Some problems of TGD as almost-topological QFT and their resolution

There are some problems involved with the precise definition of the quantum TGD as an almost-topological QFT at the partonic level and the resolution of these problems leads to new insights about the relationship between gravitational and inertial mass and a more precise definition of quantum classical correspondence.

1. Three problems

The proposed view about partonic dynamics is plagued by three problems.

1. The definition of supercanonical and super-Kac-Moody charges in M⁴ degrees of freedom poses a problem. These charges are simply vanishing since M⁴ coordinates do not appear in field equations.

2. Classical field equations for the C-S action imply that this action vanishes identically which would suggest that the dynamics does not depend at all on the value of k. The central extension parameter k determines the over-all scaling of the eigenvalues of the modified Dirac operator. 1/k- scaling occurs for the eigenvalues so that Dirac determinant scales by a finite power k^N if the number N of the allowed eigenvalues is finite for the algebraic extension considered. A constant Nlog(k) is added to the Kähler function and its effect seems to disappear completely in the normalization of states.

3. The general picture about Jones inclusions and the possibility of separate Planck constants in M⁴ and CP₂ degrees of freedom suggests a close symmetry between M⁴ and CP₂ degrees of freedom at the partonic level. Also in the construction of the geometry for the world of classical worlds the symplectic and Kähler structures of both light-cone boundary and CP₂ are in a key role. This symmetry should be somehow coded by the Chern-Simons action.

The first thing to come in mind is the extension of the Kähler form defined at the light-cone boundary to the entire M⁴ as a self dual or antiself-dual instanton. This option looks at first unattractive for several reasons. The Maxwell field in question is not covariantly constant and does not define a generalization of the CP₂ Kähler form. A charged Dirac monopole would be in question. Although its action would vanish, the energy density given by the sum of electro-static and magnetostatic energies would break Lorentz invariance of the Robertson-Walker cosmologies unless one extends the configuration space by adding to it light cones with all Lorentz transforms of the instanton field. Something more elegant is perhaps needed. I will discuss this option in a separate posting and show that it it does not exclude the option discussed here and could form part of the full story.

2. A possible resolution of the problems

A possible cure to the above described problems is based on the modification of Kähler gauge potential by adding to it a gradient of a scalar function Φ with respect to M⁴ coordinates.

1. This implies that super-canonical and super Kac-Moody charges in M⁴ degrees of freedom are non-vanishing.

2. Chern-Simons action is non-vanishing if the induced CP₂ Kähler form is non-vanishing. If the imaginary exponent of C-S action multiplies the vacuum functional, the presence of the central extension parameter k is reflected in the properties of the physical states.

3. The function Φ could code for the value of k(M⁴) via a proportionality constant

   Φ= (k(M⁴)/k(CP₂))× Φ₀ ,

   Here k(CP₂) is the central extension parameter multiplying the Chern-Simons action for CP₂ Kähler gauge potential. This tricks does just what is needed since it multiplies the Noether currents and super currents associated with M⁴ degrees of freedom with k(M⁴) instead of k(CP₂).

The obvious breaking of U(1) gauge invariance looks strange at first but it conforms with the fact that in TGD framework the canonical transformations of CP₂ acting as U(1) gauge symmetries do not give to gauge degeneracy but to spin glass degeneracy since they act as symmetries of only vacuum extremals of Kähler action.

3. How to achieve Lorentz invariance?

Lorentz invariance fixes the form of function Φ uniquely as the following argument demonstrates.

1. Poincare invariance would be broken in any case for a given light-cone in the decomposition of the configuration space to sub-configuration spaces associated with light-cones at various locations of M⁴ but since the functions Φ associated with various light cones would be related by a translation, translation invariance would not be lost.

2. The selection of Φ should not break Lorentz invariance. If Φ depends on the Lorentz proper time a only, this is achieved. Momentum currents would be proportional to m^k and become light like at the boundary of the light-cone. This fits very nicely with the interpretation that the matter emanates from the tip of the light cone in Robertson-Walker cosmology.

Lorentz invariance poses even stronger conditions to Φ.

1. Partonic four-momentum defined as Chern-Simons Noether charge is definitely not conserved and must be identified as gravitational four-momentum whose time average corresponds to the conserved inertial four-momentum assignable to the Kähler action (see this and this). This identification is very elegant since also gravitational four-momentum is well-defined although not conserved.

2. Lorentz invariance implies that mass squared is constant of motion. It is interesting to look what expression for Φ results if the gravitational mass defined as Noether charge for C-S action is conserved. The components of four-momentum for Chern-Simons action are given by

   P^k=(∂ L_C-S/∂ (∂_αa)) m^kl∂_m^la .

   Chern-Simons action is proportional to A_α=A_a∂_αa so that one has

   P^k propto ∂_aΦ ∂_m^ka=∂_aΦ m^k/a.

   The conservation of gravitational mass would give Φ propto a. Since CP₂ projection must be 2-dimensional, M⁴ projection is 1-dimensional so that mass squared is indeed conserved.

   Thus one can write

   Φ= (k(M⁴)/k(CP₂))× x×a/R,

   where R the radius of geodesic sphere of CP₂ and x a numerical constant which could be fixed by quantum criticality of the theory. Chern-Simons action density does not depend on a for this choice.

3. A rather strong prediction is that only homologically charged partonic 3-surfaces can carry gravitational four-momentum. For CP₂ type extremals, ends of cosmic strings, and wormhole contacts the non-vanishing of homological charge looks natural. In fact, for wormhole contacts 3-D CP₂ projection suggests itself and is possible only if one allows also quantum fluctuations around light-like extremals of Chern-Simons action. The interpretation could be that for a vanishing homological charge boundary conditions force X⁴ to approach vacuum extremal at partonic 3-surfaces.

4. Comment about quantum classical correspondence

The proposed general picture allows to define the notion of quantum classical correspondence more precisely. The identification of the time average of the gravitational four-momentum for C-S action as a conserved inertial four-momentum associated with the Kähler action at a given space-time sheet of a finite temporal duration (recall that we work in the zero energy ontology) is the most natural definition of the quantum classical correspondence and generalizes to all charges.

In this framework the identification of gravitational four-momentum currents as those associated with 4-D curvature scalar for the induced metric of X⁴ could be seen as a phenomenological manner to approximate partonic gravitational four-momentum currents using macroscopic currents, and the challenge is to demonstrate rigorously that this description emerges from quantum TGD.

For instance, one could require that at a given moment of time the net gravitational four-momentum of Int(X⁴) defined by the combination of the Einstein tensor and metric tensor equals to that associated with the partonic 3-surfaces. This identification, if possible at all, would certainly fix the values of the gravitational and cosmological constants and it would not be surprising if cosmological constant would turn out to be non-vanishing.

The chapter Construction of Quantum Theory: Symmetries of "Towards S-Matrix" contains this piece of text too.