The weak form of electric-magnetic duality (EWMD) turned out to make possible also the calculation of the Dirac determinant. Let us however first list some of the implications of this marvellous symmetry.
- EWMD led to the reduction to almost topological QFT defined by Chern-Simons action provided the Coulomb interaction term in Chern-Simons action vanishes in some gauge. The condition for this led to a beautiful general solution ansatz to field equations reducing field equations to the condition that isometry currents, Kähler current, and possibly also the instanton current are proportional to single Beltrami current with the property that the flow parameter for its lines extends to a global coordinate. The resulting field equations involve two scalar functions for each isometry current. The first one is common to all currents in the most restrictive ansatz and satisfies d'Alembert equation in the induced metric. The gradients of these scalar functions are orthogonal and have interpretation in terms of local polarization direction and momentum direction. Clearly, both the gauge theory interpretation and hydrodynamics interpretation with conservation laws holding separately for flow lines make sense.
- Also the modified Dirac equation in the interior reduces to ordinary differential equation along flow lines and in suitable gauge induced spinor field modes are constants along the flow lines of Kähler current. The generalized eigenvalue equation at wormhole throats and 3-D ends of the space-time surface at light-like boundaries of the causal diamond (CD) can be also solved exactly if one assumes that the extremals of Chern-Simons action are in question: this is indeed required by the effective 2-dimensionality of 3-surfaces.
The surprise was that continuity of the interior spinor field at ends gives boundary conditions which can be solved only for a discrete subset of flow lines with same length in the effective metric defined by the modified Dirac operator. Therefore one must restrict the modes to discrete to the strands of (number theoretical) braids with strands identified as flow lines of Kähler magnetic field. Therefore the notion of number theoretical braids comes out from the theory automatically and each generalized eigen value of the modified Dirac operator possibly together with its harmonics correspond to one particular number theoretic braid.
- If the number of the generalized eigenvalues is infinite as the naive expectation is (allowing all harmonics of the basic pseudo-mass) then Dirac determinant diverges if calculated as the product of the eigenvalues and one must calculate it by using some kind of regularization procedure. Zeta function regularization is the natural manner to do this and reduces to Riemann zeta function. One however ends up with concrete vision how the regularization could be avoided and a connection with infinite primes. In fact, the manifestly finite option and the option involving zeta function regularization give Kähler functions differing only by a scaling factor which is very transcdental number log(2π)/2 = .9184 (the negative of the derivative of Riemann zeta at origin). Only the manifestly finite option which does not allow harmonics of the basic pseudo-momentum satisfies number theoretical constraints coming from p-adicization.
An explicit expression for the Dirac determinant in terms of geometric data of the orbit of the partonic 2-surface emerges and is rational function of WCW coordinates defined by p1/2multiples of the minimal length Lmin of the braid strand in the effective metric defined by the modified gamma matrices.
- The challenge is to understand which p1/2 multiples of the minimum length Lmin can define allowed pseudo-mass scales giving rise to their own braids. Arithmetic quantum field theory defined by infinite primes has been already earlier proposed to characterize both quantum number spectrum and space-time surfaces and rather detailed proposals have emerged. The lines (that is light-like 3-surfaces) of the generalized Feynman graphs are geometrically characterized by infinite primes decomposing to hyper-octonionic finite primes in turn having canonical representatives as hyper-complex primes defining basic integer valued pseudo-momenta in M2.
The selection rules correlating the geometries of the lines of the generalized Feynman graphs corresponds to the conservation of the ∑ nilog(pi) of the number theoretic momenta assignable to sub-braids corresponding to different primes pi assignable to the orbit of parton. This conforms with the vision that infinite primes indeed characterize the geometry of light-like 3-surfaces and therefore also of space-time sheets. Also the proposed connection of infinite primes with hierarchy of Planck constants finds additional support and one ends up with a very concrete interpretation of infinite primes both at the level of space-time geometry and at the level of quantum states.
- The generalized eigenvalues of the modified Dirac operator - pseudo momenta- are restricted to M2 plane of M4 (number theoretic interpretation is as commutating hypercomplex plane of hyper-octonions). Pseudo-masses are proportional 1/pi1/2, where pi are the primes appearing in the definition of the p-adic prime. The resemblance with p-adic length scale hypothesis is not an accident and there are good arguments suggesting that the p-adic length scale hypothesis for pseudo-momenta is equivalent with the p-adic length scale hypothesis for real momenta. The interpretation as analogs of Higgs vacuum expectation values makes sense and is consistent with p-adic length scale hypothesis and p-adic mass calculations.
- The conjecture for the expression of the Kähler action of CP2 vacuum extremal in terms of primes deduced by heuristic arguments for 15 years ago and leading also to a formula for gravitational coupling constant and Kähler coupling strength is consistent with the formula for the Kähler function in terms of Dirac determinant. Therefore the Dirac determinant formula survives the killer test.
All this progress during last two months meaning reducing a large number of separate ideas to consequences of very few basic hypothesis is essentially due to single basic idea, namely the reduction of quantum TGD to almost topological QFT guaranteed by the Beltrami ansatz for field equations and by the weak form of electric-magnetic duality. Two other great steps of progress were the realization that the hierarchy of Planck constants follows from basic quantum TGD and the realization that generalized Feynman diagrams are manifestly finite since in zero energy ontology virtual particles can be regarded as on mass shell particles with same or opposite signs of on mass shell momenta.
For detailed arguments see the article Weak form of electric-magnetic duality and its implications.