What's new inTGD as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2008 
Zero energy ontology, self hierarchy, and the notion of time
In the previous posting I discussed the most recent view about zero energy ontology and padicization program. One manner to test the internal consistency of this framework is by formulating the basic notions and problems of TGD inspired quantum theory of consciousness and quantum biology in terms of zero energy ontology. I have discussed these topics already earlier but the more detailed understanding of the role of causal diamonds (CDs) brings many new aspects to the discussion. In consciousness theory the basic challenges are to understand the asymmetry between positive and negative energies and between two directions of geometric time at the level of conscious experience, the correspondence between experienced and geometric time, and the emergence of the arrow of time. One should also explain why human sensory experience is about a rather narrow time interval of about .1 seconds and why memories are about the interior of much larger CD with time scale of order life time. One should also have a vision about the evolution of consciousness takes place: how quantum leaps leading to an expansion of consciousness take place. Negative energy signals to geometric past  about which phase conjugate laser light represents an example  provide an attractive tool to realize intentional action as a signal inducing neural activities in the geometric past (this would explain Libet's classical findings), a mechanism of remote metabolism, and the mechanism of declarative memory as communications with the geometric past. One should understand how these signals are realized in zero energy ontology and why their occurrence is so rare. In the following my intention is to demonstrate that TGD inspired theory of consciousness and quantum TGD proper indeed seem to be in tune and that this process of comparison helps considerably in the attempt to develop the TGD based ontology at the level of details.
1 Causal diamonds as correlates for selvesQuantum jump as a moment of consciousness, self as a sequence of quantum jumps integrating to self, and self hierarchy with subselves experienced as mental images, are the basic notion of TGD inspired quantum theory of consciousness. In the most ambitious program self hierarchy reduces to a fractal hierarchy of quantum jumps within quantum jumps. It is natural to interpret CD:s as correlates of selves. CDs can be interpreted in two manners: as subsets of the generalized imbedding space or as sectors of the world of classical worlds (WCW). Accordingly, selves correspond to CD:s of the generalized imbedding space or sectors of WCW, literally separate interacting quantum Universes. The spiritually oriented reader might speak of Gods. Subselves correspond to subCD:s geometrically. The contents of consciousness of self is about the interior of the corresponding CD at the level of imbedding space. For subselves the wave function for the position of tip of CD brings in the delocalization of subWCW. The fractal hierarchy of CDs within CDs defines the counterpart for the hierarchy of selves: the quantization of the time scale of planned action and memory as T(k) = 2^{k}T_{0} suggest an interpretation for the fact that we experience octaves as equivalent in music experience.
2. Why sensory experience is about so short time interval?CD picture implies automatically the 4D character of conscious experience and memories form part of conscious experience even at elementary particle level: in fact, the secondary padic time scale of electron is T=1 seconds defining a fundamental time scale in living matter. The problem is to understand why the sensory experience is about a short time interval of geometric time rather than about the entire personal CD with temporal size of order lifetime. The obvious explanation would be that sensory input corresponds to subselves (mental images) which correspond to CD:s with T(127) @ .1 s (electrons or their Cooper pairs) at the upper lightlike boundary of CD assignable to the self. This requires a strong asymmetry between upper and lower lightlike boundaries of CD:s.
On basis of these arguments it seems that the basic conceptual framework of TGD inspired theory of consciousness can be realized in zero energy ontology. Interesting questions relate to how dynamical selves are.
3. New view about arrow of timePerhaps the most fundamental problem related to the notion of time concerns the relationship between experienced time and geometric time. The two notions are definitely different: think only the irreversibility of experienced time and the reversibility of the geometric time and the absence of future of the experienced time. Also the deterministic character of the dynamics in geometric time is in conflict with the notion of free will supported by the direct experience. In the standard materialistic ontology experienced time and geometric time are identified. In the naivest picture the flow of time is interpreted in terms of the motion of 3D time=constant surface of spacetime towards geometric future without any explanation for why this kind of motion would occur. This identification is plagued by several difficulties. In special relativity the difficulties relate to the impossibility define the notion of simultaneity in a unique manner and the only possible manner to save this notion seems to be the replacement of time=constant 3surface with past directed lightcone assignable to the worldline of observer. In general relativity additional difficulties are caused by the general coordinate invariance unless one generalizes the picture of special relativity: problems are however caused by the fact that past lightcones make sense only locally. In quantum physics quantum measurement theory leads to a paradoxical situation since the observed localization of the state function reduction to a finite spacetime volume is in conflict with the determinism of Schrödinger equation. TGD forces a new view about the relationship between experienced and geometric time. Although the basic paradox of quantum measurement theory disappears the question about the arrow of geometric time remains.
For details see chapters TGD as a Generalized Number Theory I: pAdicization Program.

The most recent vision about zero energy ontology and padicization
The generalization of the number concept obtained by fusing real and padics along rationals and common algbraics is the basic philosophy behind padicization. This however requires that it is possible to speak about rational points of the imbedding space and the basic objection against the notion of rational points of imbedding space common to real and various padic variants of the imbedding space is the necessity to fix some special coordinates in turn implying the loss of a manifest general coordinate invariance. The isometries of the imbedding space could save the situation provided one can identify some special coordinate system in which isometry group reduces to its discrete subgroup. The loss of the full isometry group could be compensated by assuming that WCW is union over subWCW:s obtained by applying isometries on basic subWCW with discrete subgroup of isometries. The combination of zero energy ontology realized in terms of a hierarchy causal diamonds and hierarchy of Planck constants providing a description of dark matter and leading to a generalization of the notion of imbedding space suggests that it is possible to realize this dream. The article TGD: What Might be the First Principles? provides a brief summary about recent state of quantum TGD helping to understand the big picture behind the following considerations. 1. Zero energy ontology briefly
Consider now the critical questions.
2. Definition of energy inzero energy ontology Can one then define the notion of energy for positive and negative energy parts of the state? There are two alternative approaches depending on whether one allows or does not allow wavefunctions for the positions of tips of lightcones. Consider first the naive option for which four momenta are assigned to the wave functions assigned to the tips of CD:s.
The less naive approach relies of super conformal structures of quantum TGD assumes fixed value of T and therefore allows the crucial quantization condition T=2^{k}T_{0}.
3. pAdic variants of the imbedding space Consider now the construction of padic variants of the imbedding space.
4. pAdic variants for the sectors of WCW One can also wonder about the most general definition of the padic variants of the sectors of the world of classical worlds.
For details see chapters TGD as a Generalized Number Theory I: pAdicization Program.

Prime Hilbert spaces and infinite primesKea told in her blog about a result of quantum information science which seems to provide an additional reason why for padic physics. Suppose that one has Ndimensional Hilbert space which allows N+1 mutually unbiased basis. This means that the moduli squared for the inner product of any two states belonging to different basis equals to 1/N. If one knows all transition amplitudes from a given state to all states of all N+1 mutually unbiased basis, one can fully reconstruct the state. For N=p^{n} dimensional N+1 unbiased basis can be found and the article of Durt gives an explicit construction of these basis by applying the properties of finite fields. Thus state spaces with p^{n} elements  which indeed emerge naturally in padic framework  would be optimal for quantum tomography. For instance, the discretization of onedimensional line with length of p^{n} units would give rise to p^{n}D Hilbert space of wave functions. The observation motivates the introduction of prime Hilbert space as as a Hilbert space possessing dimension which is prime and it would seem that this kind of number theoretical structure for the category of Hilbert spaces is natural from the point of view of quantum information theory. One might ask whether the tensor product of mutually unbiased bases in the general case could be constructed as a tensor product for the bases for prime power factors. This can be done but since the bases cannot have common elements the number of unbiased basis obtained in this manner is equal to M+1, where M is the smallest prime power factor of N. It is not known whether additional unbiased bases exists. 1. Hierarchy of prime Hilbert spaces characterized by infinite primes The notion of prime Hilbert space provides a new interpretation for infinite primes, which are in 11 correspondence with the states of a supersymmetric arithmetic QFT. The earlier interpretation was that the hierarchy of infinite primes corresponds to a hierarchy of quantum states. Infinite primes could also label a hierarchy of infiniteD prime Hilbert spaces with product and sum for infinite primes representing unfaitfully tensor product and direct sum.
2. Hilbert spaces assignable to infinite integers and rationals make also sense
3. Should one generalize the notion of von Neumann algebra? Especially interesting are the implications of the notion of prime Hilbert space concerning the notion of von Neumann algebra in particular the notion of hyperfinite factors of type II_{1} playing a key role in TGD framework. Does the prime decomposition bring in additional structure? Hyperfinite factors of type II_{1} are canonically represented as infinite tensor power of 2×2 matrix algebra having a representation as infinitedimensional fermionic Fock oscillator algebra and allowing a natural interpretation in terms of spinors for the world of classical worlds having a representation as infinitedimensional fermionic Fock space. Infinite primes would correspond to something different: a tensor product of all p×p matrix algebras from which some factors are deleted and added back as direct summands. Besides this some factors are replaced with their tensor powers. Should one refine the notion of von Neumann algebra so that one can distinguish between these algebras as physically nonequivalent? Is the full algebra tensor product of this kind of generalized hyperfinite factor and hyperfinite factor of type II_{1} corresponding to the vibrational degrees of freedom of 3surface and fermionic degrees of freedom? Could padic length scale hypothesis  stating that the physically favored primes are near powers of 2  relate somehow to the naturality of the inclusions of generalized von Neumann algebras to HFF of type II_{1}? For background see that chapter TGD as a Generalized Number Theory III: Infinite Primes. 