Negentropy Maximization Principle

Transcription

1 CONTENTS 1 Negentropy Maximization Principle M. Pitkänen, January 29, matpitka@luukku.com. Recent postal address: Köydenpunojankatu 2 D 11, 10940, Hanko, Finland. Contents 1 Introduction The notion of entanglement entropy Zero energy ontology Connection with standard quantum measurement theory Quantum classical correspondence Fusion of real and p-adic physics Dark matter hierarchy Is it possible to unify the notions of quantum jump and self? Hyper-finite factors of type II 1 and quantum measurement theory with a finite measurement resolution Basic view about NMP The general structure of quantum jump NMP and the notion of self How the contents of consciousness of self are determined Dark matter hierarchy and the notion of self NMP, self measurements, cognition, state preparation, qualia Physics as fusion of real and p-adic physics and NMP Basic definitions related to density matrix and entanglement entropy Density matrix p-adic entanglement negentropy Systems with finitely extended rational entanglement Generalization of the notion of information Number theoretic information measures at the space-time level Number theoretical Quantum Mechanics What it means to be in the intersection of real and p-adic worlds?

2 CONTENTS The general structure of U-matrix neglecting the complexities due to algebraic extensions The general structure of U-matrix when algebraic extensions of rationals are allowed Generalization of NMP to the case of hyper-finite type II 1 factors Factors of type I Factors of type II The origin of hyper-finite factors of type II 1 in TGD The new view about quantum measurement theory What happens in repeated measurements? p-adic thermodynamics with conformal cutoff and hyper-finite factors of type II Factors of type III Some consequences of NMP NMP and thermodynamics Zero energy ontology and thermodynamical ensembles Thermodynamical ensembles are 4-dimensional How second law must be modified? What do experiments say about second law? NMP and self-organization NMP and p-adic length scale hypothesis NMP and biology Life as islands of rational/algebraic numbers in the seas of real and p-adic continua? Evolution and second law Stable entanglement and quantum metabolism as different sides of the same coin NMP, consciousness, and cognition Thermodynamics for qualia Questions about various kinds of entropies The arrow of psychological time and second law Quantum jump and cognition Cognitive codes Abstraction hierarchy and genetic code The concepts of resolution and monitoring Resolution and monitoring and hyperfinite factors of type II NMP and quantum computer type systems The relationship between U-matrix and M-matrix How quantum computation in zero energy ontology differs from ordinary quantum computation Three kinds of quantum computations are possible in TGD Universe Negentropic quantum computations, fuzzy qubits, and quantum groups Generalization of thermodynamics allowing negentropic entanglement and a model for conscious information processing Beauregard s model for computer Ciriticism of the model Problems of Beauregard s model if interpreted as a model for information processing in living systems TGD based variant of Beauregard s model and generalization of thermodynamics Questions What happens in quantum jump? Modification of thermodynamics to take into account negentropic entanglement The analog of Carnot cycle for information processing in living matter Some clarifying comments About implications of generalized second law Biological implications The interpretation of generalized second law in a wider context

3 CONTENTS 3 Abstract In TGD Universe the moments of consciousness are associated with quantum jumps between quantum histories. The proposal is that the dynamics of consciousness is governed by Negentropy Maximization Principle, which states the information content of conscious experience is maximal. The formulation of NMP is the basic topic of this chapter. Negentropy Maximization Principle (NMP) codes for the dynamics of standard state function reduction and states that the state function reduction process following U-process gives rise to a maximal reduction of entanglement entropy at each step. In the generic case this implies at each step a decomposition of the system to unique unentangled subsystems and the process repeats itself for these subsystems. The process stops when the resulting subsystem cannot be decomposed to a pair of free systems since energy conservation makes the reduction of entanglement kinematically impossible in the case of bound states. The natural assumption is that self loses consciousness when it entangles via bound state entanglement. There is an important exception to this vision based on ordinary Shannon entropy. There exists an infinite hierarchy of number theoretical entropies making sense for rational or even algebraic entanglement probabilities. In this case the entanglement negentropy can be negative so that NMP favors the generation of negentropic entanglement, which need not be bound state entanglement in standard sense. Negentropic entanglement might serve as a correlate for emotions like love and experience of understanding. The reduction of ordinary entanglement entropy to random final state implies second law at the level of ensemble. For the generation of negentropic entanglement the outcome of the reduction is not random: the prediction is that second law is not a universal truth holding true in all scales. Since number theoretic entropies are natural in the intersection of real and p-adic worlds, this suggests that life resides in this intersection. The existence effectively bound states with no binding energy might have important implications for the understanding the stability of basic bio-polymers and the key aspects of metabolism. A natural assumption is that self experiences expansion of consciousness as it entangles in this manner. Quite generally, an infinite self hierarchy with the entire Universe at the top is predicted. The identification of life as a number theoretically critical phenomenon is also consistent with the idea that the transformation of intention to action corresponds to a U-process inducing leakage between different sectors. This leakage makes sense in the intersection where same mathematical expression defines both real and p-adic partonic 2-surfaces which are the fundamental objects in TGD framework. What these statements really mean requires a construction of number theoretical variant of quantum theory applying in the intersection of real and p-adic worlds. Besides number theoretic negentropies there are also other new elements as compared to the earlier formulation of NMP. Zero energy ontology modifies dramatically the formulation of NMP since U-matrix acts between zero energy states and can be regarded as a collection of M-matrices, which generalize the ordinary S-matrix and define what might be called a complex square root of density matrix so that kind of a square root of thermodynamics at single particle level justifying also p-adic mass calculations based on p-adic thermodynamics is in question. The hierarchy of Planck constants is a further new element having important implications for conciousness and biology. Hyper-finite factors of type II 1 represent an additional technical complication requiring separate treatment of NMP taking into account finite measurement resolution realized in terms of inclusions of these factors. NMP has important implications for thermodynamics. In particular, one must give up the standard view about second law and replace it with a formulation taking into accoung the hierarchy of causal diamonds assigned with zero energy ontology and dark matter hierarchy labeled partially by the values of Planck constants, as well as the effects due to negentropic entanglement. In particular, in the case of living matter breaking of second law in standard sense is expected to take place and be crucial for the understanding of evolution. Self hierarchy having the hierarchy of causal diamonds as imbedding space correlate leads naturally to a thermodynamical description of the contents of consciousness and quantum jumps is very much analogous to quantum computation. This leads to a vision about the role of bound state entanglement and negentropic entanglement in the generation of sensory qualia. Negentropic entanglement leads to a vision about cognition. Negentropically entangled state consisting of a superposition of pairs can be interpreted as a conscious abstraction or rule: negentropically entangled Schrödinger cat knows that it is better to keep the bottle closed. A connection with fuzzy qubits and quantum groups with negentropic entanglement is highly suggestive. The implications are highly non-trivial also for quantum computation, which allows three different variants in TGD context. The negentropic variant would correspond to conscious quantum computation like process.

4 1. Introduction 4 1 Introduction Quantum TGD involves holy trinity of time developments. There is the geometric time development dictated by the preferred extremal of Kähler action crucial for the realization of General Coordinate Invariance and analogous to Bohr orbit. There is the unitary time development U: Ψ i UΨ i Ψ f, associated with each quantum jump, which is the counterpart of the Schrödinger time evolution U( t, t ). There is however no actual Schrödinger equation involved: situation is in practice same also in quantum field theories. Quantum jump sequence itself defines what might be called subjective time development. Some dynamical principle governing subjective time evolution should exist and explain state function reduction with the characteristic one-one correlation between macroscopic measurement variables and quantum degrees of freedom and state preparation process. Negentropy Maximization Principle is the candidate for this principle, which I have been developing during last fifteen years. The evolution of ideas related to NMP has been slow and tortuous process characterized by misinterpretations, overgeneralizations, and unnecessarily strong assumptions, and has been basically evolution of ideas related to the anatomy of quantum jump and of quantum TGD itself. 1.1 The notion of entanglement entropy 1. The first form of NMP was rather naive. There was no idea about the anatomy of quantum jump and NMP only stated that the allowed quantum jumps are such that the information gain of conscious experience measured by the reduction of entanglement entropy resulting in the reduction of entanglement between the subsystem of system and its complement is maximal. Later it became clear that quantum jump has a complex anatomy consisting of unitary process U followed by the TGD counterpart of state function reduction serving as a state preparation for the next quantum jump. 2. The attempts to formulate NMP in p-adic physics led to the realization that one can distinguish between three kinds of information measures. (a) In real physics the negative of the entanglement entropy defined by the standard Shannon formula defines a natural information measure, which is always non-positive. (b) In p-adic physics one can generalize this information measure to p-adic valued information measure by replacing the logarithms of p-adic valued probabilities with the p-based logarithms log p ( P p ) which are integer valued and can be interpreted as p-adic numbers. This p-adic valued entanglement entropy can be mapped to a non-negative real number by the so called canonical identification x = n x np n n x np n. In both cases a non-positive information measure results. (c) When the entanglement probabilities are rational numbers or at most finitely algebraically extended rational numbers one can still define logarithms of probabilities as p-based logarithms log p ( P p ) and interpret the entropy as a rational or algebraic number. In this case the entropy can be however negative and positive definite information measure is possible. Irrespective of number field one can in this case define entanglement entropy as a maximum of number theoretic entropies S p over the set of primes. The first proposal was that the algebraic entanglement corresponds to bound state entanglement turned out to be wrong. 3. At some stage the importance of the almost trivial fact that bound state entanglement must be kinematically stable against NMP became obvious. One can imagine that the state function reduction proceeds step by step by reducing the state to two parts in such a manner that the reduction of entanglement entropy is maximal. (a) If a resulting subsystem corresponds to a bound state having no decomposition to free subsystems the process stops for this subsystem. The natural assumption is that subsystems lose their consciousness when U process leads to bound state entanglement whereas bound state itself can be conscious. (b) If the entanglement is negentropic (and thus rational or algebraic) a more natural interpretation consistent with the teaching of spiritual practices is that subsystems experience

5 1.2 Zero energy ontology 5 a fusion to a larger conscious entity. The negentropic entanglement between free states is stabilized by NMP and negentropically entangled states need not reside at the bottom of potential well forbidding the reduction of entanglement. This makes possible new kinds of correlated states for which binding energy can be negative. Bound state entanglement would be like the jail of organized marriage and negentropic entanglement like a love marriage in which companions are free to leave but do not what it. The existence of this kind of negentropic entanglement is especially interesting in living matter, where metabolism (high energy phosphate bond in particular) and the stability of DNA and other highly charged polymers is poorly understood physically: negentropic entanglement could be responsible for stabilization making possible the transfer of metabolic energy [11]. 4. For the negentropic entanglement the outcome of the state function reduction ceases to be random as it is for the standard definition of entanglement entropy. Note however that U process as a creative act yielding superposition of possibibilities from which state function reduction selects leaves means non-determinism. This has far reaching consequences. Ordinary state function reductions for an ensemble of systems lead to a generation of thermodynamical entropy and this explains the second law of thermodynamics. In the case of negentropic entanglement situation changes and the predicted breaking of second law of thermodynamics provides a new view to understand self-organization [22], and living matter could be identified as something residing in the intersection of real and p-adic worlds where p-adic intentions can be transformed to real actions. 5. One particular choice involved with state function reduction process could be the choice between generic entanglement and number theoretic entanglement possible only in the intersection of p-adic and real WCWs. If the choice is the generic entanglement, system ends up either to an unentangled state with maximal conscious freedom or to a bound state with a loss of consciousness. If the choice is algebraic entanglement, system ends up to negentropic entanglement and correlations with external world and experiences an expansion of consciousness. Maybe ethical choices are basically choices between these two options. Also positive emotions like love and experience of understanding could directly relate to various aspects of the negentropic entanglement. 1.2 Zero energy ontology Zero energy ontology changes considerably the interpretation of the unitary process. In zero energy ontology quantum states are replaced with zero energy states defined as a superpositions of pairs of positive and negative energy states identified as counterparts of initial and final states of a physical event such as particle scattering. The matrix defining entanglement between positive and negative - christened as M-matrix- is the counterpart of the ordinary S-matrix but need not be unitary. It can be identified as a complex square root of density matrix expressible as a product of positive square root of diagonal density matrix and unitary S-matrix. Quantum TGD can be seen as defining a square root of thermodynamics, which thus becomes an essential part of quantum theory. U-matrix is defined between zero energy states and cannnot therefore be equated with the S- matrix used to describe particle scattering events. Unitary conditions however imply that U-matrix can be seen as a collection of M-matrices labelled by zero energy states so that the knowledge of U-matrix implies the knowledge of M-matrices. The unitarity conditions will be discussed later. A natural guess is that U is directly related to consciousness and the description of intentional actions. For positive energy ontology state function reduction would serve as a state preparation for the next quantum jump. In zero energy ontology state function preparation and reduction can be assigned to the positive and negative energy states defining the initial and final states of the physical event. The reduction of the time-like entanglement during the state function reduction process corresponds to the measurement of the scattering matrix. In the case of negentropic time-like entanglement the reduction process is not random anymore and the resulting dynamics is analogous to that of cellular automata providing a natural description of the dynamics of self-organization in living matter. Zero energy ontology leads to a precise identification of the subsystem at space-time level. General coordinate invariance in 4-D sense means that 3-surfaces related by 4-D diffeomorphisms are physically equivalent. It is conventient to perform a gauge fixing by a introducing a natural choice for the representatives of the equivalence classes formed by diffeo-related 3-surfaces.

6 1.3 Connection with standard quantum measurement theory 6 1. Light-like 3-surfaces identified as surfaces at which the Minkowskian signature of the induced space-time metric changes to Euclidian one - wormhole contacts- are excellent candidates in this respect. The intersections of these surfaces with the light-like boundaries of CD defined 2-D partonic surfaces. Also the 3-D space-like ends of space-time sheets at the light-like boundaries of CDs are very natural candidates for preferred 3-surfaces. 2. The condition that the choices are mutually consistent implies effective 2-dimensionality. The intersections of these surfaces defining partonic 2-surface plus the distribution of 4-D tangent spaces at its points define the basic dynamical objects with 4-D general coordinate invariance reduced to 2-dimensional one. This effective 2-dimensionality was clear from the very beginning but is only apparent since also the data about 4-D tangent space distribution is necessary to characterize the geometry of WCW and quantum states. The descriptions in terms of 3-D light-like or space-like surfaces and even in terms of 4-D surfaces are equivalent but redundant descriptions. As far as consciousness is considered effective 2-dimensionality means holography and could relate to the fact that at least our visual experience is at least effectively 2-dimensional. 1.3 Connection with standard quantum measurement theory TGD allows to deduce the standard quantum measurement theory involving the notion of classical variables and their correlation with quantum numbers in an essential manner. Configuration space (or world of classical worlds, briefly W CW ) is a union over zero modes labelling infinite-dimensional symmetric spaces having interpretation as classical non-quantum fluctuating classical variables such as the pointer of a measurement apparatus essential for the standard quantum measurement theory [4]. Quantum holography states that partonic 2-surfaces at the light-like boundaries of CDs plus the corresponding distributions of 4-D tangent spaces of space-time surfaces at carry the information about quantum state and space-time sheet. The distribution of values of induced Kähler form of CP 2 at these surfaces defines zero modes whereas quantum fluctuating degrees of freedom correspond to the deformations of space-time surface by the flows induced by Hamiltonians associated with the degenerate symplectic structure of δm± 4 CP 2. There exists no well-defined metric integration measure in the infinite-dimensional space of zero modes, which by definition do not contribute to the line element of W CW. This does not lead to difficulties if one assumes that a complete localization in zero modes occurs in each quantum jump. A weaker condition is that wave functions are localized to discrete subsets in the space of zero modes. An even weaker and perhaps the most realistic condition is that a localization to a finite-dimensional 2n-dimensional manifold with induced symplectic form defining a positive definite integration volume takes place. The fundamental formulation of quantum TGD in terms of the modified Dirac action [2, 10] containing a measurement interaction term guarantees quantum classical correspondence in the sense that the geometry of the space-time surface correlates with the values of conserved quantum numbers. The resulting correlation of zero modes with the values of quantum numbers can be interpreted as an abstract form of quantum entanglement reduced in quantum jump for the standard definition of the entanglement entropy. This reproduces standard quantum measurement theory. 1.4 Quantum classical correspondence Quantum classical correspondence has served as a guideline in the evolution of the ideas and the identification of the geometric correlates of various quantum notions at the level of imbedding space and space-time surfaces has been an important driving force in the progress of ideas. 1. In zero energy ontology causal diamonds (CDs) identified roughly as intersections of future and past directed light-cones are in key role. At imbedding space level CD is a natural correlate for self and sub-cds serve as correlates of sub-selves identified as mental images. At space-time level the space-time sheets having their ends at the light-like boundaries of CD serve as correlates for self. For a system characterized by a primary p-adic length scale L p 2 k/2 the size scale of CD is secondary p-adic scale L p,2 = pl p 2 k. p-adic length scale hypothesis follows if the

7 1.5 Fusion of real and p-adic physics 7 proper time distance between the tips of CDs is quantized in powers of 2. This quantization should relate directly to almost equivalence of octaves associated with music experience. 2. At the level of space-time the identification of join along boundaries bonds between space-time sheets (more precisely, between partonic 2-surfaces) as a correlate for bound state entanglement suggests itself. Join along boundaries bonds correspond typically to magnetic flux tubes in the TGD inspired quantum model of living matter. The size scale of the magnetic body of system is given by the size scale of CD and much larger than the size of the system itself. 3. The space-time sheets in the intersection of the real and p-adic W CW s characterized by the property that the mathematical representation of the partonic 2-surfaces at the ends representing holograpically the state allows interpretation in both real and p-adic sense would correspond to the correlates for negentropic entanglement. Rational and algebraic 2-surfaces (in preferred coordinates) would be the common points of realities and p-adicities. Quantum classical correspondence allows also to generate new views about quantum theory itself. Many-sheeted space-time and p-adic length scale hierarchy force to generalize the notion of sub-system. The space-time correlate for the negentropic and bound state entanglement is the formation of join along boundaries bonds connecting two space-time sheets. The basic realization is that two disjoint space-time sheets can contain smaller space-time sheets topologically condensed at them and connected by join along boundaries bonds. Thus systems un-entangled at a given level of p-adic hierarchy -that is in the measurement resolution defined by the level considered - can contain entanglement subsystems at lower level not visible in the resolution used. In TGD inspired theory of consciousness this makes possible sharing and fusion of mental images by entanglement. The resolution dependence for the notions of sub-system and entanglement means that the entanglement between sub-systems is not seen in the length scale resolution of unentangled systems. This phenomenon does not result as an idealization of theoretician but is a genuine physical phenomenon. Obviously this generalized view about sub-system poses further challenges to the detailed formulation of NMP. Note that the resulting mental image should depend on whether subselves are entangled by bound state entanglement or negentropic entanglement. 1.5 Fusion of real and p-adic physics The fusion of real and p-adic physics to a larger structure has been a long standing challenge for TGD. The motivations come both from elementary particle physics and TGD inspired theory of consciousness, in particular from the attempt to model how intentions proposed to have p-adic space-time sheets as space-time correlates are transformed to actions having real space-time sheets as correlates. The basic idea is that various number fields are fused to a larger structure by gluing them along rationals and common algebraic numbers. The challenge is to imagine what quantum jump and NMP could mean in this framework. The first question is how the unitary process acts. 1. U-process acts in spinorial degrees of freedom of WCW (fermionic Fock space for a given 3- surface) and in WCW degrees of freedom (the space of partonic 2-surfaces roughly). The transformation of intention to action would correspond to a leakage from p-adic to real sector of W CW. 2. At the level of WCW one can only speak about classical spinor fields and the idea about tensor product of states corresponding to different sectors of WCW does not look reasonable at the first glance. Rather, a quantum superposition of WCW spinor fields localized at various sectors would look more appropriate. Therefore the WCW spinor field would be in fixed number field after state function reduction if it involves localization in this sense. This does not look sensible. The tensor product for fermionic Fock spaces is indeed very natural and strongly suggested also by the interpretation of the 3-surfaces as particles. One can indeed consider CDs and their unions and it would seem reasonable to assign to the unions of CDs tensor products of the corresponding WCW spinor fields. Let us assume this. 3. Let us assume that the initial zero energy state state represents an un-entangled tensor product of states in various number fields. The simplest assumption is that U process can induce a leakage between different sectors only in the intersection of real and p-adic worlds. This would

8 1.6 Dark matter hierarchy 8 also hold true as far as entanglement between different number fields is considered. This would allow to realize intentional action geometrically as a p-adic-to-real transition. The p-adic and real variants of a state quantum entangled with a third (say real) state would define the entangled system and state function reduction would select either p-adic or real variant of the state. The selection would be whether to transform action to its cognitive representation or intention to action. Also a transformation of a real zero energy state to its cognitive representation in p-adic sense is possible as also transformations between p-adic cognitive representations characterized by different primes. 4. For partonic 2-surfaces the quantum superposition of quantum states belonging to different number fields in the intersection would mean a quantum superposition of real and various p- adic variants of the surface with given mathematical representation forming tensor products with the states of second system, which could be real for instance. U-matrix could lead to this kind of quantum superposition. U-matrix between different number fields should be expressible using only the geometric data from the intersection of the real and p-adic variants of the partonic surface- that is rational points and common algebraic points, whose number is expected to be finite. Some kind of number theoretic quantum field theory should describe the U-matrix. State function reduction would involve the selection of whether the outcome is action or intention (or cognitive representation). Note that if the real-real entanglement is non-algebraic the NMP leads to a final state with algebraic entanglement between real system and p-adic cognitive representation of the other system. If real-real entanglement is algebraic, the reduction can lead from intention to action as a more negentropic final state. 5. It has been assumed that entanglement and matrix elements of U between different number fields are possible only in the intersection of the real and p-adic worlds. This is natural if entanglement coefficients between different number fields are represented in terms of the data provided by the intersection of the real and p-adic variants of partonic 2-surfaces involved and consisting of rational points and some algebraic points. Outside the intersection real and p-adic worlds would evolve independently. One could criticize this picture as raising the intersection of real and p-adic worlds to a singular position. Life is however something very special and the interpretation in terms of number theoretical criticality justifies this singular character. 1.6 Dark matter hierarchy The identification of dark matter as phases having large value of Planck constant [23, 9, 6] led to a vigorous evolution of ideas. Entire dark matter hierarchy with levels labelled by increasing values of Planck constant is predicted, and in principle TGD predicts the values of Planck constant if physics as a generalized number theory vision is accepted [9]. The hierarchy of Planck constants is realized in terms of a generalization of the causal diamond CD CP 2, where CD is defined as an intersection of the future and past directed light-cones of 4-D Minkowski space M 4. CD CP 2 is generalized by gluing singular coverings and factor spaces of both CD and CP 2 together like pages of book along common back, which is 2-D sub-manifold which is M 2 for CD and homologically trivial geodesic sphere S 2 for CP 2 [9]. The value of the Planck constant characterizes partially the given page and arbitrary large values of are predicted so that macroscopic quantum phases are possible since the fundamental quantum scales scale like. The most general spectrum comes in rational multiples of standard value of Planck constant which corresponds to the unit of rationals. For CDs the scaling of Planck constants means scaling of the size of CD. This could explain why the rational multiples of the fundamental frequency are so special for music experience. All particles in the vertices of Feynman diagrams have the same value of Planck constant so that particles at different pages cannot have local interactions. Thus one can speak about relative darkness in the sense that only the interactions mediated by the exchange of particles and by classical fields are possible between different pages. Dark matter in this sense can be observed, say through the classical gravitational and electromagnetic interactions. It is in principle possible to photograph dark matter by the exchange of photons which leak to another page of book, reflect, and leak back. This leakage corresponds to changing phase transition occurring at quantum criticality and living matter is expected carry out these phase transitions routinely in bio-control. This picture leads to no obvious contradictions with what is really known about dark matter and to my opinion the basic difficulty in understanding of dark matter (and living matter) is the blind belief in standard quantum theory.

9 1.7 Is it possible to unify the notions of quantum jump and self? 9 These observations motivate the tentative identification of the macroscopic quantum phases in terms of dark matter and also of dark energy with gigantic gravitational Planck constant. It seems safe to conclude that the dark matter hierarchy with levels labelled by the values of Planck constants explains the macroscopic and macro-temporal quantum coherence naturally. That this explanation is consistent with the explanation based on spin glass degeneracy is suggested by the following observations. First, the argument supporting spin glass degeneracy as an explanation of the macro-temporal quantum coherence does not involve the value of at all. Secondly, the failure of the perturbation theory assumed to lead to the increase of Planck constant and formation of macroscopic quantum phases could be precisely due to the emergence of a large number of new degrees of freedom due to spin glass degeneracy. Thirdly, the phase transition increasing Planck constant has concrete topological interpretation in terms of many-sheeted space-time consistent with the spin glass degeneracy. At least dark matter could be a key player in quantum biology. 1. Dark matter hierarchy and p-adic length scale hierarchy would provide a quantitative formulation for the self hierarchy. To a given p-adic length scale one can assign a secondary p-adic time scale as the temporal distance between the tips of the CD. For electron this time scale is.1 second, the fundamental bio-rhythm. For a given p-adic length scale dark matter hierarchy gives rise to additional time scales coming as / 0 multiples of this time scale. 2. The predicted breaking of second law of thermodynamics chacterizing living matter - if identified as something in the intersection of real and p-adic words - would be always below the time scale of CD considered but would take place in arbitrary long time scales at appropriate levels of the hierarchy. The scaling up of also scales up the time scale for the breaking of the second law. 3. The hypothesis that magnetic body is the carrier of dark matter in large phase has led to models for EEG predicting correctly the band structure and even individual resonance bands and also generalizing the notion of [1] [7]. Also a generalization of the notion of genetic code emerges resolving the paradoxes related to the standard dogma [15, 7]. A particularly fascinating implication is the possibility to identify great leaps in evolution as phase transitions in which new higher level of dark matter emerges [7]. 1.7 Is it possible to unify the notions of quantum jump and self? An important step in the process was the realization that the generation of macro-temporal quantum coherence means effective gluing of quantum jumps of quantum jump sequence of sub-system defining mental images to single quantum jump. This means that in appropriate degrees of freedom state function reduction and state preparation cease to occur during macro-temporal quantum coherence. This makes sense if macro-temporal quantum coherence means generation of negentropic or bound state entanglement stable under subsequent U-processes. The hierarchy of Planck constants and p-adic length scale hypothesis lead to the view that there is an entire hierarchy of durations for effective quantum jumps and this forces to ask whether the quantum jumps sequence decomposes into a hierarchy of effective quantum jumps of increasingly long duration just like physical systems form a hierarchy starting from the level of elementary particles and continuing through hadronic, nuclear, atomic and molecular physics up to level where astrophysical objects take the role of particles. The usually un-noticed fact that hadrons can be regarded as quantum objects in long length and time scales whereas quark description treats hadrons as dissipative systems forces to ask whether state function reductions and preparations associated with the hierarchy of CDs form a hierarchy and whether the dissipative processes in short scales could occur in quantum parallel manner in longer scales so that one would have quantum superposition of parallel dissipative Universes? Using quantum computer language this would mean the possibility of quantum superposition of classical dissipative quantum computations. These hierarchies suggest that the notions of self and quantum jump could be identified. Self would correspond to single quantum jump at the highest level and at the lowest levels to sequences of quantum jumps in accordance with the geometric representation in terms of CDs.

10 1.8 Hyper-finite factors of type II 1 and quantum measurement theory with a finite measurement resolution Hyper-finite factors of type II 1 and quantum measurement theory with a finite measurement resolution The realization that the von Neumann algebra known as hyper-finite factor of type II 1 is tailor made for quantum TGD has led to a considerable progress in the understanding of the mathematical structure of the theory and these algebras provide a justification for several ideas introduced earlier on basis of physical intuition. Hyper-finite factor of type II 1 has a canonical realization as an infinite-dimensional Clifford algebra and the obvious guess is that it corresponds to the algebra spanned by the gamma matrices of WCW. Also the local Clifford algebra of the imbedding space H = M 4 CP 2 in octonionic representation of gamma matrices of H is important and the entire quantum TGD emerges from the associativity or co-associativity conditions for the sub-algebras of this algebra which are local algebras localized to maximal associative or co-associate sub-manifolds of the imbedding space identifiable as space-time surfaces. The notion of inclusion for hyper-finite factors provides an elegant description for the notion of measurement resolution absent from the standard quantum measurement theory. 1. The included sub-factor creates in zero energy ontology states not distinguishable from the original one and the formally the coset space of factors defining quantum spinor space defines the space of physical states modulo finite measument resolution. 2. The quantum measurement theory for hyperfinite factors differs from that for factors of type I since it is not possible to localize the state into single ray of state space. Rather, the ray is replaced with the sub-space obtained by the action of the included algebra defining the measurement resolution. The role of complex numbers in standard quantum measurement theory is taken by the non-commutative included algebra so that a non-commutative quantum theory is the outcome. 3. This leads also to the notion of quantum group. For instance, the finite measurement resolution means that the components of spinor do not commute anymore and it is not possible to reduce the state to a precise eigenstate of spin. It is however perform a reduction to an eigenstate of an observable which corresponds to the probability for either spin state. As already explained, the topology of the many-sheeted space-time encourages the generalization of the notion of quantum entanglement in such a manner that unentangled systems can possess entangled sub-systems. One can say that the entanglement between subselves is not visible in the resolution characterizing selves. This makes possible sharing and fusion of mental images central for TGD inspired theory of consciousness. These concepts find a deeper justification from the quantum measurement theory for hyper-finite factors of type II 1 for which the finite measurement resolution is basic notion. Also the notions of resolution and monitoring pop up naturally in this framework. p-adic probabilities relate very naturally to hyper-finite factors of type II 1 and extend the expressive power of the ordinary probability theory. p-adic thermodynamics with conformal cutoff is very natural for hyperfinite factors of type II 1 and explains p-adic length scale hypothesis p 2 k, k prime characterizing exponentially smaller p-adic length scale. 2 Basic view about NMP The following represents a brief overall view about the notions of quantum jump, self, and NMP. 2.1 The general structure of quantum jump It has gradually become clear that TGD involves holy trinity of dynamics. 1. The dynamics defined by the preferred extremals of Kähler action identifiable as counterparts of Bohr orbits corresponds to the dynamics of material existence, with matter defined as res extensa, three-surfaces.

11 2.1 The general structure of quantum jump The dynamics defined by the action of the unitary time development operator U can be regarded as informational time development occurring at the level of objective existence. U brings in mind the time evolution operator U( t, t), t associated with the scattering solutions of Schrödinger equation. It seems however un-necessary and also impossible to assign Schrödinger equation with U. Furthermore, U acts between zero energy states in zero energy ontology and is more naturally assigned with intentional action rather than to the description of particle scattering. 3. The dynamics of quantum jumps governed by U and by NMP corresponds to the dynamics of subjective existence. In accordance with this, quantum jump decomposes into informational time development Ψ i UΨ i, followed by a sequence of self measurements (generalization of state function reduction) Ψ f0 Ψ f1... Ψ f governed by NMP. At given step subsystem the decomposition to two un-entangled systems is such that maximum reduction of entanglement entropy is achieved. This means that the reduction process proceeds as a binary tree. If subsystem does not allow a decomposition to a pair of free subsystems with entropic entanglement the process stops. Zero energy ontology means that one must distinguish between M-matrix and U-matrix. M- matrix characterizes the time like entanglement between positive and negative energy parts of zero energy state and is measured in particle scattering experiments. M-matrix need not be unitary and can be identified as a complex square root of density matrix representable as a product of its real and positive square root and of unitary S-matrix so that thermodynamics becomes part of quantum theory with thermodynamical ensemble being replaced with a zero energy state. The unitary U-matrix describes quantum transitions between zero energy states and is therefore something genuinely new. It is natural to assign the statistical description of intentional action with U-matrix since quantum jump occurs between zero energy states. U process is in zero energy ontology something totally new and can be seen as representing an act of genuine re-creation of the Universe. The following metaphors might help to understand what is involved. 1. A good metaphor for the quantum jump is as Djinn leaving the bottle (U) fulfilling the wish realized as a choice between various option that is state function reduction. In the case that final state has negentropic entanglement wish is realized in different manner. 2. A second useful metaphor is as generation of infinite number of quantum parallel potentialities in which entire universe is in a totally entangled holistic state of oneness followed by state function reduction and self measurement cascade analyzing the state into maximally unentangled subsystems. NMP states that the analysis produces maximum amount of conscious information. For irreducible selves analysis process do not continue and the sequences of quantum jumps effectively take the role of single quantum jump. A further element is the expansion of consciousness when negentropic entangelment is generated. Therefore this structure characterizes also conscious experience in macro-temporal time scales. Clearly, quantum measurement theory has fascinating parallels with Krishnamurti s philosophy of consciousness which underlines the competing holistic and reductionistic aspects of consciousness. 3. A third metaphor comes from particle physics. Moment of consciousness can be seen as elementary particle of consciousness and selves as the atoms, molecules,...galaxies,... of consciousness. Fractality hypothesis allows to get general vision about structure of consciousness even in the time scale of human life. If quantum jump occurs between two different time evolutions of Schrödinger equation (understood here in very metaphoral sense) rather than interfering with single deterministic Schrödinger evolution, the basic problem of quantum measurement theory finds a resolution. The interpretation of quantum jump as a moment of consciousness means that volition and conscious experience are outside spacetime and state space and that quantum states and space-time surfaces are zombies.

12 2.2 NMP and the notion of self NMP and the notion of self Negentropy Maximization Principle (NMP) codes for the dynamics of standard state function reduction and states that the state function reduction process following U-process gives rise to a maximal reduction of entanglement entropy at each step. In the generic case this implies decomposition of the system to unique unentangled systems and the process repeats itself for these systems. The process stops when the resulting subsystem cannot be decomposed to a pair of free systems since energy conservation makes the reduction of entanglement kinematically impossible in the case of bound states. Intuitively self corresponds to a sequence of quantum jumps which somehow integrates to a larger unit much like many-particle bound state is formed from more elementary building blocks. It also seems natural to assume that self stays conscious as long as it can avoid bound state entanglement with the environment in which case the reduction of entanglement is energetically impossible. One could say that everything is conscious and consciousness can be only lost when the system forms bound state entanglement with environment. There is an important exception to this vision based on ordinary Shannon entropy. There exists an infinite hierarchy of number theoretical entropies making sense for rational or even algebraic entanglement probabilities. In this case the entanglement negentropy can be negative so that NMP favors the generation of negentropic entanglement, which need not be bound state entanglement in standard sense. Negentropic entanglement might serve as a correlate for emotions like love and experience of understanding. The reduction of ordinary entanglement entropy to random final state implies second law at the level of ensemble. For the generation of negentropic entanglement the outcome of the reduction is not random: the prediction is that second law is not universal truth holding true in all scales. Since number theoretic entropies are natural in the intersection of real and p-adic worlds, this suggests that life resides in this intersection. The existence effectively bound states with no binding energy might have important implications for the understanding the stability of basic bio-polymers and the key aspects of metabolism [11]. A natural assumption is that self experiences expansion of consciousness as it entangles in this manner. Quite generally, an infinite self hierarchy with the entire Universe at the top is predicted. If one accepts the hierarchy of Planck constants [9], it might be un-necessary to distinguish between self and quantum jump. The hierarchy of Planck constants interpreted in terms of dark matter hierarchy predicts a hierarchy of quantum jumps such that the size of space-time region contributing to the contents of conscious experience scales like. Also the hierarchy of space-time sheets labeled by p-adic primes suggests the same. That sequence of sub-selves/sub-quantum jumps are experienced as separate mental images explains why we can distinguish between digits of phone number. The irreducible component of self (pure awareness) would correspond to the highest level in the personal hierarchy of quantum jumps and the sequence of lower level quantum jumps would be responsible for the experience of time flow. Entire life cycle would correspond to single quantum jump at the highest(?) level of the personal self hierarchy and pure awareness would prevail during sleep: this would make it possible to experience directly that I existed yesterday. Whether these two definitions of self are in some sense equivalent will be discussed later How the contents of consciousness of self are determined The hypothesis that the experiences of self associated with the quantum jumps occurred after the last wake-up sum up to single experience, implies that self can have memories about earlier moments of consciousness. Therefore self becomes an extended object with respect to subjective time and has a well defined personal history. If temporal binding of experiences involves kind of averaging, quantum statistical determinism makes the total experience defined by the heap of the experiences associated with individual quantum jumps reliable. Subjective memory has natural identification as a short term memory. A given self S behaves essentially as a separate sub-universe with respect to NMP. If one postulates that the conscious experiences of sub-selves S i of an self S integrate with the self experience of S to single experience, one obtains a filtered hierarchy of conscious experiences with increasingly richer contents and at the top of the hierarchy is entire universe, God, enjoying eternal self-consciousness since it cannot get entangled with any larger system. An attractive hypothesis is that the experience of self is abstraction in the sense that the experiences of sub-selves S ij of S i are abstracted to average experience S ij. This implies that the experiences