- Conservative approach takes some axiomatics as a starting point and deduces theorems. In the highly competitive academic environment this approach is good for survival. The best juggler wins but theorems about physics of say K3 surface are not very useful physically.
- Second approach might be called radical. The rebel tries to detect the weak points of existing paradigms and propose new visions generalizing the old ones. This approach is intellectually extremely rewarding but does not lead to academic success. There is a continual battle between the two approaches and the first approach has been the winner for about four decades since the establishment of standard model (and perhaps has lead to the recent dead end).
The rough physical idea was that in classical theory space-times are 4-surfaces in H=M4× CP2 satisfying some variational principle: this allows to lift Poincare invariance to the level of H have Poincare charges as Noether charges so that the energy problem of general relativity is circumvented. My personal conviction is that the loss of Noether charges is the deep reason for the failure to quantize general relativity: the success of string models would reflect the fact that in string models Poincare invariance is obtained.
The challenge is to quantize this theory.
- I of course started by trying to apply canonical quantisation taking GRT as a role model. After its failure I tried path integral in fashion at that time: every young theoretician wanted to perform a really difficult path integral explicitly and become the hero! It turned out that these methods failed completely by the non-linearity of any general coordinate invariant variational principle dictating the dynamics of space-time surface.
I had good luck since this failure forced me quite soon to deeper waters: at this point my path deviated radically from that still followed by colleagues. Note that canonical quantization relies also on Newtonian time as also the notion of unitary time evolution and this is conceptually highly unsatisfactory in the new framework.
- Around 1985 I indeed realized that much more radical approach is required and around 1990 I had finally solved the problem at general level. TGD must be formulated as a geometrization of not only classical physics but also of quantum theory by geometrizing the infinite-D "world of classical worlds" consisting of 3-surfaces.
WCW must be endowed with Kähler geometry - already in case of much simpler loop spaces Freed showed that Kähler geometry is unique. Physics would be unique from the mere mathematical existence of WCW geometry!
Also superstringers had this idea for some time but during the wandering in landscape they began to see the total loss of predictivity as a blessing. After LHC they speak only about formal string theory so that the conservative approach has taken the lead.
Kähler function would correspond to action for a preferred extremal of Kähler action and have interpretation as analog of Bohr orbit. Classical physics in the sense of Bohr orbitology would be exact part of quantum theory rather than a limit. This follows from general coordinate invariance (GCI) only.
- Physical states would correspond to spinor fields in WCW: for given a 3-surface they correspond to fermionic Fock states. An important point is that WCW spinor fields are formally purely classical: "quantization without quantization" would Wheeler say.
Induced spinor fields are quantized at space-time surface and the gamma matrices of WCW are linear combinations of fermionic oscillator operators so that at space-time level quantization of free fermions is needed and anticommutativity has geometric interpretation at WCW level: fermionic statistics is geometrized.
- The generalisation of S-matrix in zero energy ontology (ZEO) - also needed - is associated with the modes of WCW Dirac operator satisfying analog of Dirac equation as conditions stating supersymplectic invariance (formally analogous to Super Virasoro conditions in string models). One just solves the free Dirac equation in WCW!
Childishly simple at the general conceptual level but the practical construction of the S-matrix as coefficients from zero energy state identified as a bilinear formed by positive and negative energy states is a really hard task!
- General Coordinate Invariance is what led to the idea that WCW geometry assigns to 3-surface (more precisely, to a a pair 3-surfaces at boundaries of causal diamond) a space-time surface unique space-time surface as preferred
extremal of Kähler action (most plausible guess). Kähler function is the value of Kähler action for regions of space-time surface with Euclidian signature of induced metric and Minkowskian regions give imaginary contribution as analog of QFT action. Mathematically ill-defined path integral transforms to a well-defined functional integral over 3-surfaces.
- Infinite dimensional WCW geometry requires maximal symmetries. Four-dimensionality of M4 factor and space-time surface is necessary condition since 3-D light-like surfaces by their metric 2-dimensionality allow ab extension of ordinary 2-D conformal invariance. M4 and CP2 are unique in that they allow twistor space with Kähler structure. The twistorial lift of TGD predicts cosmological term as an additional term besides Kähler action in the dimensional reduction of 6-D Kähler action and also predicts the value of Planck length as radius of the sphere of twistor bundle having sphere as fiber and space-time surface as base. I am still not sure whether twistorial lift is really necessary or whether cosmological constant and gravitational constant emerge in some other manner.
- Infinite number of conditions stating the vanishing of classical super-symplectic Noether charges (not all of them) is satisfied and guarantee strong form of holography (SH) implied by strong form of general coordinate invariance (SGCI): space-time dynamics is coded by 2-D string world sheets and partonic 2-surfaces serving as "space-time genes": a close connection with string models is obtained. These conditions are satisfied also by quantal counterparts of super-symplectic charges and there is a strong formal resemblance with super Virasoro conditions. These conditions include also the analog of Dirac equation in WCW.
- Feynman/twistor/scattering diagrammatics (last one is the best choice) is something real and must be generalized and diagrams have a concrete geometric and topological interpretation at space-time level: it must be emphasized and thickly underlined that also this is something completely new. By strong form of holography this diagrammatics reduces to a generalization of string diagrammatics and 8-D generalization of twistor diagrammatics based on octonionic representation of sigma matrices is highly suggestive. Recall that massive particles are a nuisance for twistor approach and massive momenta in 4-D sense would correspond to massless momenta in 8-D sense.
Twistor approach suggests a Yangian generalization of super-symplectic symmetries with polylocal generators (polylocal with respect to partonic 2-surfaces). Yangian symmetries should dictate the S-matrix as in twistor Grassmann approach. The symmetries are therefore monstrous and the formidable challenge is to understand them mathematically.
- One has half dozen of new deep principles: physics as WCW spinor geometry and quantum physics formulated in terms of modes of classical WCW spinor fields so that the only genuinely quantal aspect of quantum theory would be state function reduction; SGCI implying SH; maximal isometry group of WCW to guarantee existence of its Kähler geometry and its extension to Yangian symmetry, ZEO; number theoretic vision and extension of physics to adelic physics by number theoretic universality; hierarchy of Planck constants assignable to the fractal hierarchy of super-symplectic algebras and inclusions of hyperfinite factors of type II1.
I also learned that one cannot proceed without a proper quantum measurement theory and this led to a theory of consciousness and applications in quantum biology: the most important implication is the new view about the relationship of experienced and geometric time.
What remarkable, that all this has followed during almost four decades from a childishly simple observation: the notion of energy in GRT is ill-defined since Noether theorem does not apply, and one can cure the situation by assuming that space-times are 4-surfaces in space-time having M4 as Cartesian factor. This should show how important it is for a theoretician to be keenly aware of the weak points of the theory.
To get some perspective a comparison with what most mathematical physicists are doing is in order. They start from a given axiomatics since they want to deduce theorems to be the most skillful juggler and get the reward. My goals have been different. I have used these years to identify the axioms of a theory allowing to lift quantum theory to a real unified theory of fundamental interactions. I have been also forced to use every bit of experimental information whereas mathematical physicist could not care less of anomalies.
For a summary of earlier postings see Latest progress in TGD.