My contribution to the Unified Theories conference represents a brief summary of various competing visions about the basic principles of quantum Topological Geometrodynamics (TGD) and about tensions between them and with bottom-up approach to physics with emphasis on the recent developments occurred after the first 2006 conference.
1. TGD as a unified theory There are several strongly overlapping approaches to quantum TGD and the process of writing the summary helped to see new connections between them.
- The mathematical realization of Equivalence Principle is the weak point of General Relativity and led to the proposal that physical space-times are identifiable as 4-D surfaces in 8-D imbedding space M4×CP2. This choice explains standard model quantum numbers and provides a geometrization of classical fields. This does not resolve completely interpretational problems and the outcome is what I call zero energy ontology. Quantum states are identified as events: the initial resp. final state of event is identified as positive resp. negative energy part of zero energy state (all conserved quantum numbers vanish) residing at light-like boundaries of a causal diamond consisting of future and past directed light-cones. The basic principle is that everything is creatable from vacuum by intentional action in quantum jump: there is no conflict with basic conservation laws.
- Physics as the classical spinor field geometry of the world of classical worlds (configuration space CH) identified as 3-surfaces in M4×CP2 is the oldest and best developed approach to TGD and means a generalization of Einstein's program of geometrizing classical physics so that it applies to the entire quantum physics. This approach geometrizes the basic concepts of quantum field theory: hermitian conjugation is realized in terms of Kähler metric, general coordinate invariance implies Bohr quantized version of classical physics as an exact part of quantum theory, quantum criticality condition fixes the values of the basic coupling parameter of the theory as an analog of critical temperature, quantum states correspond to spinor fields in CH, and fermionic statistics is realized in terms of Clifford algebra of CH. Configuration space Clifford algebra, and by supersymmetry presumably also operators creating CH spinor fields, correspond as algebra to a direct sum of hyperfinite factors of type II1 (HFF), a von Neumann algebra unifying large number of basic mathematical structures of modern physics (quantum groups, non-commutative physics, conformal field theories, topological quantum field theories for classifying knots and braids,...).
- Parton level formulation of quantum TGD as an almost topological quantum field theory (almost TQFT) using light-like 3-surfaces as fundamental objects allows a detailed understanding of super-conformal symmetries generalizing those of super string models. The important attribute älmost" is due to the fact that by light-likeness condition for 3-surfaces the notions of metric, length, and mass leak in: otherwise one would have mere TQFT not very interesting physically. A category theoretical interpretation of M-matrix generalizing S-matrix as a functor is possible. This picture has tight connections to the physics as configuration space geometry approach since the identification of light-like 3-surfaces as basic objects is implied by general coordinate invariance one hand and almost TQFT approach allows to construct CH geometry.
- Physics as generalized number theory represents a third vision about TGD. There are three threads in this vision.
- Number theoretic universality meaning a fusion of real and p-adic physics to single coherent whole as a completion of rational physics - much like rational numbers are completed to reals and various p-adic number fields - forces a generalization of number concept by fusing reals and various p-adic number fields together along rationals and common algebraic numbers to form a book like structure. This means a similar generalization of imbedding space concept. p-Adic space-time sheets are identified as correlates for intentions and actions, the "mind stuff" of Descartes. A formulation of quantum TGD in terms of so called number theoretic braids emerges utilizing only data from discrete set of algebraic points with interpretation in terms of finite resolution of cognition, sensory experience, and quantum measurement.
- Standard model symmetries allow interpretation in terms of the symmetries of classical number fields (reals, complex numbers, quaternions, octonions) and a number theoretic interpretation of the imbedding space M4×CP2 emerges. Associativity condition in number theoretic sense would define laws of classical and quantum TGD. The outcome is what might be called number theoretical compactification: 8-D Minkowski space M8 identified as hyper-octonions - subspace of complexified octonions - and M4×CP2 provide equivalent formulation of quantum TGD and the basic gauge condition of gauge theories and string models emerges as a purely number theoretical condition (commutativity). The space-time surface associated with a given partonic 3-surface (Bohr orbit) is uniquely determined and a remaining "must-be-true" is that it correspond to a preferred extremal of the Kähler action implied by the geometric vision.
- The notion of infinite prime (infinite only in real topology!) defines a third thread in the braid of number theoretical ideas and it is now possible to give a surprisingly realization for the number theoretic Brahman=Atman identity (algebraic holography) based on the generalization of the number concept by allowing infinite number of real units representable as ratios of infinite integers having interpretation as representations for physical states of super-symmetric arithmetic QFT. The infinitely rich number theoretic anatomy for the points of number theoretic braids allow to represent the information about zero energy states associated with given causal diamond remaining below measurement resolution as Schrödinger amplitudes in the infinite-dimensional space of real units associated with the 8 coordinates of imbedding space.
- The string model inspired extension of CH spinor fields to hyper-octonion valued conformal fields having values in HFF gives allows to deduce most of the speculative "must-be-true's" of quantum TGD. In particular, associativity and commutativity conditions imply a unique notion of number theoretic braid required by almost TQFT vision and implied by a fusion of real and p-adic physics but not uniquely.
- The idea about hierarchy of Planck constants does not represent alternative approach to TGD but a generalization of other approaches and was inspired by certain empirical facts. The hierarchy leads to a further generalization of the notion of imbedding space emerges naturally from the requirement that the choice of quantization axes has a geometric correlate at the level of imbedding space. Generalized imbedding space has a book like structure: the particles at different pages of book cannot appear in the same vertex of Feynman diagram but interact via classical gauge fields and exchange of particles experiencing hbar changing phase transition. Particles at different pages behave like dark matter with respect to each other. This weak notion of darkness is enough to explain what is known about dark matter and inspires the identification of dark matter in terms of a hierarchy of macroscopically quantum coherent phases with quantized values of Planck constant having arbitrarily large values and playing a key role, not only in biology but also in astrophysics and cosmology of TGD Universe. The hierarchy of Planck constants can be seen as necessary for the realization of quantum criticality: quantum criticality means criticality against phase transitions changing the value of the Planck constant. The generalization of imbedding space is also essential for the construction of the Kähler function of configuration space.
- A further vision about quantum TGD is that the mere finiteness of measurement resolution fixes the scattering matrix of quantum TGD. In zero energy ontology S-matrix must be generalized to M-matrix identified as time-like entanglement coefficients between positive and negative energy parts of zero energy states. M-matrix can be regarded as a "complex square root" of density matrix expressible as product of a real square root of density matrix and unitary S-matrix: thermodynamics becomes part of quantum theory. Hyper-finite factors of type II1 (HFFs) emerge naturally through Clifford algebra of CH and allow a formulation of quantum measurement theory with a finite measurement resolution. The notion of finite measurement resolution expressed in terms of inclusion of HFFs with included algebra defining the measurement resolution leads to an identification of M-matrix in terms of Connes tensor product and a simple argument shows that M-matrix is unique apart from the presence of the square root of density matrix needed by thermodynamics. Coupling constant evolution corresponds to a hierarchy of measurement resolutions and p-adic coupling constant hypothesis follows as a consequence with an additional prediction assigning to particles an additional time scale characterizing temporal distance between positive and negative energy parts of corresponding zero energy state: for electron this time scale is .1 second, fundamental biorhythm. Thus zero energy ontology implies a direct connection between elementary particle physics and biology.