diffeo structure, the standard diffeo structure can be said to have point-like defects analogous to lattice defects.
- Remarkably, this problem is encountered only in the space-time dimension 4 (see this)! Physical intuition suggests that it must be possible to turn this problem from a disaster to victory. In TGD, this is what actually happens: these point-like diffeo-defects can be identified as interaction vertices, the fermion turns back in the direction of time. Pair creation would be possible only in space-time dimension 4!
A generalization of the classical fermion pair creation vertex has the same general form as in QFT. As a special case the pair can correspond to a boson as a fermion-antifermion bound state. This vertex also has geometric variants in different dimensions. A fermion line, string world sheet, the orbit of a partonic 2-surface and also the Bohr orbit of 3 surface can turn backwards in time and the fermion states associated with the induced spinor fields do the same.
- Is the creation of a pair actually the only vertex or is it possible to have a geometric 3-vertex and is it really needed? At the fermion level only the 2-vertex described above is not possible, but for the topological reactions of surfaces one could think of 3-vertices and in the earlier picture I thought these are needed. They do not seem to be necessary however.
If so, the theory would be extremely simple compared to quantum field theories. There dangerous genuine 3-vertices would be absent and diffeo defects defining 2 vertices, which give all that is needed! At the geometric level, monopole fluxes would replicate and break and join. Intriguingly, this is what would happen at the magnetic bodies of DNA and induce similar reactions at the level of DNA molecules! Maybe biology has been doing its best to tell us what the fundamental particle dynamics is!
- Since only the induced electroweak gauge potentials couple to fermions, the question arises whether color and strong interactions are obtained. How is it possible to have strong interactions without parity violation when basic vertices involve weak parity violation? I have already discussed this question (see this).
There is still one crucially important question left. Is it possible and what would happen in it? Can one obtain a vertex, where the analog for a contraction Tαβδ gαβ of energy-momentum tensor with the deviation of the metric from the Minkowski metric appears?
- In TGD all elementary particles, also gravitons, are identified as closed 2-sheeted monopole flux tubes with two wormhole contacts at its "ends" and opposite wormhole throats carrying fermions and antifermions (see this and this). For gravitation one has 1 fermion or antifermion for each wormhole throat.
The graviton emission vertex should correspond to a splitting of flux tubes. Mopole flux tubes with fermion-antifermion pairs assignable to both wormhole contacts should appear. The fermion and antifermion should reside at the opposite throats of each wormhole contact. This should happen in the splitting of a monopole flux tube and second monopole flux tube would correspond to graviton. That two bosonic vertices are involved with the emission, brings to mind the proposal that gravitation is in some sense a square of gauge theory.
- The vertex is the same as for gauge boson emission and for a creation of a fermion-antifermion pair. The definition of the modified gamma matrices as Γα= TαkΓk appearing in the modified Dirac action (see this), involving the modified Dirac operator ΓμDμ makes it possible to identify the gravitational part of the vertex. Here the quantities Tαk=∂ L/∂(∂αhk) are canonical momentum currents associated with the action defining the space-time surface and also the analog of the energy-momentum tensor.
Modified gamma matrices are required by hermiticity forcing the vanishing of the divergence of the vector Γα giving classical field equations for space-time surfaces. This implies a supersymmetry between the dynamics of fermions and 3-surfaces. The gravitational interaction would correspond to the deviation of the induced metric from the induced metric defined by induced CP2 metric. CP2 radius must correspond to Planck length lP. This requires that the CP2 as R≈ 104lP must correspond to h= nh0, n≈ 107 as found already earlier.
- The cosmological term in GRT has coefficient 1/8π GΛ== 1/R4 so that the modified gamma matrices would contain a term proportional to 1/R4 plus a term coming from the Kähler action. In the TGD framework (see this and this) cosmological constant Λ depends on the p-adic length scale, which is assumed to correspond to a ramified prime for an extension of rationals associated with the polynomial P determining to high degree the space-time surface and approaches to zero in cosmic scales. The cosmological value corresponds to R≈ 10-4 meters, i.e. cell length scale and a scale near neutrino Compton length.
In the general coordinate invariant formalism, one does not assign dimension to the coordinates or to covariant derivative Dα. Metric has dimension 2. The scale dimension of Tαkg1/2 is the same dimension of Lg1/2 and thus vanishes. Γα has scale dimension -1. The modified Dirac action must be dimensionless so that the induced spinors must have scale dimension 1/2.
- The cosmological constant as the coefficient of the action depends on the p-adic length scale unlike. This term contributes to the string tension of string-like objects an additional term, which among other things can explain hadronic string tension. This term is visible also in the interaction vertices. The Kähler part of the bosonic action terms comes from the deviation of the induced metric from the flat metric and should give the usual gravitational interactions with matter.
- Holomorphy hypothesis allows any general coordinate invariant action constructible in terms of the induced geometry. Although preferred extremals are always minimal surfaces, the properties of the action are visible via classical conservation laws, via the field equation at singular 3-surfaces involving the entire action, and via the vertices.
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.