One can however argue that this argument is quantum field-theoretic and does not apply in TGD since holography changes the very concept of perturbation theory. There is no path integral to worry about. Path integral is indeed such a fundamental concept that one expects it to have some approximate counterpart also in the TGD Universe. Bohr orbits are not completely deterministic: could the sum over the Bohr orbits however translate to an approximate description as a path integral at the QFT limit? The dynamics of light-like partonic orbits is indeed non-deterministic and could give rise to an analog of path integral as a finite sum.
- The dynamics implied by Chern-Simons-Kähler action assignable to the partonic 3-surface with light-one coordinate in the role of time, is very topological in that the partonic orbits is light-like 3-surface and has 2-D CP2 and M4 projections unless the induced M4 and CP2 Kähler forms sum up to zero. The light-likeness of the projection is a very loose condition and and the sum over partonic orbits as possible representation of holographic data analogous to initial values (light-likeness!) is therefore analogous to the sum over all paths appearing as a representation of Schrödinger equation in wave mechanics.
One would have an analog of 1-D QFT. This means that the infinities of quantum field theories are absent but for a large enough coupling strength g2/4πℏ the perturbation series fails to converge. The increase of heff would resolve the problem. For instance, Dirac equation in atomic physics makes unphysical predictions when the value of nucler charge is larger than Z≈ 137.
- I have also considered a discretized variant of this picture. The light-like orbits would consist of pieces of light-like geodesics. The points at which the direction of segment changes would correspond to points at which energy and momentum transfer between the partonic orbit and environment takes place. This kind of quantum number transfer might occur at least for the fermionic lines as boundaries of string world sheets. They could be described quantum mechanically as interactions with classical fields in the same way as the creation of fermion pairs as a fundamental vertex (see this). The same universal 2-vertex would be in question.
- What is intriguing, that the light-likeness of the projection of the CP2 type extremals in M4 leads to Virasoro conditions assignable to M4 coordinates and this eventually led to the idea of conformal symmetries as isometries as WCW. In the case of the partonic orbits, the light-like curve would be in M4× CP2 but it would not be surprising if the generalization of the Virasoro conditions would emerge also now.
One can write M4 and CP2 coordinates for the light-like curve as Fourier expansion in powers of exp(it), where t is the light-like coordinate. This gives hk= ∑ hkn exp(int). If the CP2 projection of the orbits of the partonic 2-surface is geodesic circle, CP2 metric skl is constant, the light-likeness condition hkl∂thk∂lthl=0 gives Re(hkl∑m hkn-mhlm=0). This does not give Virasoro conditions.
The condition d/dt(hkl∂thk∂thl=0)=0 however gives the standard Virasoro conditions stating that the normal ordered operators Ln= Re(hkl∑m (n-m) hkn-mhlm) annihilate the physical states. What is interesting is that the latter condition also allows time-like (and even space-like) geodesics.
- Could massivation mean a failure of light-likeness? For piecewise light-like geodesics the light-likeness condition would be true only inside the segments. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals. For piecewise light-like geodesics the condition would be trivially true inside the segments and therefore discretized. By taking Fourier transform one expects to obtain Virasoro conditions with a cutoff analogous to the momentum cutoff in condensed matter physics for crystals.
- In TGD the Virasoro, Kac-Moody algebras and symplectic algebras are replaced by half-algebras and the gauge conditions are satisfied for conformal weights which are n-multiples of fundamentals with with n larger than some minimal value. This would dramatically reduce the effects of the non-determinism and could make the sum over all paths allowed by the light-likeness manifestly finite and reduce it to a sum with a finite number of terms. This cutoff in degrees of freedom would correspond to a genuinely physical cutoff due to the finite measurement resolution coded to the number theoretical anatomy of the space-time surfaces. This cutoff is analogous to momentum cutoff and could at the space-time picture correspond to finite minimum length for the light-like segments of the orbit of the partoic 2-surface.
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.