- Pp. 157-232 (76)Matti Pitkanen
In this chapter the classical field equations associated with the Kahler action are studied. </p><p> 1. Are all extremals actually "preferred"? </p><p> The notion of preferred extremal has been central concept in TGD but is there really compelling need to pose any condition to select preferred extremals in zero energy ontology (ZEO) as there would be in positive energy ontology? In ZEO the union of the space-like ends of space-time surfaces at the boundaries of causal diamond (CD) are the first guess for 3-surface. If one includes to this 3-surface also the light-like partonic orbits at which the signature of the induced metric changes to get analog of Wilson loop, one has good reasons to expect that the preferred extremal is highly unique without any additional conditions apart from non-determinism of Kahler action proposed to correspond to sub-algebra of conformal algebra acting on the light-like 3-surface and respecting light-likeness. One expects that there are nite number n of conformal equivalence classes and n corresponds to n in h<sub>eff</sub> = nh. These objects would allow also to understand the assignment of discrete physical degrees of freedom to the partonic orbits as required by the assignment of hierarchy of Planck constants to the non-determinism of Kahler action. </p><p> 2. Preferred extremals and quantum criticality </p><p> The identification of preferred extremals of Kahler action defining counterparts of Bohr orbits has been one of the basic challenges of quantum TGD. By quantum classical correspondence the non-deterministic space-time dynamics should mimic the dissipative dynamics of the quantum jump sequence. </p><p> The space-time representation for dissipation comes from the interpretation of regions of space-time surface with Euclidian signature of induced metric as generalized Feynman diagrams (or equivalently the light-like 3-surfaces de ning boundaries between Euclidian and Minkowskian regions). Dissipation would be represented in terms of Feynman graphs representing irreversible dynamics and expressed in the structure of zero energy state in which positive energy part corresponds to the initial state and negative energy part to the final state. Outside Euclidian regions classical dissipation should be absent and this indeed the case for the known extremals. </p><p> The non-determinism should also give rose to space-time correlate for quantum criticality. The study of Kahler-Dirac equations suggests how to de ne quantum criticality. Noether currents assignable to the Kahler-Dirac equation are conserved only if the first variation of Kahler-Dirac operator D<sub>K</sub> defined by Kahler action vanishes. This is equivalent with the vanishing of the second variation of Kahler action - at least for the variations corresponding to dynamical symmetries having interpretation as dynamical degrees of freedom which are below measurement resolution and therefore efectively gauge symmetries.....