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TGD Inspired Theory of Consciousness

Note: Newest contributions are at the top!

Year 2016


ER=EPR correspondence proposed by Leonard Susskind and Juan Maldacena in 2014 (see also this) has become the most fashionable fashion in theoretical physics. Even the idea that space-time could emerge from ER-EPR has been proposed.


ER (Einstein-Rosen) bridge in turn is purely classical notion associated with general relativity theory (GRT). ER bridge is illustrated in terms of a fold of space-time. Locally there are two sheets near to each other and connected by a wormhole: these sheets are actually parts of the same sheet. Along the bridge the distance between two systems can be very short. Along folded sheet it can be very long. This suggest some kind of classical non-locality in the sense that the physics around the two throats of wormhole can be strongly correlated: the non-locality would be implied by topology. This is not in accordance with the view of classical physics in Minkowski space-time.


EPR (Einstein-Podolsky-Rosen) paradox states that it is possible to measure both position and momentum of two particles more accurately than Heisenberg Uncertainty Principle allows unless the measurement involves instantaneous transfer of information between particles denied by special relativity. The conclusion of EPR was that quantum theory is incomplete and should be extended by introducing hidden variables. The argument was based on classical physics view a bout microcausality.

Later the notion of quantum entanglement became an established notion and it became clear that no classical superluminal transfer of information is needed. If one accepts the basic rules of quantum measurement theory - in particular tensor products of distant systems - EPR paradox disappears. Entanglement is of course a genuinely non-nonlocal phenomenon not encountered in classical physics and one could wonder whether it might have classical sace-time correlate after all. State function reduction becomes the problem and has remained the ugly duckling of quantum theory. Unfortunately, this ugly duckling has become a taboo and is surrounded by a thick cloud of messy interpretations. Hence the situation is still far from settled.

At time EPR and ER were proposed, there was no idea about possible connection between these two ideas. Both notions involve unexpected non-locality and one might however ask whether there might be a connection.


In some sense ER=EPR could be seen as kind of victory for Einstein. There could be after all a classical space-time correlate for entanglement and for what happens state function reduction for a system induces state function reduction in distant entangled system. It however seems that quantum theory does not allow a signal travelling along the wormhole throat connecting the entangled systems.

What ER= EPR says that maximal entanglement for blackholes is somehow dual to Einstein-Rosen bridge (wormhole). Susskind and Maldacena even suggests that this picture generalizes to entanglement between any kind of systems and that even elementary particles are connected by Planckian wormholes.

The next step has been to argue that entanglement is more fundamental than space-time, and that space-time would emerge. The attempts to realize the idea involve holography and already this means introduction of 2-D surfaces in 3-D space so that the argument becomes circular. To my opinion the emergence of space-time is doomed to remain one of the many fashions of theoretical physics, which last few years and are then lost to sands of time. These fashions reflect the deep crisis of theoretical physics, which has lasted for four decades, and are as such a good sign telling that people at least try.

The motivation for following TGD inspired arguments was one of the arguments against ER=EPR: ER=EPR does not conform with the linearity of quantum mechanics. The state pairs in the superposition defining entangled state are unentangled (separable) and there should be no wormhole connecting the systems in this case. In an entangled state there should be wormhole. This makes sense only if the space-time geometry couples to quantum dynamics so that one must give up the idea that one has Schödinger amplitudes in fixed background and linear superposition for them. This looks weird even in GRT space-time.

Some background about TGD

Before discussing what ER-EPR corresponds in TGD few words about quantum TGD are in order.

  1. The formulation of TGD in terms of geometry of "world of classical worlds" (WCW) consisting of 3-surfaces, which are holographically related to 4-D space-time surfaces. This holography is implied by General Goordinate Invariance (GCI). One can say that space-time surfaces as preferred extremal of Kähler action are analogous to Bohr orbits and that classical theory is an exact part of quantum theory.

    What I call strong form of GCI (SGCI) implies strong form of holography (SH) stating that string world sheets and partonic 2-surfaces dictate the dynamics. A slightly weaker form of SH is that the light-like orbits of partonic 2-surfaces, which are metrically 2-dimensional and lead to a generalization of conformal invariance dictate the dynamics. The additional degrees of freedom would be discrete and label conformal equivalence classes of the light-like orbits.

  2. Quantum states are described as spinor fields in WCW - WCW spinors correspond to fermionic Fock states. Zero energy ontology (ZEO) is an important element of the picture and means that physical states are replaced by analogs of physical events- pairs of states whose members reside at the boundaries of causal diamond (CD) with opposite conserved quantum numbers: this guarantees conservation laws. CD is obtained from causal diamond of M4 defined as intersection of future and past directed light-cones by replacing its points with CP2 and has light-like boundaries. Quantum measurement theory based on ZEO resolves the basic paradox of quantum measurement theory and extends it to a theory of consciousness.
  3. Quantum classical correspondence (QCC) is an essential element of quantum TGD and relates to quantum measurement theory: the results of measurements are always interpreted classically. In particular, space-time surfaces as preferred extemals of Kähler action (the lift of Kähler action to twistor space brings in cosmological constant term to 4-D Kähler action in dimensional reduction) define classical correlates for quantum states. Conserved fermionic quantum numbers identified as eigenvalues for Cartan algebra of symmetries are equal to the corresponding classical charges assignable to Kähler action. Already this implies that space-time interior is different for unentangled fermion resp. entangled fermion pairs.

The counterpart of ER=EPR in TGD framework

The TGD variant of ER=EPR has been part of TGD for two decades but have remained un-noticed since superstring hegemony has dominated the theory landscape. There are still many profound ideas to be re-discovered but their realization in the framework of GRT is practically impossible since they relate closely the vision about space-times as 4-surfaces in M4× CP2. What ER=EPR then corresponds in TGD.

  1. In TGD framework one gets rid of blackholes. One can say that they are replaced by the regions of space-time with Euclidian signature of the induced metric. This is of course something completely new from GRT viewpoint. One can say that these regions correspond to 4-D counterparts for the lines of scattering diagrams. Minkowskian and Euclidian space-time regions are separated by light-like 3-surfaces at which the induced 4-metric is singular in the sense that its determinant vanishes. The 4-D tangent space of space-time surface becomes locally 3-D. These surfaces can be identified as light-like orbits of partonic 2-surfaces starting from and and ending at the light-like boundaries of CD.
  2. The orbits of partonic 2-surfaces replace blackhole horizons and can be regarded as carriers of fundamental fermionic quantum numbers and therefore elementary particle numbers. For instance, elementary particles can be seen as pairs of wormhole contacts connected at both sheets by a magnetic flux tube carrying monopole flux so that a closed flux tube results. SH implies that all data about quantum state can be assigned with these 2-D surfaces at future and past ends of CD. There could be wave function in discrete degrees of freedom assignable to the light-like orbits (their conformal equivalence classes).
  3. Wormholes of GRT are replaced with the magnetic flux tubes, which can be homologically trivial or non-trivial. In the latter case wormhole throat behaves effectively as magnetic charge and these are expected to be relevant for elementary particles. The magnetic flux tubes, which are homologically trivial are nearly vacuum extemals and gravitational interactions are expected to be mediated along them.
  4. The counterpart of ER=EPR is that magnetic flux tubes serve as spacetime correlates of entanglement long scales. In CP2 scales wormhole contacts serve in the same role: for instance, gauge bosons correspond to entangled fermion-antifermion pairs at opposite throats of the wormhole of length about CP2 size.

    This should follow from QCC and the challenge is to understand why un-entangled wormhole throats are not connected by magnetic flux tube but entangled ones are.

    The key point is SH. The linearity of quantum theory need to hold true only at the orbits of partonic 2-surfaces and at string world sheets for second quantized induced spinor fields. In the interior of space-time it need not hold true. As a matter, fact it cannot be true since QCC demands that different fermionic Fock states correspond to different space-time interiors.

    The dependence of fermionic Cartan charges on fermionic quantum numbers and entanglement implies the dependence of corresponding classical conserved charges on fermion state. The natural conjecture is that entanglement demands fermionic strings connecting the partonic 2-surfaces assignable to magnetic flux tubes. Interior degrees of freedom would code for the conserved charges of fermionic states.

  5. In TGD framework there is no need to assume a signal between two systems during state function reduction even classically. The magnetic flux tubes fuse the wormhole throats to single system behaving like single particle. Indeed, TGD as a generalization of string model replaces point-like particles with 3-D surfaces and by SH these are replaced with the (conformal equivalence classes of the orbits of) partonic 2-surfaces.
  6. This picture does not imply the emergence of space-time. The entanglement between fermionic states associated with different partonic 2-surfaces breaks the effective 2-dimensionality of the theory predicted otherwise (note that discrete degrees of freedom associated with light-like 3-surfaces are however possible). Entanglement forces genuine 3-dimensionality of the dynamics rather than emergence of 3-space.
The conclusion is that due to SH at space-time level the superposition for fermionic Fock states (also that in orbital WCW degrees of freedom) is consistent with QCC. Notice that fundamental space-time spinor fields identified as induced spinor fields are localized at string world sheets having boundaries at the orbits of partonic 2-surfaces (besides SH and number theoretical vision also the well-definedness of em charge for spinor modes demands this) and therefore cannot as such correspond to the spinor fields of QFT limit. These correspond to the modes of the classical imbedding space spinor fields characterizing the ground states for the representations of super-symplectic algebra acting as isometries of WCW and its extension to Yangian algebra with genetors multi-local with respect to partonic surfaces and generating naturally strongly (perhaps maximally) entangled states. In fact, in TGD framework the entanglement would be always algebraic by number theoretic universality and would be maximally negentropic in p-adic sense although it need not be maximal in real sense.

See the chapter Negentropy Maximization Principle. See also the article ER=EPR and TGD.

Cloning of maximally negentropic states is possible: DNA replication as cloning of this kind of states?

In Facebook discussion with Bruno Marchal and Stephen King the notion of quantum cloning as copying of quantum state popped up and I ended up to ask about approximate cloning and got a nice link about which more below. From Wikipedia one learns some interesting facts cloning. No-cloning theorem states that the cloning of all states by unitary time evolution of the tensor product system is not possible. It is however possible clone orthogonal basis of states. Does this have some deep meaning?

As a response to my question I got a link to an article of Lamourex et al showing that cloning of entanglement - to be distinguished from the cloning of quantum state - is not possible in the general case. Separability - the absence of entanglement - is not preserved. Approximate cloning generates necessarily some entanglement in this case, and the authors give a lower bound for the remaining entanglement in case of an unentangled state pair.

The cloning of maximally entangled state is however possible. What makes this so interesting is that maximally negentropic entanglement for rational entanglement probabilities in TGD framework corresponds to maximal entanglement - entanglement probabilities form a matrix proportional to unit matrix- and just this entanglement is favored by Negentropy Maximization Principle . Could maximal entanglement be involved with say DNA replication? Could maximal negentropic entanglement for algebraic extensions of rationals allow cloning so that DNA entanglement negentropy could be larger than entanglement entropy?

What about entanglement probabilities in algebraic extension of rationals? In this case real number based entanglement entropy is not maximal since entanglement probablities are different. What can one say about p-adic entanglement negentropies: are they still maximal under some reasonable conditions? The logarithms involved depend on p-adic norms of probabilities and this is in the generic case just inverse of the power of p. Number theoretical universality suggests that entanglement probabilities are of form

Pi= ai/N

with ∑ ai= N with algebraic numbers ai not involving natural numbers and thus having unit p-adic norm.

With this assumption p-adic norms of Pi reduce to those of 1/N as for maximal rational entanglement. If this is the case the p-adic negentropy equals to log(pk) if pk divides N. The total negentropy equals to log(N) and is maximal and has the same value as for rational probabilities equal to 1/N.

The real entanglement entropy is now however smaller than log(N), which would mean that p-adic negentropy is larger than the real entropy as conjectured earlier (see this). For rational entanglement probabilities the generation of entanglement negentropy - conscious information during evolution - would be accompanied by a generation of equal entanglement entropy measuring the ignorance about what the negentropically entangled states representing selves are.

This conforms with the observation of Jeremy England that living matter is entropy producer (for TGD inspired commentary see this). For algebraic extensions of rationals this entropy could be however smaller than the total negentropy. Second law follows as a shadow of NMP if the real entanglement entropy corresponds to the thermodynamical entropy. Algebraic evolution would allow to generate conscious information faster than the environment is polluted, one might concretize! The higher the dimension of the algebraic extension rationals, the larger the difference could be and the future of the Universe might be brighter than one might expect by just looking around! Very consolating! One should however show that the above described situation can be realized as NMP strongly suggests before opening a bottle of champaigne.

The impossibility of cloning of entanglement in the general case makes impossible the transfer of information as any kind of entanglement. Maximal entanglement - and maybe be even negentropic entanglement maximal in p-adic sectors - could however make the communication without damaging the information at the source. Since conscious information is associated with p-adic sectors responsible for cognition, one could even allow the modification of the entanglement probabilities and thus of the real entanglement entropy in the communication process since the maximal p-adic negentropy depends only weakly on the entanglement probabilities.

NE is assigned with conscious experiences with positive emotional coloring: experience of understanding, experience of love, etc... There is an old finnish saying, which can be translated to "Shared joy is double joy!". Could the cloning of NE make possible generation of entanglement by loving attitude so that living entities would not be mere thieves trying to steal NE by killing and eating each other?

For background see the chapter Negentropy Maximization Principle. See also the article Is the sum of p-adic negentropies equal to real entropy?.

Wigner's friend and Schrödinger's cat

I encountered in Facebook discussion Wigner's friend paradox (see this and this). Wigner leaves his friend to the laboratory together with Schrödinger's cat and the friend measures the state of cat: the outcome is "dead" or "alive". Wigner returns and learns from his friend what the state of the cat is. The question is: was the state of cat fixed already earlier or when Wigner learned it from his friend. In the latter case the state of friend and cat would have been superposition of pairs in which cat was alive and friend new this and cat was dead also now friend new this. Entanglement between cat and bottle would have been transferred to that between cat+bottle and Wigner's friend. Recall that this kind of information transfer occur in quantum computation and quantum teleportation allows to transfer arbitrary quantum state but destroys the original.

The original purpose of Wigner was to demonstrate that consciousness is involved with the state function collapse. TGD view is that the state function collapse can be seen as moment consciousness. Or more precisely, self as conscious entity corresponds to the repeated state function reduction sequence to the same boundary of causal diamond (CD). One might say that self is generalized Zeno effect in Zero Energy Ontology (ZEO). The first reduction to the opposite boundary of CD means death of self and re-incarnation at opposite boundary as time reversed self. The experiencet flow of time corresponds to the shift of the non-fixed boundary of self reduction by reduction farther from the fixed boundary - also the state at it changes. Thus subjective time as sequence of reductions is mapped to clock time identifiable as the temporal distance between the tips of CD. Arrow of time is generated but changes in death-reincarnation.

In TGD inspired theory of consciousness the intuitive answerto the question of Wigner looks obvious. If the friend measured the state of cat, it was indeed dead or alive already before Wigner arrived. What remains is the question what it means for Wigner, the "ultimate observer", to learn about the state of the cat from his friend. The question is about what conscious communications are.

Consider first the situation in the framework of standard quantum information theory.

  1. Quantum teleportation could make it possible to transfer arbitrary quantum state from the brain of Wigner's friend to Wigner's brain. Quantum teleportation involves generation of Bell state of qubits assignable with Wigner's friend (A) and Wigner (B).
  2. This quantum state can be constructed by a joint measurement of component of spin in same direction at both A and B. One of the four eigenstates of (by convention) the operator Qz= Jx1)⊗ Jy2)-Jy1)⊗ Jx2) is the outcome. For spinors the actions of Jx and Jy change the sign of Jz eigenvalue so that it becomes possible to construct the Bell states as eigenstates of Qz.
  3. After that Wigner's friend measures both the qubit representing cat's state, which is to be communicated and the qubit at A. The latter measurement does not allow to predict the state at B. Wigner's friend communicates the two bits resulting from this measurement to Wigner classically. On basis of these two classical bits his friend performs some unitary operation to the qubit at his end and transforms it to qubit that was to be communicated.
This allows to communicate the qubit representing measurement outcome (alive/dead). But what about meaning? What guarantees that the meaning of the bit representing the state of the cat is the same for Wigner and his friend? One can also ask how the joint measurement can be realized: its seems to require the presence of system containing A⊗ B. To answer these questions one must introduce some notions of TGD inspired theory of consciousness: self hierarchy and subself=mental image identification.

TGD inspired theory of consciousness predicts that during communication Wigner and his friend form a larger entangled system: this makes possible sharing of meaning. Directed attention means that subject and object are entangled. The magnetic flux tubes connecting the two systems would serve as a correlate for the attention. This mechanism would be at work already at the level of molecular biology. Its analog would be wormholes in ER-EPR corresponence proposed by Maldacena and Susskind. Note that directed attention brings in mind the generation of the Bell entangled pair A-B. It would make also possible quantum teleportation.

Wigner's friend could also symbolize the "pointer of the measurement apparatus" constructed to detect whether cats are dead of alive. Consider this option first. If the pointer is subsystem defining subself of Wigner, it would represent mental image of Wigner and there would be no paradox. If qubit in the brain in the brain of Wigner's friend replaces the pointer of measurement apparatus then during communication Wigner and his friend form a larger entangled system experiencing this qubit. Perhaps this temporary fusion of selves allows to answer the question about how common meaning is generated. Note that this would not require quantum teleportation protocol but would allow it.

Negentropically entangled objects are key entities in TGD inspired theory of consciousness and the challenge is to understand how these could be constructed and what their properties could be. These states are diametrically opposite to unentangled eigenstates of single particle operators, usually elements of Cartan algebra of symmetry group. The entangled states should result as eigenstates of poly-local operators. Yangian algebras involve a hierarchy of poly-local operators, and twistorial considerations inspire the conjecture that Yangian counterparts of super-symplectic and other algebras made poly-local with respect to partonic 2-surfaces or end-points of boundaries of string world sheet at them are symmetries of quantum TGD. Could Yangians allow to understand maximal entanglement in terms of symmetries?

  1. In this respect the construction of maximally entangled states using bi-local operator Qz=Jx⊗ Jy - Jy⊗ Jx is highly interesting since entangled states would result by state function. Single particle operator like Jz would generate un-entangled states. The states obtained as eigenstates of this operator have permutation symmetries. The operator can be expressed as Qz=fzijJi⊗ Jj, where fABC are structure constants of SU(2) and could be interpreted as co-product associated with the Lie algebra generator Jz. Thus it would seem that unentangled states correspond to eigenstates of Jz and the maximally entangled state to eigenstates of co-generator Qz. Kind of duality would be in question.
  2. Could one generalize this construction to n-fold tensor products? What about other representations of SU(2)? Could one generalize from SU(2) to arbitrary Lie algebra by replacing Cartan generators with suitably defined co-generators and spin 1/2 representation with fundamental representation? The optimistic guess would be that the resulting states are maximally entangled and excellent candidates for states for which negentropic entanglement is maximized by NMP.
  3. Co-product is needed and there exists a rich spectrum of algebras with co-product (quantum groups, bialgebras, Hopf algebras, Yangian algebras). In particular, Yangians of Lie algebras are generated by ordinary Lie algebra generators and their co-generators subject to constraints. The outcome is an infinite-dimensional algebra analogous to one half of Kac-Moody algebra with the analog of conformal weight N counting the number of tensor factors. Witten gives a nice concrete explanation of Yangian for which co-generators of TA are given as QA= ∑i<j fABC TBi ⊗ TCj, where the summation is over discrete ordered points, which could now label partonic 2-surfaces or points of them or points of string like object. For a practically totally incomprehensible description of Yangian one can look at the Wikipedia article .
  4. This would suggest that the eigenstates of Cartan algebra co-generators of Yangian could define an eigen basis of Yangian algebra dual to the basis defined by the totally unentangled eigenstates of generators and that the quantum measurement of poly-local observables defined by co-generators creates entangled and perhaps even maximally entangled states. A duality between totally unentangled and completely entangled situations is suggestive and analogous to that encountered in twistor Grassmann approach where conformal symmetry and its dual are involved. A beautiful connection between generalization of Lie algebras, quantum measurement theory and quantum information theory would emerge.
For background see the chapter Negentropy Maximization Principle.

Is the sum of p-adic negentropies equal to real entropy?

I ended almost by accident to a fascinating and almost trivial theorem. Adelic theorem for information would state that conscious information represented as sum of p-adic negentropies (entropies, which are negative) is equal to real entropy. The more conscious information, the larger the chaos in the environment as everyone can verify by just looking around;-)

This looks bad! Luckily, it turned out that this statement is true for rational probabilities only. For algebraic extensions it cannot be true as is easy to see. That negentropic entanglement is possible only for algebraic extensions of rationals conforms with the vision that algebraic extensions of rationals characterize evolutionary hierarchy. The rationals represent the lowest level at which there either conscious information vanishes or if equal to p-adic contribution to negentropy is companied by equally large real entropy.

It is not completely obvious that the notion of p-adic negentropy indeed makes sense for algebraic extensions of rationals. A possible problem is caused by the fact that the decomposition of algebraic integer to primes is not unique. Simple argument however strongly suggests that the various p-adic norms of the factors do not depend on the factorization. Also a formula for the difference of the total p-adic negentropy and real entropy is deduced.

p-Adic contribution to negentropy equals to real entropy for rational probabilities but not for algebraic probabilities

The following argument shows that p-adic negentropy equals to real entropy for rational probabilities.

  1. The fusion of real physics and various p-adic physics (identified as correlates for cognition, imagination, and intentionality) to single coherent whole leads to what I call adelic physics. Adeles associated with given extension of rationals are Cartesian product of real number field with all p-adic number fields extended by the extension of rationals. Besides algebraic extensions also the extension by any root of e is possible since it induces finite-dimensional p-adic extension. One obtains hierarchy of adeles and of corresponding adelic physics interpreted as an evolutionary hierarchy.

    An important point is that p-adic Hilbert spaces exist only if one restricts the p-adic numbers to an algebraic extension of rationals having interpretation as numbers in any number field. This is due to the fact that sum of the p-adic valued probabilities can vanish for general p-adic numbers so that the norm of state can vanish. One can say that the Hilbert space of states is universal and is in the algebraic intersection of reality and various p-adicities.

  2. Negentropy Maximization Principle (NMP) is the variational principle of consciousness in TGD framework reducing to quantum measurement theory in Zero Energy Ontology assuming adelic physics. One can define the p-adic counterparts of Shannon entropy for all finite-dimensional extensions of p-adic numbers, and the amazing fact is that these entropies can be negative and thus serve as measures for information rather than for lack of it. Furthermore, all non-vanishing p-adic negentropies are positive and the number of primes contributing to negentropy is finite since any algebraic number can be expressed using a generalization of prime number decomposition of rational number. These p-adic primes characterize given systen, say elementary particle.

    NMP states that the negentropy gain is maximal in the quantum jump defining state function reduction. How does one define the negentropy? As the sum of p-adic negentropies or as the sum of real negative negentropy plus the sum of p-adic negentropies? The latter option I proposed for some time ago without checking what one obtains.

  3. The adelic theorem says that the norm of rational number is equal to the product of the inverses of its p-adic norms. The statement that the sum of real and p-adic negentropies is zero follows more or less as a statement that the logarithms of real norm and the product of p-adic norms for prime factors of rational sum up to zero.

    The core formula is adelic formula stating that the real norm of rational number is product of its p-adic norms. This implies that the logarithm of the rational number is sum over the logarithms of its p-adic norms. Since in p-adic entropy assigned to prime p logarithms of probabilities are replaced by their p-adic norms, this implies that for rational probabilities the real entropy equals to p-adic negentropy. If real entropy is counted as conscious information, the negentropy vanishes identically for rational probabilities.

    It would seem that the negentropy appearing in the definition of NMP must be the sum of p-adic negentropies and real entropy should have interpretation as a measure for ignorance about the state of either entangled system. The sum of p-adic negentropies would serve as a measure for the information carried by a rule with superposed state pairs representingt the instances of the rule. The information would be conscious information and carried by the negentropically entangled system.

  4. What about probabilities in algebraic extensions? The probabilities are now algebraic numbers. Below an argument is develoed that the p-adic norms of of probabilities are uniquely defined and are always powers of primes so that the adelic formula cannot be true since on the real side one has logarithms of algebraic numbers and on the p-adic side only logarithms of primes.

    What could be the interpretation?

    1. If conscious information corresponds to N-P, it accompanies the emergence of algebraic extensions of rationals at the level of Hilbert space.
    2. If N corresponds to conscious information, then at the lowest level conscious information is necessary accompanied by entropy but for algebraic extensions N-P could be positive since N is maximized. This option looks more plausible.
    One however expects that the value of real entropy correlates strongly with the value of negentropy. One expects that the value of real entropy correlates strongly with the value ofp-adic total negentropy. This would conform with the observation that large entropy seems to be a prerequisite for life by providing large number of states with degenerate energies providing large representative capacity. For instance, Jeremy England has made this proposal: I have commented this proposal from TGD point of view.

Formula for the difference of total p-adic negentropy and real entanglement entropy

In the following some non-trivial details related to the definition of p-adic norms for the rationals in the algebraic extension of rationals are discussed.

The induced p-adic norm Np(x) for n-dimensional extension of Q is defined as the determinant det(x) of the linear map defined by multiplication with x. det(x) is rational number. The corresponding p-adic norm is defined as the n:th root Np(det(x))1/n of the ordinary p-adic norm. Root guarantees that the norm co-incides with the ordinary p-adic norm for ordinary p-adic integers. One must perform now a factorization to algebraic primes. Below an argument is given that although the factorization to primes is not always unique, the product of p-adic norms for given algebraic rational defined as ratio of algebraic integers is unique.

Can one write an explicit formula the difference of total p-adic entanglement negentropy (positive) and real entanglement entropy using prime factorization in finite dimensional algebraic extension (note that for algebraic numbers defining infinite-dimensional extension of rationals factorization does not even exist since one can write a=a1/2a1/2=...)? This requires that total p-adic entropy is uniquely defined. There is a possible problem due to the non-uniqueness of the prime factorization.

  1. For Dedekind rings, in particular rings of integers, there exists by definition a unique factorization of proper ideals to prime ideals (see this). In contrast, the prime factorization in the extensions of Q is not always unique. Already for Q((-5)1/2) one has 6=2× 3= (1+(-5)1/2)(1-(-5)1/2) and the primes involved are not related by multiplication with units.

    Various factorizations are characterized by so called class group and class field theory is the branch of number theory studying factorizations in algebraic extensions of integer rings. Factorization is by definition unique for Euclidian domains. Euclidian domains allow by definition so called Euclidian function f(x) having values in R+ with the property that for any a and b one has either a=qb or a= qb+r with f(r)<f(b). It seems that one cannot restrict to Euclidian domains in the recent situation.

  2. Even when the factorization in the extension is not unique, one can hope that the product of various p-adic norms for the factors is same for all factorizations. Since the p-adic norm for the extensions of primes is induced by ordinary p-adic number this requires that the p-adic prime for which the induced p-adic norm differs from unity are same for all factorizations and that the products of p-adic norms differing from unity are same. This independence on the representative for factorization would be analogous to gauge invariance in physicist's conceptualization.

    The probabilities Pk belongs to a unique product of ideals labelled by primes of extension. The ideals are characterized by norms and if this norm is product of p-adic norms for any prime factorization as looks natural then the independence on the factorization follows. Number theorist can certainly immediately tell whether this is true. What is encouraging that for Q((-5)1/2) z=x+(-5)1/2y has determinant det(z)=x2+5y2 and for z==1+/- (-5)1/2 one has has det(z)=6 so that for the products of p-adic norms for the factorizations 6=2× 3 and (1+(-5)1/2)(1-(-5)1/2) are equal.

  3. If this guess is true, one can write the the difference of total p-adic negentropy N and real entanglement entropy S as

    N-S= ∑ Pk log(Pk/∏p Np(Pk)) .

    Here ∏p Np(Pk) would not depend on particular factorization. The condition ∑ Pk=1 poses an additional condition. It would be nice to understand whether N-S≥ 0 holds true generally and if not, what are the conditions guaranteeing this. The p-adic numbers of numerators of rationals involved give positive contributions to N-S as the example Pk=1/N in rational case shows.

For background see the chapter Negentropy Maximization Principle.

Re-incarnation as a basic prediction of TGD inspired theory of consciousness

Life has been hard for skeptics during last two decades. A typical skeptic has as building bricks of his ego the items in the list of notions that they regard as pseudoscientific. This allows to attack the people who have the gift of imagination and passion for genuine understanding, which skeptics unfortunately do not possess. What makes attacks easy that no arguments based on contents are needed and the skeptic need not waste his time by trying to understand the arguments of the person to be labelled as pseudoscientist or crackpot.

The typical rhetoric tricks used begin from replacement of Dr X with Mr X and end up with the "conclusion" that the work of Mr X is totally incomprehensible. I have learned that rather often skeptic of this kind is an academic dropout who never managed to do his MsC. Obviously, the role of skeptic became a manner to survive socially and retain the illusion "I am a scientist". During last decades the list of pseudoscientific notions has shortened item by item as quantum biology and quantum consciousness have emerged as respected branches of science. The notion of re-incarnation has been certainly regarded as one of safest pillars supporting the ego of skeptic but even this pillar is in danger to fall down. Poor skeptics.

It is indeed amusing how fast the attitudes change as ideas evolve and experimental data emerge. Only few years ago I could not say anything definite about reincarnation in the framework of TGD inspired theory of consciousness. Now it has become an unavoidable prediction of zero energy ontology (ZEO), which itself is a "must" in TGD framework.

The prediction related to re-incarnation is however not quite what one might have expected. In death of self a reincarnation as time reversed conscious entity takes place. For time reversed self subjective time evolution corresponds to evolution in a reverse direction of geometric time. The next death/reincarnation after this re-incarnation gives rise a mental image for which the arrow of geometric time is the original one.

Can one test this prediction? If one accepts the predicted fractal self hierarchy in which sub-selves correspond to mental images of self, this is possible. I am too lazy to retype basics about ZEO, CDs, and about how self as generalized Zeno effect emerges and just assume that reader knows the basic concepts or sees to trouble to refresh her knowledge about them.

  1. Self hierarchy predicts that also our mental images are conscious entities. Motor-sensory dichotomy naturally corresponds to sub-self and time reversed sub-self. That is sensory mental image and that associated with motor action induced by sensory input. Motor action initiated in the geometric past at the opposite boundary of causal diamond (CD) (this explains Libet's finding that conscious decision is preceded by neural activity in geometric time). Note that motor action does not proceed from brain to muscles but in reversed time direction from muscles to brain! This conforms with the vision in which magnetic body is intentional agent.
  2. To proceed one must identify EEG correlates for the sub-selves (mental images) and their time reversed re-incarnates. Here the work of Fingelkurts brothers working in Finland helps. They postulate what they call operational architecture of brain (OA) having operations (O) and operational modules (OM) as building bricks. Quasi-stationary EEG segments are assumed to serve as correlates for operations and synchrony of these segments associated with various locations in brain tells that they belong to the same OM.

    Synchrony means spatio-temporal coherence - not only spatial - and is very natural concept in ZEO, where 4-D CDs and space-time surfaces inside them serve as geometric correlates of selves. Synchrony implies that these EEG segments at different spatial locations begin and end at the same time. Between EEG segments there is rapid transition period (RTP) allowing to distinguish segments from each other. Quasi-stationary segments of EEG have average duration is about .3 seconds.

    The translation of this picture to TGD framework is rather straighforward. Operations correspond to sub-selves and OMs to collections of them forming sub-selves of self. CDs (sub-CDs) in turn serve as geometric correlates for selves (sub-selves). The quasi-stationary segments of EEG become correlates for sub-selves/mental images. Operational module corresponds to a self/CD having sub-selves/sub-CDs with synchronous EEG segments. The average duration of mental image would be about .3 seconds.

    Two sub-sequent quasi-stationary segments separated by RTP would correspond to sub-self and its re-incarnation in the original time direction. Note that a very brief period of geometric time defined by the duration of RTP identifiable as the duration of a unitary time evolution between two sub-sequent state function reductions at the same boundary of CD corresponds to a finite duration of experienced time - the lifetime of the time reversed mental image!

    The testable prediction is that the segment corresponding to time-reversed sub-self is located in geometric past and runs in opposite direction of geometric time. This EEG segment should be assignable to motor response accompanying sensory mental image. This is a highly non-trivial prediction testing the new view about time.

  3. One can check whether these EEG segments appear as pairs with first member assignable to sensory mental image and second one to motor mental image. Time reversal implies that second law is obeyed in "wrong" time direction for EEG segment assignable to the motor output and this can be tested. Already Fantappie discovered that both directions of (geometric) time appear in living matter and introduced the notion of syntropy as time reversal of entropy. Spontaneous molecular self-assembly is a basic example of a syntropic process and identifiable as a decay process in reverse direction of geometric time. Phase conjugation is known to occur for phase conjugate laser light and sound. Does a process analogous to self-assembly occur for segments of EEG associated with motor actions: is the motor part of EEG time reversed? To answer this question one needs phase information about EEG besides power spectrum. In principle this information is contained in EEG.

For background see the chapter About the nature of time.

Inverse Research on Decisions Shows Instinct Makes Us Behave Like Cyborgs, not Robots: Really?

I learned about an interesting work, which relates to the relationship of experienced time and geometric time but ortodoxically assumes that these two and one and the same thing. The title of the popular article was Inverse Research on Decisions Shows Instinct Makes Us Behave Like Cyborgs, not Robots (see this). It tells about the work of Adam Bear and Paul Bloom. The article claims that our mind for some mysterious-to-me reason tricks us to believe that were are responsible for totally automatic or reflexive behaviours. In fact, these behaviors by definition are such that we do not feel of being responsible for them. Bear how allows us some subconscious free will so that we are not programmed robots but cyborgs whatever that might mean.

This work is an excellent example about how a dominating paradigm, which is wrong, leads to wrong interpretation of experimental findings, which as such are correct. The standard belief in neuroscience and standard physics is that causal effects propagate always in the same direction of the geometric time. This interpretation follows from the identification of geometric time (time of physicist) with subjective time. This despite the fact that these times have very different properties: consider only reversibility viz. irreversibility, existence of both future and past viz. only past exists.

The classical experiments of Libet challenge this dogma. Person decides to raise finger but neuro-activity begins fraction of second earlier. Mainstream neuroscientist of interprets this by saying that there is no free will. Second proposed interpretation is that the decision is made earlier at subconscious level and at our level the experience of free will is an illusion. One can of course wonder why this illusion.

The third manner to interpret the situation respects our immediate experience that we indeed have free will but in order to avoid mathematical contradictions must be accompanied by a new more general view about quantum physics accepting as a fact that there are two causalities: that of free will and that of deterministic laws of field equations. In TGD framework Zero Energy Ontology realizes this view. The outcome is prediction of signals which can propagate in both directions of geometric time. If the conscious decision generates a signal to geometric past it initiates a neural activity in geometric past. An excellent tool for survival in jungle or in modern market economy full or merciless predators.

In the experiment considered subject persons saw five dots and selected one. One of the dots became red with a varying time lag but subject person did not know when. Subject person had to tell whether her choice had been correct, wrong, or whether she had failed to make any choice at all before the change took place.

The surprising observation was that the shorter the time interval from the guess to change of color to red was, the better the reported ability to guess correctly was and in conflict with statistical model based on fixed arrow of time. If information can travel backwards in geometric time, the natural interpretation would be the same as in Libet's experiments and in the experiments of say Radin and Bierman claimed to demonstrate precognition. This is possible in zero energy ontology (ZEO). ZEO allows also a slightly different interpretation relies. In ZEO in which mental images correspond to causal diamonds (CDs). For sensory mental images their time scale would be of order .1 seconds so that below this scale one cannot anymore put events in precise time order and one indeed has precognition. What this means that one does not know whether the sensory input corresponds to the "upper" or "lower" boundary of CD so that these interpretations are equivalent.

Neuroscientist cannot of course publicly utter the word "precognition" associating immediately with really dirty word "paranormal". The orthodox conclusion is that subject persons are "cheating" themselves without knowing it. Very bizarre interpretation - if taken completely seriously it forces to question all our knowledge! One can also ask, why the subjects would tend to cheat themselves when the change occurred immediately after their choice: why not always? Certainly the interpretation is a heroic attempt to save the standard world view.

A simple modification of the experiment would be an addition of a keystroke telling the choice when it was done and before the change in color. This would immediately tell whether something like precognition was involved.

For background see the chapter About the nature of time.

Number Theoretical Feats and TGD Inspired Theory of Consciousness

Number theoretical feats of some mathematicians like Ramanujan remain a mystery for those believing that brain is a classical computer. Also the ability of idiot savants - lacking even the idea about what prime is - to factorize integers to primes challenges the idea that an algorithm is involved. In this article I discuss ideas about how various arithmetical feats such as partitioning integer to a sum of integers and to a product of prime factors might take place. The ideas are inspired by the number theoretic vision about TGD suggesting that basic arithmetics might be realized as naturally occurring processes at quantum level and the outcomes might be "sensorily perceived". One can also ask whether zero energy ontology (ZEO) could allow to perform quantum computations in polynomial instead of exponential time.

The indian mathematician Srinivasa Ramanujan is perhaps the most well-known example about a mathematician with miraculous gifts. He told immediately answers to difficult mathematical questions - ordinary mortals had to to hard computational work to check that the answer was right. Many of the extremely intricate mathematical formulas of Ramanujan have been proved much later by using advanced number theory. Ramanujan told that he got the answers from his personal Goddess. A possible TGD based explanation of this feat relies on the idea that in zero energy ontology (ZEO) quantum computation like activity could consist of steps consisting quantum computation and its time reversal with long-lasting part of each step performed in reverse time direction at opposite boundary of causal diamond so that the net time used would be short at second boundary.

The adelic picture about state function reduction in ZEO suggests that it might be possible to have direct sensory experience about prime factorization of integers (see this). What about partitions of integers to sums of primes? For years ago I proposed that symplectic QFT is an essential part of TGD. The basic observation was that one can assign to polygons of partonic 2-surface - say geodesic triangles - Kähler magnetic fluxes defining symplectic invariance identifiable as zero modes. This assignment makes sense also for string world sheets and gives rise to what is usually called Abelian Wilson line. I could not specify at that time how to select these polygons. A very natural manner to fix the vertices of polygon (or polygons) is to assume that they correspond ends of fermion lines which appear as boundaries of string world sheets. The polygons would be fixed rather uniquely by requiring that fermions reside at their vertices.

The number 1 is the only prime for addition so that the analog of prime factorization for sum is not of much use. Polygons with n=3,4,5 vertices are special in that one cannot decompose them to non-degenerate polygons. Non-degenerate polygons also represent integers n>2. This inspires the idea about numbers 3,4,5 as "additive primes" for integers n>2 representable as non-degenerate polygons. These polygons could be associated many-fermion states with negentropic entanglement (NE) - this notion relate to cognition and conscious information and is something totally new from standard physics point of view. This inspires also a conjecture about a deep connection with arithmetic consciousness: polygons would define conscious representations for integers n>2. The splicings of polygons to smaller ones could be dynamical quantum processes behind arithmetic conscious processes involving addition.

For details see the chapter Conscious Information and Intelligence or the article Number Theoretical Feats and TGD Inspired Theory of Consciousness.

NMP and adelic physics

In given p-adic sector the entanglement entropy (EE) is defined by replacing the logarithms of probabilities in Shannon formula by the logarithms of their p-adic norms. The resulting entropy satisfies the same axioms as ordinary entropy but makes sense only for probabilities, which must be rational valued or in an algebraic extension of rationals. The algebraic extensions corresponds to the evolutionary level of system and the algebraic complexity of the extension serves as a measure for the evolutionary level. p-Adically also extensions determined by roots of e can be considered. What is so remarkable is that the number theoretic entropy can be negative.

A simple example allows to get an idea about what is involved. If the entanglement probabilities are rational numbers Pi=Mi/N, ∑i Mi=N, then the primes appearing as factors of N correspond to a negative contribution to the number theoretic entanglement entropy and thus to information. The factors of Mi correspond to negative contributions. For maximal entanglement with Pi=1/N in this case the EE is negative. The interpretation is that the entangled state represents quantally concept or a rule as superposition of its instances defined by the state pairs in the superposition. Identity matrix means that one can choose the state basis in arbitrary manner and the interpretation could be in terms of "enlightened" state of consciousness characterized by "absence of distinctions". In general case the basis is unique.

Metabolism is a central concept in biology and neuroscience. Usually metabolism is understood as transfer of ordered energy and various chemical metabolites to the system. In TGD metabolism could be basically just a transfer of NE from nutrients to the organism. Living systems would be fighting for NE to stay alive (NMP is merciless!) and stealing of NE would be the fundamental crime.

TGD has been plagued by a longstanding interpretational problem: can one apply the notion of number theoretic entropy in the real context or not. If this is possible at all, under what conditions this is the case? How does one know that the entanglement probabilities are not transcendental as they would be in generic case? There is also a second problem: p-adic Hilbert space is not a well-defined notion since the sum of p-adic probabilities defined as moduli squared for the coefficients of the superposition of orthonormal states can vanish and one obtains zero norm states.

These problems disappear if the reduction occurs in the intersection of reality and p-adicities since here Hilbert spaces have some algebraic number field as coefficient field. By SH the 2-D states states provide all information needed to construct quantum physics. In particular, quantum measurement theory.

  1. The Hilbert spaces defining state spaces has as their coefficient field always some algebraic extension of rationals so that number theoretic entropies make sense for all primes. p-Adic numbers as coefficients cannot be used and reals are not allowed. Since the same Hilbert space is shared by real and p-adic sectors, a given state function reduction in the intersection has real and p-adic space-time shadows.
  2. State function reductions at these 2- surfaces at the ends of causal diamond (CD) take place in the intersection of realities and p-adicities if the parameters characterizing these surfaces are in the algebraic extension considered. It is however not absolutely necessary to assume that the coordinates of WCW belong to the algebraic extension although this looks very natural.
  3. NMP applies to the total EE. It can quite well happen that NMP for the sum of real and p-adic entanglement entropies does not allow ordinary state function reduction to take place since p-adic negative entropies for some primes would become zero and net negentropy would be lost. There is competition between real and p-adic sectors and p-adic sectors can win! Mind has causal power: it can stabilize quantum states against state function reduction and tame the randomness of quantum physics in absence of cognition! Can one interpret this causal power of cognition in terms of intentionality? If so, p-adic physics would be also physics of intentionality as originally assumed.
A fascinating question is whether the p-adic view about cognition could allow to understand the mysterious looking ability of idiot savants (not only of them but also of some greatest mathematicians) to decompose large integers to prime factors. One possible mechanism is that the integer N represented concretely is mapped to a maximally entangled state with entanglement probabilities Pi=1/N, which means NE for the prime factors of Pi or N. The factorization would be experienced directly.

One can also ask, whether the other mathematical feats performed by idiot savants could be understood in terms of their ability to directly experience - "see" - the prime composition (adelic decomposition) of integer or even rational. This could for instance allow to "see" if integer is - say 3rd - power of some smaller integer: all prime exponents in it would be multiples of 3. If the person is able to generate an NE for which probabilities Pi=Mi/N are apart from normalization equal to given integers Mi, ∑ Mi=N, then they could be able to "see" the prime compositions for Mi and N. For instance, they could "see" whether both Mi and N are 3rd powers of some integer and just by going through trials find the integers satisfying this condition.

For details see the chapter Negentropy Maximization Principle or the article TGD Inspired Comments about Integrated Information Theory of Consciousness.

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