**Authors:** Matti Pitkänen

This book is devoted to a detailed representation of the recent state of quantum TGD. </p><p> The first part of the book summarizes quantum TGD in its recent form. </p><p> <OL> <LI> General coordinate invariance and generalized super-conformal symmetries are the basic symmetries of TGD and Equivalence Principle can be generalized using generalized coset construction. <LI> In zero energy ontology the basis of classical WCW spinors fields forms unitary U-matrix having M-matrices as its orthogonal rows. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy states. M-matrix is a product of a hermitian density matrix and unitary S-matrix commuting with it. The hermitian density matrices define infinite-dimensional Lie-algebra extending to a generalization of Kac-Moody type algebra with generators defined as products of hermitian density matrices and powers of S-matrix. Yangian type algebra is obtained if only non-negative powers of S are allowed. The interpretation is in terms of the hierarchy of causal diamonds with size scales coming as integer multiples of CP<sub>2</sub> size scale. Zero energy states define their own symmetry algebra. For generalized Feynman diagrams lines correspond to light-like 3-surfaces and vertices to 2-D surfaces. <LI> Finite measurement resolution realized using fractal hierarchy of causal diamonds (CDs) inside CDs implies a stringy formulation of quantum TGD involving replacement of 3-D light-like surfaces with braids representing the ends of strings. Category theoretical formulation leads to a hierarchy of algebras forming an operad. <LI> Twistors emerge naturally in TGD framework and several proposal for twistorialization of TGD is discussed in two chapters devoted to the topic. Twistorial approach combined with zero energy ontology, bosonic emergence, and the properties of the Chern-Simons Dirac operator leads to the conjecture that all particles -also string like objects- can be regarded as bound states of massless particles identifiable as wormhole throats. Also virtual particles would consist of massles wormhole throats but bound state property is not assumed anymore and the energies of wormhole throats can have opposite signs so that space-like momentum exchanges become possible. This implies extremely strong constraints on loop momenta and manifest finiteness of loop integrals. </p><p> An essential element of the formulation is exact Yangian symmetry obtained by replacing the loci of multilocal symmetry generators of Yangian algebra with partonic 2-surfaces so that conformal algebra of Minkowski space is extened to infinite-dimensional algebra bringing in also the conformal algebra assigned to the partonic 2-surfaces. Yangian symmetry requires the vanishing of both UV and IR divergences achieved if the physical particles are bound states of massless wormhole throats. </p><p> Rather general arguments suggest the formulation of TGD in terms of holomorphic 6-surfaces in the product CP<sub>3</sub>× CP<sub>3</sub> of twistor spaces leading to a unique partial differential equations determining these surfaces in terms of homogenous polynomials of the projective complex coordinates of the two twistor spaces. </OL> </p><p> Second part of the book is devoted to hyper-finite factors and hierarchy of Planck constants. </p><p> <OL> <LI> The Clifford algebra of WCW is hyper-finite factor of type II<sub>1</sub>. The inclusions provide a mathematical description of finite measurement resolution. The included factor is analogous to gauge symmetry group since the action of the included factor creates states not distinguishable from the original one. TGD Universe would be analogous to Turing machine able to emulate any internally consistent gauge theory (or more general theory) so that finite measurement resolution would provide TGD Universe with huge simulational powers. <LI> In TGD framework dark matter corresponds to ordinary particles with non-standard value of Planck constant. The simplest view about the hierarchy of Planck constants is as an effective hierarchy describable in terms of local, singular coverings of the imbedding space. The basic observation is that for Kähler action the time derivatives of the imbedding space coordinates are many-valued functions of canonical momentum densities. If all branches for given values of canonical momentum densities are allowed, one obtains the analogs of many-sheeted Riemann surfaces with each sheet giving same contribution to the Kähler action so that Planck constant is effectively a multiple of the ordinary Planck constant. Dark matter could be in quantum Hall like phase localized at light-like 3-surfaces with macroscopic size and analogous to black-hole horizons. </OL>

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