Challenging some details of the recent view of TGD

The development of the mathematical TGD has been a sequence of simplifications and generalizations. Holography = holomorphy vision removes path integral from quantum physics and together with the number theoretic vision might make the bosonic action unnecessary. This means that this vision allows us to solve field equations explicitly and the solution does not depend on the bosonic action.

TGD allows to get rid of primary bosonic fields and fermions are free free fermions at the level of the imbedding space and their localization to space-time surfaces makes them interaction. Pair creation is made possible by the presence of exotic smooth structures possible only in 4-D space-time.

This however leads to a problem with the sign of energy. This problem disappears when one realizes that fundamental fermions can have tachyonic momenta and that only the physical l states as their bound states, which are Galois singlets, have non-negative mass squared and positive energy.

Could the classical bosonic action completely disappear from TGD?

Number theoretic vision of TGD and holography = holomorphy principle (see this and this) forces to challenge the necessity of the classical bosonic action.

1. Any general coordinate action defining the K\"ahler function K and constructible in terms of the induced geometry gives the same minimal space-time surfaces as extremals and only the boundaries and partonic orbits depend on the action since the boundary conditions stating conservation laws depend on the action. Spinor lift suggests K\"ahler action for the 6-D twistor surfaces as a unique action principle. But is it necessary?
2. The conjecture exp(K) ∝ Dⁿ, n an integer, or its generalization to exp(K) ∝ (DD^*)ⁿ, where D is a product of discriminants for the polynomials assignable to partonic 2-surfaces define a discrete set of points as their roots, would allow to express vacuum functional completely in terms of number theory. Coupling parameters would be present but evolve in such a way that the condition would hold true.
3. The discriminant D is defined also when the roots assignable to the partonic 2-surfaces are real or even complex numbers. This would conform with the strong form of holography. One could get completely rid of the bosonic action principle. The holomorphy = holography principle would automatically give the non-linear counterpart of massless fields satisfied by the space-time surfaces as minimal surfaces. Could the classical action completely disappear from the theory?

Could the fermionic interaction vertices be independent of the bosonic action principle?

Could the interaction vertices for fermions be independent of the bosonic action principle?

1. The long-held idea is (see this, this, and this), the vertices appearing in the scattering amplitudes are determined by the modified Dirac equation (see this) determined by the bosonic action associated with the partonic orbits as couplings to the induced gauge potentials. Twistor lift suggests that this action contains volume term and K\"ahler action.

   But is the modified Dirac action necessary or even physically plausible? The problem is that for a general bosonic action the modified gamma matrices, defined in terms of canonical momentum currents, do not commute to the induced metric unlike the modified Dirac action determined by the mere volume term of the bosonic action. This led to the proposal that this option, consistent also with the fact that, irrespective of the bosonic action, space-time surfaces are minimal surfaces outside singularities at which generalized holomorphy fails, is more plausible.

2. Fermion pair creation (and emission of bosons as Galois singlet bound states of fermions and antifermions is possible only for 4-D space-time surfaces. The existence of exotic smooth structures in dimension D=4 (see this) makes possible pair creation vertices (see this and this). A given exotic smooth structure corresponds to the unique standard ordinary smooth structure with defects and vertices would correspond to defects at which the fermion line turns backwards in time. The defects would be associated with partonic 2-surfaces at which the generalized holomorphy of the function pair (f₁,f₂) with respect to generalized complex coordinates of H (one of them is hypercomplex coordinate) fails, perhaps only at the defect.
3. There is an objection against this proposal. The creation of fermion pairs with opposite sign of single fermionic energy suggests that a given light-like boundary of CD can contain fermions with both signs of energy. This does not conform with the assumption that the sign of the single particle energy is fixed and opposite for the opposite boundaries of CD. Should one only require that the total energy has a fixed sign at a given boundary of the CD?

   Could one only require that the sign of the energy is fixed only for physical states formed as many-fermions states and identified as Galois singlets and that the physical states can also contain negative energy tachyonic fermions or antifermions. Could this make sense mathematically?

Extension of the fermionic state space to include tachyonic fundamental fermions as analogs of virtual fermions

I recently received from Paul Kirsch a link to an interesting article about the possibility to describing tachyons in a mathematically consistent way (see this). The basic problem is that for tachyons the number of positive energy particles is not well-defined since Lorentz transformation can change positive energy tachyons to negative energy tachyons and vice versa. The proposed solution of the problem is the doubling of the Hilbert space which includes both incoming and outgoing states. To me this looks like a mathematically sensible idea and might make sense also physically.

Surprisingly, this proposal has a rather concrete connection with zero energy ontology (ZEO).

1. In the simplest formulation of ZEO (see this and this), the fermionic vacua at the passive resp. active boundaries of CD correspond to the fermionic vacua annihilated by annihilation operators resp. creation operators as their hermitian conjugates. In the standard QFT only the second vacuum is accepted and this allows only a single arrow of geometric time.
2. ZEO allows both arrows and a given zero energy state is a state pair for which the fermionic state at the passive boundary of CD remains fixed during the sequence of small state function reductions (SSFRs) and corresponding time evolution which lead to the increase of CD in a statistical sense. The state at the active boundary changes and this corresponds to the subjective time evolution of a conscious entity, self. SSFRs are the TGD counterparts of repeated measurements for observables which commute with the observables whose eigenstates the states at the passive boundary are.
3. The doubled state space is highly analogous to the space of fermionic states in ZEO involving positive and negative energy physical particles at the opposite boundaries of CD. If one also allows single fermion tachyonic states then one could have fermions with wrong sign of energy at a given boundary of CD. If bosons correspond to fermion-antifermion pairs such that either fermion or antifermion is tachyonic, one obtains boson emission and physical bosons can have correct sign of mass squared. In the vertex identified as a defect of the standard spinor structure, either fermion or antifermion would be tachyonic. Since several vertices involving the change of the sign of the fermion or antifermion momentum are possible, outgoing physical fermions and antifermions with a correct sign or energy can be produced. Recall that both the physical leptons and quarks involve fermion-antifermion pairs in the recent picture based on closed monopole flux tubes associated with a pair of Minkowskian space-time sheets.
4. Tachyonic single fundamental fermion states (quarks or leptons) are natural in the number theoretic vision of TGD (see this and this). The components of the fermionic momenta for a given extension of rationals are algebraic integers and mass squared for them can be tachyonic. These states are analogs of virtual fermions of the standard QFT which also can have tachyonic momenta. Physical states are assumed to be Galois singlets so that the total momentum for a bound state of fermions and antifermions has integer valued components and mass squared is integer. The condition that mass squared energy have a fixed sign for the physical states at a given boundary of the CD is natural and has been made.

See the article TGD as it is towards end of 2024: part II or the 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.