Wednesday, April 29, 2015

What could be the origin of p-adic length scale hypothesis?

The argument would explain the existence of preferred p-adic primes. It does not yet explain p-adic length scale hypothesis stating that p-adic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3-adic time scale hierarchies in living matter (see this) and in music both 2-adicity and 3-adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this).

The weak form of NMP might come in rescue here.

  1. Entanglement negentropy for a negentropic entanglement characterized by n-dimensional projection operator is the log(Np(n) for some p whose power divides n. The maximum negentropy is obtained if the power of p is the largest power of prime divisor of p, and this can be taken as definition of number theoretic entanglement negentropy. If the largest divisor is pk, one has N= k× log(p). The entanglement negentropy per entangled state is N/n=klog(p)/n and is maximal for n=pk. Hence powers of prime are favoured which means that p-adic length scale hierarchies with scales coming as powers of p are negentropically favored and should be generated by NMP. Note that n=pk would define a hierarchy of heff/h=pk. During the first years of heff hypothesis I believe that the preferred values obey heff=rk, r integer not far from r= 211. It seems that this belief was not totally wrong.

  2. If one accepts this argument, the remaining challenge is to explain why primes near powers of two (or more generally p) are favoured. n=2k gives large entanglement negentropy for the final state. Why primes p=n2= 2k-r would be favored? The reason could be following. n=2k corresponds to p=2, which corresponds to the lowest level in p-adic evolution since it is the simplest p-adic topology and farthest from the real topology and therefore gives the poorest cognitive representation of real preferred extremal as p-adic preferred extermal (Note that p=1 makes formally sense but for it the topology is discrete).

  3. Weak form of NMP suggests a more convincing explanation. The density matrix of the state to be reduced is a direct sum over contributions proportional to projection operators. Suppose that the projection operator with largest dimension has dimension n. Strong form of NMP would say that final state is characterized by n-dimensional projection operator. Weak form of NMP allows free will so that all dimensions n-k, k=0,1,...n-1 for final state projection operator are possible. 1-dimensional case corresponds to vanishing entanglement negentropy and ordinary state function reduction isolating the measured system from external world.

  4. The negentropy of the final state per state depends on the value of k. It is maximal if n-k is power of prime. For n=2k=Mk+1, where Mk is Mersenne prime n-1 gives the maximum negentropy and also maximal p-adic prime available so that this reduction is favoured by NMP. Mersenne primes would be indeed special. Also the primes n=2k-r near 2k produce large entanglement negentropy and would be favored by NMP.

  5. This argument suggests a generalization of p-adic length scale hypothesis so that p=2 can be replaced by any prime.

This argument together with the hypothesis that preferred prime is ramified would correlate the character of the irreducible extension and character of super-conformal symmetry breaking. The integer n characterizing super-symplectic conformal sub-algebra acting as gauge algebra would depends on the irreducible algebraic extension of rational involved so that the hierarchy of quantum criticalities would have number theoretical characterization. Ramified primes could appear as divisors of n and n would be essentially a characteristic of ramification known as discriminant. An interesting question is whether only the ramified primes allow the continuation of string world sheet and partonic 2-surface to a 4-D space-time surface. If this is the case, the assumptions behind p-adic mass calculations would have full first principle justification.

For details see the article The Origin of Preferred p-Adic Primes?.

For a summary of earlier postings see Links to the latest progress in TGD.

Tuesday, April 28, 2015

How preferred p-adic primes could be determined?

p-Adic mass calculations allow to conclude that elementary particles correspond to one or possible several preferred primes assigning p-adic effective topology to the real space-time sheets in discretization in some length scale range. TGD inspired theory of consciousness leads to the identification of p-adic physics as physics of cognition. The recent progress leads to the proposal that quantum TGD is adelic: all p-adic number fields are involved and each gives one particular view about physics.

Adelic approach plus the view about evolution as emergence of increasingly complex extensions of rationals leads to a possible answer to th question of the title. The algebraic extensions of rationals are characterized by preferred rational primes, namely those which are ramified when expressed in terms of the primes of the extensions. These primes would be natural candidates for preferred p-adic primes.

1. Earlier attempts

How the preferred primes emerges in this framework? I have made several attempts to answer this question.

  1. Classical non-determinism at space-time level for real space-time sheets could in some length scale range involving rational discretization for space-time surface itself or for parameters characterizing it as a preferred extremal correspond to the non-determinism of p-adic differential equations due to the presence of pseudo constants which have vanishing p-adic derivative. Pseudo- constants are functions depend on finite number of pinary digits of its arguments.

  2. The quantum criticality of TGD is suggested to be realized in in terms of infinite hierarchies of super-symplectic symmetry breakings in the sense that only a sub-algebra with conformal weights which are n-multiples of those for the entire algebra act as conformal gauge symmetries. This might be true for all conformal algebras involved. One has fractal hierarchy since the sub-algebras in question are isomorphic: only the scale of conformal gauge symmetry increases in the phase transition increasing n. The hierarchies correspond to sequences of integers n(i) such tht n(i) divides n(i+1). These hierarchies would very naturally correspond to hierarchies of inclusions of hyper-finite factors and m(i)= n(i+1)/n(i) could correspond to the integer n characterizing the index of inclusion, which has value n≥ 3. Possible problem is that m(i)=2 would not correspond to Jones inclusion. Why the scaling by power of two would be different? The natural question is whether the primes dividing n(i) or m(i) could define the preferred primes.

  3. Negentropic entanglement corresponds to entanglement for which density matrix is projector. For n-dimensional projector any prime p dividing n gives rise to negentropic entanglement in the sense that the number theoretic entanglement entropy defined by Shannon formula by replacing pi in log(pi)= log(1/n) by its p-adic norm Np(1/n) is negative if p divides n and maximal for the prime for which the dividing power of prime is largest power-of-prime factor of n. The identification of p-adic primes as factors of n is highly attractive idea. The obvious question is whether n corresponds to the integer characterizing a level in the hierarchy of conformal symmetry breakings.

  4. The adelic picture about TGD led to the question whether the notion of unitary could be generalized. S-matrix would be unitary in adelic sense in the sense that Pm=(SS)mm=1 would generalize to adelic context so that one would have product of real norm and p-adic norms of Pm. In the intersection of the realities and p-adicities Pm for reals would be rational and if real and p-adic Pm correspond to the same rational, the condition would be satisfied. The condition that Pm≤ 1 seems however natural and forces separate unitary in each sector so that this options seems too tricky.

These are the basic ideas that I have discussed hitherto.

2. Could preferred primes characterize algebraic extensions of rationals?

The intuitive feeling is that the notion of preferred prime is something extremely deep and the deepest thing I know is number theory. Does one end up with preferred primes in number theory? This question brought to my mind the notion of ramification of primes (see this) (more precisely, of prime ideals of number field in its extension), which happens only for special primes in a given extension of number field, say rationals. Could this be the mechanism assigning preferred prime(s) to a given elementary system, such as elementary particle? I have not considered their role earlier also their hierarchy is highly relevant in the number theoretical vision about TGD.

  1. Stating it very roughly (I hope that mathematicians tolerate this language): As one goes from number field K, say rationals Q, to its algebraic extension L, the original prime ideals in the so called integral closure (see this) over integers of K decompose to products of prime ideals of L (prime is a more rigorous manner to express primeness).

    Integral closure for integers of number field K is defined as the set of elements of K, which are roots of some monic polynomial with coefficients, which are integers of K and having the form xn+an-1xn-1+...+a0 . The integral closures of both K and L are considered. For instance, integral closure of algebraic extension of K over K is the extension itself. The integral closure of complex numbers over ordinary integers is the set of algebraic numbers.

  2. There are two further basic notions related to ramification and characterizing it. Relative discriminant is the ideal divided by all ramified ideals in K and relative different is the ideal of L divided by all ramified Pi:s. Note that te general ideal is analog of integer and these ideas represent the analogous of product of preferred primes P of K and primes Pi of L dividing them.

  3. A physical analogy is provided by decomposition of hadrons to valence quarks. Elementary particles becomes composite of more elementary particles in the extension. The decomposition to these more elementary primes is of form P= ∏ Pie(i), where ei is the ramification index - the physical analog would be the number of elementary particles of type i in the state (see this). Could the ramified rational primes could define the physically preferred primes for a given elementary system?

In TGD framework the extensions of rationals (see this) and p-adic number fields (see this) are unavoidable and interpreted as an evolutionary hierarchy physically and cosmological evolution would have gradually proceeded to more and more complex extensions. One can say that string world sheets and partonic 2-surfaces with parameters of defining functions in increasingly complex extensions of prime emerge during evolution. Therefore ramifications and the preferred primes defined by them are unavoidable. For p-adic number fields the number of extensions is much smaller for instance for p>2 there are only 3 quadratic extensions.
  1. In p-adic context a proper definition of counterparts of angle variables as phases allowing definition of the analogs of trigonometric functions requires the introduction of algebraic extension giving rise to some roots of unity. Their number depends on the angular resolution. These roots allow to define the counterparts of ordinary trigonometric functions - the naive generalization based on Taylors series is not periodic - and also allows to defined the counterpart of definite integral in these degrees of freedom as discrete Fourier analysis. For the simplest algebraic extensions defined by xn-1 for which Galois group is abelian are are unramified so that something else is needed. One has decomposition P= ∏ Pie(i), e(i)=1, analogous to n-fermion state so that simplest cyclic extension does not give rise to a ramification and there are no preferred primes.

  2. What kind of polynomials could define preferred algebraic extensions of rationals? Irreducible polynomials are certainly an attractive candidate since any polynomial reduces to a product of them. One can say that they define the elementary particles of number theory. Irreducible polynomials have integer coefficients having the property that they do not decompose to products of polynomials with rational coefficients. It would be wrong to say that only these algebraic extensions can appear but there is a temptation to say that one can reduce the study of extensions to their study. One can even consider the possibility that string world sheets associated with products of irreducible polynomials are unstable against decay to those characterize irreducible polynomials.

  3. What can one say about irreducible polynomials? Eisenstein criterion states following. If Q(x)= ∑k=0,..,n akxk is n:th order polynomial with integer coefficients and with the property that there exists at least one prime dividing all coefficients ai except an and that p2 does not divide a0, then Q is irreducible. Thus one can assign one or more preferred primes to the algebraic extension defined by an irreducible polynomial Q - in fact any polynomial
    allowing ramification. There are also other kinds of irreducible polynomials since Eisenstein's condition is only sufficient but not necessary.

  4. Furthermore, in the algebraic extension defined by Q, the primes P having the above mentioned characteristic property decompose to an n :th power of single prime Pi: P= Pin. The primes are maximally/completely ramified. The physical analog P=P0n is Bose-Einstein condensate of n bosons. There is a strong temptation to identify the preferred primes of irreducible polynomials as preferred p-adic primes.

    A good illustration is provided by equations x2+1=0 allowing roots x+/-=+/- i and equation x2+2px+p=0 allowing roots x+/-= -p+/-p1/2p-11/2. In the first case the ideals associated with +/- i are different. In the second case these ideals are one and the same since x+= =- x- +p: hence one indeed has ramification. Note that the first example represents also an example of irreducible polynomial, which does not satisfy Eisenstein criterion. In more general case the n conditions on defined by symmetric functions of roots imply that the ideals are one and same when Eisenstein conditions are satisfied.

  5. What does this mean in p-adic context? The identity of the ideals can be stated by saying P= P0n for the ideals defined by the primes satisfying the Eisenstein condition. Very loosely one can say that the algebraic extension defined by the root involves n:th root of p-adic prime p. This does not work! Extension would have a number whose n:th power is zero modulo p. On the other hand, the p-adic numbers of the extension modulo p should be finite field but this would not be field anymore since there would exist a number whose n:th power vanishes. The algebraic extension simply does not exist for preferred primes. The physical meaning of this will be considered later.

  6. What is so nice that one could readily construct polynomials giving rise to given preferred primes. The complex roots of these polymials could correspond to the points of partonic 2-surfaces carrying fermions and defining the ends of boundaries of string world sheet. It must be however emphasized that the form of the polynomial depends on the choices of the complex coordinate. For instance, the shift x→ x+1 transforms (xn-1)/(x-1) to a polynomial satisfying the Eisenstein criterion. One should be able to fix allowed coordinate changes in such a manner that the extension remains irreducible for all allowed coordinate changes.

    Already the integral shift of the complex coordinate affects the situation. It would seem that only the action of the allowed coordinate changes must reduce to the action of Galois group permuting the roots of polynomials. A natural assumption is that the complex coordinate corresponds to a complex coordinate transforming linearly under subgroup of isometries of the imbedding space.

In the general situation one has P= ∏ Pie(i), e(i)≥ 1 so that aso now there are prefered primes so that the appearance of preferred primes is completely general phenomenon.

3. A connection with Langlands program?

In Langlands program (see this) the great vision is that the n-dimensional representations of Galois groups G characterizing algebraic extensions of rationals or more general number fields define n-dimensional adelic representations of adelic Lie groups, in particular the adelic linear group Gl(n,A). This would mean that it is possible to reduce these representations to a number theory for adeles. This would be highly relevant in the vision about TGD as a generalized number theory. I have speculated with this possibility earlier (see this) but the mathematics is so horribly abstract that it takes decade before one can have even hope of building a rough vision.

One can wonder whether the irreducible polynomials could define the preferred extensions K of rationals such that the maximal abelian extensions of the fields K would in turn define the adeles utilized in Langlands program. At least one might hope that everything reduces to the maximally ramified extensions.

At the level of TGD string world sheets with parameters in an extension defined by an irreducible polynomial would define an adele containing various p-adic number fields defined by the primes of the extension. This would define a hierarchy in which the prime ideals of previous level would decompose to those of the higher level. Each irreducible extension of rationals would correspond to some physically preferred p-adic primes.

It should be possible to tell what the preferred character means in terms of the adelic representations. What happens for these representations of Galois group in this case? This is known.

  1. For Galois extensions ramification indices are constant: e(i)=e and Galois group acts transitively on ideals Pi dividing P. One obtains an n-dimensional representation of Galois group. Same applies to the subgroup of Galois group G/I where I is subgroup of G leaving Pi invariant. This group is called inertia group. For the maximally ramified case G maps the ideal P0 in P=P0n to itself so that G=I and the action of Galois group is trivial taking P0 to itself, and one obtains singlet representations.

  2. The trivial action of Galois group looks like a technical problem for Langlands program and also for TGD unless the singletness of Pi under G has some physical interpretation. One possibility is that Galois group acts as like a gauge group and here the hierarchy of sub-algebras of super-symplectic algebra labelled by integers n is highly suggestive. This raises obvious questions. Could the integer n characterizing the sub-algebra of super-symplectic algebra acting as conformal gauge transformations, define the integer defined by the product of ramified primes? P0n brings in mind the n conformal equivalence classes which remain invariant under the conformal transformations acting as gauge transformiations. . Recalling that relative discriminant is an of K ideal divisible by ramified prime ideals of K, this means that n would correspond to the relative discriminant for K=Q.

    Are the preferred primes those which are "physical" in the sense that one can assign to the states satisfying conformal gauge conditions?

4. A connection with infinite primes?

Infinite primes are one of the mathematical outcomes of TGD. There are two kinds of infinite primes. There are the analogs of free many particle states consisting of fermions and bosons labelled by primes of the previous level in the hierarchy. They correspond to states of a supersymmetric arithmetic quantum field theory or actually a hierarchy of them obtained by a repeated second quantization of this theory. A connection between infinite primes representing bound states and and irreducible polynomials is highly suggestive.

  1. The infinite prime representing free many-particle state decomposes to a sum of infinite part and finite part having no common finite prime divisors so that prime is obtained. The infinite part is obtained from "fermionic vacuum" X= ∏kpk by dividing away some fermionic primes pi and adding their product so that one has X→ X/m+ m, where m is square free integer. Also m=1 is allowed and is analogous to fermionic vacuum interpreted as Dirac sea without holes. X is infinite prime and pure many-fermion state physically. One can add bosons by multiplying X with any integers having no common denominators with m and its prime decomposition defines the bosonic contents of the state. One can also multiply m by any integers whose prime factors are prime factors of m.

  2. There are also infinite primes, which are analogs of bound states and at the lowest level of the hierarchy they correspond to irreducible polynomials P(x) with integer coefficients. At the second levels the bound states would naturally correspond to irreducible polynomials Pn(x) with coefficients Qk(y), which are infinite integers at the previous level of the hierarchy.

  3. What is remarkable that bound state infinite primes at given level of hierarchy would define maximally ramified algebraic extensions at previous level. One indeed has infinite hierarchy of infinite primes since the infinite primes at given level are infinite primes in the sense that they are not divisible by the primes of the previous level. The formal construction works as such. Infinite primes correspond to polynomials of single variable at the first level, polynomials of two variables at second level, and so on. Could the Langlands program could be generalized from the extensions of rationals to polynomials of complex argument and that one would obtain infinite hierarchy?

  4. Infinite integers in turn could correspond to products of irreducible polynomials defining more general extensions. This raises the conjecture that infinite primes for an extension K of rationals could code for the algebraic extensions of K quite generally. If infinite primes correspond to real quantum states they would thus correspond the extensions of rationals to which the parameters appearing in the functions defining partonic 2-surfaces and string world sheets.

    This would support the view that partonic 2-surfaces associated with algebraic extensions defined by infinite integers and thus not irreducible are unstable against decay to partonic 2-surfaces which corresponds to extensions assignable to infinite primes. Infinite composite integer defining intermediate unstable state would decay to its composites. Basic particle physics phenomenology would have number theoretic analog and even more.

  5. According to Wikipedia, Eisenstein's criterion (see this) allows generalization and what comes in mind is that it applies in exactly the same form also at the higher levels of the hierarchy. Primes would be only replaced with prime polynomials and the there would be at least one prime polynomial Q(y) dividing the coefficients of Pn(x) except the highest one such that its square would not divide P0. Infinite primes would give rise to an infinite hierarchy of functions of many complex variables. At first level zeros of function would give discrete points at partonic 2-surface. At second level one would obtain 2-D surface: partonic 2-surfaces or string world sheet. At the next level one would obtain 4-D surfaces. What about higher levels? Does one obtain higher dimensional objects or something else. The union of n 2-surfaces can be interpreted also as 2n-dimensional surface and one could think that the hierarchy describes a hierarchy of unions of correlated partonic 2-surfaces. The correlation would be due to the preferred extremal property of Kähler action.

    One can ask whether this hierarchy could allow to generalize number theoretical Langlands to the case of function fields using the notion of prime function assignable to infinite prime. What this hierarchy of polynomials of arbitrary many complex arguments means physically is unclear. Do these polynomials describe many-particle states consisting of partonic 2-surface such that there is a correlation between them as sub-manifolds of the same space-time sheet representing a preferred extremals of Kähler action?

This would suggest strongly the generalization of the notion of p-adicity so that it applies to infinite primes.
  1. This looks sensible and maybe even practical! Infinite primes can be mapped to prime polynomials so that the generalized p-adic numbers would be power series in prime polynomial - Taylor expansion in the coordinate variable defined by the infinite prime. Note that infinite primes (irreducible polynomials) would give rise to a hierarchy of preferred coordinate variables. In terms of infinite primes this expansion would require that coefficients are smaller than the infinite prime P used. Are the coefficients lower level primes? Or also infinite integers at the same level smaller than the infinite prime in question? This criterion makes sense since one can calculate the ratios of infinite primes as real numbers.

  2. I would guess that the definition of infinite-P p-adicity is not a problem since mathematicians have generalized the number theoretical notions to such a level of abstraction much above of a layman like me. The basic question is how to define p-adic norm for the infinite primes (infinite only in real sense, p-adically they have unit norm for all lower level primes) so that it is finite.

  3. There exists an extremely general definition of generalized p-adic number fields (see this). One considers Dedekind domain D, which is a generalization of integers for ordinary number field having the property that ideals factorize uniquely to prime ideals. Now D would contain infinite integers. One introduces the field E of fractions consisting of infinite rationals.

    Consider element e of E and a general fractional ideal eD as counterpart of ordinary rational and decompose it to a ratio of products of powers of ideals defined by prime ideals, now those defined by infinite primes. The general expression for the p-adic norm of x is x-ord(P), where n defines the total number of ideals P appearing in the factorization of a fractional ideal in E: this number can be also negative for rationals. When the residue field is finite (finite field G(p,1) for p-adic numbers), one can take c to the number of its elements (c=p for p-adic numbers.

    Now it seems that this number is not finite since the number of ordinary primes smaller than P is infinite! But this is not a problem since the topology for completion does not depend on the value of c. The simple infinite primes at the first level (free many-particle states) can be mapped to ordinary rationals and q-adic norm suggests itself: could it be that infinite-P p-adicity corresponds to q-adicity discussed by Khrennikov about p-adic analysics. Note however that q-adic numbers are not a field.

Finally, a loosely related question. Could the transition from infinite primes of K to those of L takes place just by replacing the finite primes appearing in infinite prime with the decompositions? The resulting entity is infinite prime if the finite and infinite part contain no common prime divisors in L. This is not the case generally if one can have primes P1 and P2 of K having common divisors as primes of L: in this case one can include P1 to the infinite part of infinite prime and P2 to finite part.

For details see the article The Origin of Preferred p-Adic Primes?.

For a summary of earlier postings see Links to the latest progress in TGD.

Monday, April 27, 2015

Could adelic approach allow to understand the origin of preferred p-adic primes?

The comment of Crow to the posting Intentions, cognitions, and time stimulated rather interesting ideas about the adelization of quantum TGD.

First two questions.

  1. What is Adelic quantum TGD? The basic vision is that scattering amplitudes are obtained by algebraic continuation
    to various number fields from the intersection of realities and p-adicities (briefly intersection in what follows) represented at the space-time level by string world sheets and partonic 2-surfaces for which defining parameters (WCW coordinates) are in rational or in in some algebraic extension of p-adic numbers. This principle is a combination of strong form of holography and algebraic continuation as a manner to achieve number theoretic universality.

  2. Why Adelic quantum TGD? Adelic approach is free of the earlier assumptions, which require mathematics which need not exist: transformation of p-adic space-time surfaces to real ones as a realization of intentional actions was the questionable assumption, which is un-necessary if cognition and matter are two different aspects of existence as already the success of p-adic mass calculations strongly suggests. It always takes years to develop ability to see things from bigger perspective and distill discoveries from clever inventions. Now adelicity is totally obvious. Being a conservative radical - not radical radical or radical conservative - is the correct strategy which I have been gradually learning. This particular lesson was excellent!
For some years ago Crow sent to me the book of Lapidus about adelic strings. Witten wrote for long time ago an article in which the demonstrated that that the product of real stringy vacuum amplitude and its p-adic variants equals to 1. This is a generalisation of the adelic identity for a rational number stating that the product of the norm of rational number with its p-adic norms equals to one.

The real amplitude in the intersection of realities and p-adicities for all values of parameter is rational number or in an appropriate algebraic extension of rationals. If given p-adic amplitude is just the p-adic norm of real amplitude, one would have the adelic identity. This would however require that p-adic variant of the amplitude is real number-valued: I want p-adic valued amplitudes. A further restriction is that Witten's adelic identity holds for vacuum amplitude. I live in Zero Energy Ontology (ZEO) and want it for entire S-matrix, M-matrix, and/or U-matrix and for all states of the basis in some sense.

Consider first the vacuum amplitude. A weaker form of the identity would be that the p-adic norm of a given p-adic valued amplitude is same as that p-adic norm for the rational-valued real amplitude (this generalizes to algebraic extensions, I dare to guess) in the intersection. This would make sense and give a non-trivial constraint: algebraic continuation would guarantee this constraint. In particular, the p-adic norm of the real amplitude would be inverse of the product of p-adic norms of p-adic amplitudes. Most of these amplitudes should have p-adic norm equal to one in other words, real amplitude is product of finite number of powers of prime. This because the p-adic norms must approach rapidly to unity as p-adic prime increases and for large p-adic primes this means that the norm is exactly unity. Hence the p-adic norm of p-adic amplitude equals to 1 for most primes.

In ZEO one must consider S-, M-, or U-matrix elements. U and S are unitary. M is product of hermitian square root of density matrix times unitary S-matrix. Consider next S-matrix.

  1. For S-matrix elements one should have pm=(SS)mm=1. This states the unitarity of S-matrix. Probability is conserved. Could it make sense to generalize this condition and demand that it holds true only adelically that only for the product of real and p-adic norms of pm equals to one: NR(pm)(R)∏p Np(pm(p))=1. This could be actually true identically in the intersection if algebraic continuation principle holds true. Despite the triviality of the adelicity condition, one need not have anymore unitarity separately for reals and p-adic number fields. Notice that the numbers pm would be arbitrary
    rationals in the most general cased.

  2. Could one even replace Np with canonical identification or some form of it with cutoffs reflecting the length scale cutoffs? Canonical identification behaves for powers of p like p-adic norm and means only
    more precise map of p-adics to reals.

  3. For a given diagonal element of unit matrix characterizing particular state m one would have a product of real norm and p-adic norms. The number of the norms, which differ from unity would be finite. This condition would give finite number of exceptional p-adic primes, that is assign to a given quantum state m a finite number of preferred p-adic primes! I have been searching for a long time the underlying deep reason for this assignment forced by the p-adic mass calculations and here it might be.

  4. Unitarity might thus fail in real sector and in a finite number of p-adic sectors (otherwise the product of p-adic norms would be infinite or zero). In some sense the failures would compensate each other in the adelic picture. The failure of course brings in mind p-adic thermodynamics, which indeed means that adelic SS, or should it be called MM, is not unitary but defines the density matrix defining the p-adic thermal state! Recall that M-matrix is defined as hermitian square root of density matrix and unitary S-matrix.

  5. The weakness of these arguments is that states are assumed to be labelled by discrete indices. Finite measurement resolution implies discretization and could justify this.
The p-adic norms of pm or the images of pm under canonical identification in a given number field would define analogs of probabilities. Could one indeed have ∑m pm=1 so that SS would define a density matrix?
  1. For the ordinary S-matrix this cannot be the case since the sum of the probabilities pm equals to the dimension N of the state space: ∑ pm=N. In this case one could accept pm>1 both in real and p-adic sectors. For this option adelic unitarity would make sense and would be highly non-trivial condition allowing perhaps to understand how preferred p-adic primes emerge at the fundamental level.

  2. If S-matrix is multiplied by a hermitian square root of density matrix to get M-matrix, the situation changes and one indeed obtains ∑ pm=1. MM†=1 does not make sense anymore and must be replaced with MM†=ρ, in special case a projector to a N-dimensional subspace proportional to 1/N. In this case the numbers p(m) would have p-adic norm larger than one for the divisors of N and would define preferred p-adic primes. For these primes the sum Np(p(m)) would not be equal to 1 but to NNp(1/N.

  3. Situation is different for hyper-finite factors of type II1 for which the trace of unit matrix equals to one by definition and MM=1 and ∑ pm=1 with sum defined appropriately could make sense. If MM† could be also a projector to an infinite-D subspace. Could the M-matrix using the ordinary definition of dimension of Hilbert space be equivalent with S-matrix for the state space using the definition of dimension assignable to HFFs? Could these notions be dual of each other? Could the adelic S-matrix define the counterpart of M-matrix for HFFs?

This looks like a nice idea but usually good looking ideas do not live long in the crossfire of counter arguments. The following is my own. The reader is encouraged to invent his or her own objections.
  1. The most obvious objection against the very attractive direct algebraic continuation} from real to p-adic sector is that if the real norm or real amplitude is small then the p-adic norm of its p-adic counterpart is large so that p-adic variants of pm(p) can become larger than 1 so that probability interpretation fails. As noticed there is no actually no need to pose probability interpretation. The only way to overcome the "problem" is to assume that unitarity holds separately in each sector so that one would have p(m)=1 in all number fields but this would lead to the loss of preferred primes.

  2. Should p-adic variants of the real amplitude be defined by canonical identification or its variant with cutoffs? This is mildly suggested by p-adic thermodynamics. In this case it might be possible to satisfy the condition pm(R)∏p Np(pm(p))=1. One can however argue that the adelic condition is an ad hoc condition in this

To sum up, if the above idea survives all the objections, it could give rise to a considerable progress. A first principle understanding of how preferred p-adic primes are assigned to quantum states and thus a first principle justification for p-adic thermodynamics. For the ordinary definition of S-matrix this picture makes sense and also for M-matrix. One would still need the justification of canonical identification map playing a key role in p-adic thermodynamics allowing to map p-adic mass squared to its real counterpart.

Sunday, April 26, 2015

Hierarchies of conformal symmetry breakings, quantum criticalities, Planck constants, and hyper-finite factors

TGD is characterized by various hierarchies. There are fractal hierarchies of quantum criticalities, Planck constants and hyper-finite factors and these hierarchies relate to hierarchies of space-time sheets, and selves. These hierarchies are closely related and this article describes these connections. In this article the recent view about connections between various hierarchies associated with quantum TGD are described.

For details see the article Hierarchies of conformal symmetry breakings, quantum criticalities, Planck constants, and hyper-finite factors.

For a summary of earlier postings see Links to the latest progress in TGD.

Updated View about Kähler geometry of WCW

Quantum TGD reduces to a construction of Kähler geometry for what I call the "World of Classical Worlds. It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model type description of gravitation emerges and also the TGD based view about dark matter becomes more precise. String tension is however dynamical rather than pregiven and the hierarchy of Planck constants is necessary in order to understand the formation of gravitationally bound states. Also the proposal that sparticles correspond to dark matter becomes much stronger: sparticles actually are dark variants of particles.

A crucial element of the construction is the assumption that super-symplectic and other super-conformal symmetries having the same structure as 2-D super-conformal groups can be seen a broken gauge symmetries such that sub-algebra with conformal weights coming as n-ples of those for full algebra act as gauge symmetries. In particular, the Noether charges of this algebra vanish for preferred extremals- this would realize the strong form of holography implied by strong form of General Coordinate Invariance. This gives rise to an infinite number of hierarchies of conformal gauge symmetry breakings with levels labelled by integers n(i) such that n(i) divides n(i+1) interpreted as hierarchies of dark matter with levels labelled by the value of Planck constant heff=n× h. These hierarchies define also hierarchies of quantum criticalities and are proposed to give rise to inclusion hierarchies of hyperfinite factors of II1 having interpretation in terms of finite cognitive resolution. These hierarchies would be fundamental for the understanding of living matter.

For details see the article Updated view about Kähler geometry of WCW.

For a summary of earlier postings see Links to the latest progress in TGD.

Intentions, Cognition, and Time

Intentions involve time in an essential manner and this led to the idea that p-adic-to-real quantum jumps could correspond to a realization of intentions as actions. It however seems that this hypothesis posing strong additional mathematical challenges is not needed if one accepts adelic approach in which real space-time time and its p-adic variants are all present and quantum physics is adelic. I have already earlier developed the first formulation of p-adic space-time surfaces as cognitive charges of real space-time surfaces and also the ideas related to the adelic vision.

The recent view involving strong form of holography would provide dramatically simplified view about how these representations are formed as continuations of representations of strings world sheets and partonic 2-surfaces in the intersection of real and p-adic variants of WCW ("World of Classical Worlds") in the sense that the parameters characterizing these representations are in the algebraic numbers in the algebraic extension of p-adic numbers involved.

For details see the article Intentions, Cognition, and Time

For a summary of earlier postings see Links to the latest progress in TGD.

Saturday, April 25, 2015

Good and Evil, Life and Death

In principle the proposed conceptual framework allows already now a consideration of the basic questions relating to concepts like Good and Evil and Life and Death. Of course, too many uncertainties are involved to allow any definite conclusions, and one could also regard the speculations as outputs of the babbling period necessarily accompanying the development of the linguistic and conceptual apparatus making ultimately possible to discuss these questions more seriously.

Even the most hard boiled materialistic sceptic mentions ethics and moral when suffering personal injustice. Is there actual justification for moral laws? Are they only social conventions or is there some hard core involved? Is there some basic ethical principle telling what deeds are good and what deeds are bad?

Second group of questions relates to life and biological death. How should on define life? What happens in the biological death? Is self preserved in the biological death in some form? Is there something deserving to be called soul? Are reincarnations possible? Are we perhaps responsible for our deeds even after our biological death? Could the law of Karma be consistent with physics? Is liberation from the cycle of Karma possible?

In the sequel these questions are discussed from the point of view of TGD inspired theory of consciousness. It must be emphasized that the discussion represents various points of view rather than being a final summary. Also mutually conflicting points of view are considered. The cosmology of consciousness, the concept of self having space-time sheet and causal diamond as its correlates, the vision about the fundamental role of negentropic entanglement, and the hierarchy of Planck constants identified as hierarchy of dark matters and of quantum critical systems, provide the building blocks needed to make guesses about what biological death could mean from subjective point of view.

For details see the article Good and Evil, Life and Death.

For a summary of earlier postings see Links to the latest progress in TGD.

Friday, April 24, 2015

Variation of Newton's constant and of length of day

J. D. Anderson et al have published an article discussing the observations suggesting a periodic variation of the measured value of Newton constant and variation of length of day (LOD) (see also this). This article represents TGD based explanation of the observations in terms of a variation of Earth radius. The variation would be due to the pulsations of Earth coupling via gravitational interaction to a dark matter shell with mass about 1.3× 10-4ME introduced to explain Flyby anomaly: the model would predict Δ G/G= 2Δ R/R and Δ LOD/LOD= 2Δ RE/RE with the variations pf G and length of day in opposite phases. The expermental finding Δ RE/RE= MD/ME is natural in this framework but should be deduced from first principles.

The gravitational coupling would be in radial scaling degree of freedom and rigid body rotational degrees of freedom. In rotational degrees of freedom the model is in the lowest order approximation mathematically equivalent with Kepler model. The model for the formation of planets around Sun suggests that the dark matter shell has radius equal to that of Moon's orbit. This leads to a prediction for the oscillation period of Earth radius: the prediction is consistent with the observed 5.9 years period. The dark matter shell would correspond to n=1 Bohr orbit in the earlier model for quantum gravitational bound states based on large value of Planck constant. Also n>1 orbits are suggestive and their existence would provide additional support for TGD view about quantum gravitation.

For details see the chapter Cosmology and Astrophysics in Many-Sheeted Space-Time or the article Variation of Newton's constant and of length of day.

For a summary of earlier postings see Links to the latest progress in TGD.

Tuesday, April 21, 2015

Connection between Boolean cognition and emotions

Weak form of NMP allows the state function reduction to occur in 2n-1 manners corresponding to subspaces of the sub-space defined by n-dimensional projector if the density matrix is n-dimensional projector (the outcome corresponding to 0-dimensional subspace and is excluded). If the probability for the outcome of state function reduction is same for all values of the dimension 1≤m ≤n, the probability distribution for outcome is given by binomial distribution B(n,p) for p=1/2 (head and tail are equally probable) and given by p(m)= b(n,m)× 2-n= (n!/m!(n-m)!)×2-n . This gives for the average dimesion E(m)= n/2 so that the negentropy would increase on the average. The world would become gradually better.

One cannot avoid the idea that these different degrees of negentropic entanglement could actually give a realization of Boolean algebra in terms of conscious experiences.

  1. Could one speak about a hierarchies of codes of cognition based on the assignment of different degrees of "feeling good" to the Boolean statements? If one assumes that the n:th bit is always 1, all independent statements except one correspond at least two non-vanishing bits and corresponds to negentropic entanglement. Only of statement (only last bit equal to 1) would correspond 1 bit and to state function reduction reducing the entanglement completely (brings in mind the fruit in the tree of Good and Bad Knowlege!).

  2. A given hierarchy of breakings of super-symplectic symmetry corresponds to a hierarchy of integers ni+1= ∏k≤ i mk. The codons of the first code would consist of sequences of m1 bits. The codons of the second code consists of m2 codons of the first code and so on. One would have a hierarchy in which codons of previous level become the letters of the code words at the next level of the hierarchy.
In fact, I ended up with almost Boolean algebra for decades ago when considering the hierarchy of genetic codes suggested by the hierarchy of Mersenne primes M(n+1)= MM(n), Mn= 2n-1.
  1. The hierarchy starting from M2=3 contains the Mersenne primes 3,7,127,2127-1 and Hilbert conjectured that all these integers are primes. These numbers are almost dimensions of Boolean algebras with n=2,3,7,127 bits. The maximal Boolean sub-algebras have m=n-1=1,2,6,126 bits.

  2. The observation that m=6 gives 64 elements led to the proposal that it corresponds to a Boolean algebraic assignable to genetic code and that the sub-algebra represents maximal number of independent statements defining analogs of axioms. The remaining elements would correspond to negations of these statements. I also proposed that the Boolean algebra with m=126=6× 21 bits (21 pieces consisting of 6 bits) corresponds to what I called memetic code obviously realizable as sequences of 21 DNA codons with stop codons included. Emotions and information are closely related and peptides are regarded as both information molecules and molecules of emotion.

  3. This hierarchy of codes would have the additional property that the Boolean algebra at n+1:th level can be regarded as the set of statements about statements of the previous level. One would have a hierarchy representing thoughts about thoughts about.... It should be emphasized that there is no need to assume that the Hilbert's conjecture is true.

    One can obtain this kind of hierarchies as hierarchies with dimensions m, 2m, 22m,... that is n(i+1)= 2n(i). The conditions that n(i) divides n(i+1) is non-trivial only for at the lowest step and implies that m is power of 2 so that the hierarchies starting from m=2k. This is natural since Boolean algebras are involved. If n corresponds to the size scale of CD, it would come as a power of 2.

    p-Adic length scale hypothesis has also led to this conjecture. A related conjecture is that the sizes of CDs correspond to secondary p-adic length scales, which indeed come as powers of two by p-adic length scale hypothesis. In case of electron this predicts that the minimal size of CD associated with electron corresponds to time scale T=.1 seconds, the fundamental time scale in living matter (10 Hz is the fundamental bio-rhythm). It seems that the basic hypothesis of TGD inspired partly by the study of elementary particle mass spectrum and basic bio-scales (there are 4 p-adic length scales defined by Gaussian Mersenne primes in the range between cell membrane thickness 10 nm and and size 2.5 μm of cell nucleus!) follow from the proposed connection between emotions and Boolean cognition.

  4. NMP would be in the role of God. Strong NMP as God would force always the optimal choice maximizing negentropy gain and increasing negentropy resources of the Universe. Weak NMP as God allows free choice so that
    entropy gain is not be maximal and sinners populate the world. Why the omnipotent God would allow this? The reason is now obvious. Weak form of NMP makes possible the realization of Boolean algebras in terms of degrees of "feels good"! Without the God allowing the possibility to do sin there would be no emotional intelligence!

Hilbert's conjecture relates in interesting manner to space-time dimension. Suppose that Hilbert's conjecture fails and only the four lowest Mersenne integers in the hierarchy are Mersenne primes that is 3,7,127, 2127-1. In TGD one has hierarchy of dimensions associated with space-time surface coming as 0,1,2,4 plus imbedding space dimension 8. The abstraction hierarchy associated with space-time dimensions would correspond discretization of partonic 2-surfaces as point set, discretization of 3-surfaces as a set of strings connecting partonic 2-surfaces characterized by discrete parameters, discretization of space-time surfaces as a collection of string world sheet with discretized parameters, and maybe - discretization of imbedding space by a collection of space-time surfaces. Discretization means that the parameters in question are algebraic numbers in an extension of rationals associated with p-adic numbers.

In TGD framework it is clear why imbedding space cannot be higher-dimensional and why the hierarchy does not continue. Could there be a deeper connection between these two hierarchies. For instance, could it be that higher dimensional manifolds of dimension 2×n can be represented physically only as unions of say n 2-D partonic 2-surfaces (just like 3×N dimensional space can be represented as configuration space of N point-like particles)? Also infinite primes define a hierarchy of abstractions. Could it be that one has also now similar restriction so that the hierarchy would have only finite number of levels, say four. Note that the notion of n-group and n-algebra involves an analogous abstraction hierarchy.

For details see the article Good and Evil, Life and Death.

For a summary of earlier postings see Links to the latest progress in TGD.

Monday, April 20, 2015

Can one identify quantum physical correlates of ethics and moral?

TGD inspired theory of consciousness involves a bundle of new concepts making it possible to seriously discuss quantum physical correlates of ethics and moral assuming that we live in TGD Universe. In the following I summarize the recent understanding. I do not guarantee that I will agree with myself tomorrow since I am just going through this stuff in the updating of TGD inspired theory of consciousness and quantum biology.

Quantum ethics very briefly

Could physics generalized to a theory of consciousness allow to undersand the physical correlates of ethics and moral. The proposal is that this is the case. The basic ethical principle would be that good deeds help evolution to occur. Evolution should correspond to the increase of negentropic entanglement resources, defining negentropy sources, which I have called Akashic records.

This idea can be criticized.

  1. If strong form of NMP prevails, one can worry that TGD Universe does not allow Evil at all, perhaps not even genuine free will! No-one wants Evil but Evil seems to be present in this world.

  2. Could one weaken NMP so that it does not force but only allows to make a reduction to a final state characterized by density matrix which is projection operator? Self could choose whether to perform a projection to some sub-space of this subspace, say 1-D ray as in ordinary state function reduction. NMP would be like Christian God allowing the sinner to choose between Good and Evil. The final entanglement negentropy would be measure for the goodness of the deed. This is so if entanglement negentropy is a correlate for love. Deeds which are done with love would be good. Reduction of entanglement would in turn mean loneliness and separation.

  3. Or could could think that the definition of good deed is as a selection between deeds, which correspond to the same maximal increase of negentropy so that NMP cannot tell what happens. For instance the density matrix operator is direct sum of projection operators of same dimension but varying coefficients and there is a selection between these. It is difficult to imagine what the criterion for a good deed could be in this case. And how self can know what is the good deed and what is the bad deed.
Good deeds would support evolution. There are many manners to interpret evolution in TGD Universe.
  1. p-Adic evolution would mean a gradual increase of the p-adic primes characterizing individual partonic 2-surfaces and therefore their size. The identification of p-adic space-time sheets as representations for cognitions gives additional concreteness to this vision. The earlier proposal that p-adic--real-phase transitions correspond to realization of intentions and formations of cognitions seems however to be wrong. Instead, adelic view that both real and p-adic sectors are present simultaneously and that fermions at string world sheets correspond to the intersection of realities and p-adicities seems more realistic.

    The inclusion of phases q=exp(i2π/n) in the algebraic extension of p-adics allows to define the notion of angle in p-adic context but only with a finite resolution since only finite number of angles are represented as phases for a given value of n. The increase of the integers n could be interpreted as the emergence of higher algebraic extensions of p-adic numbers in the intersection of the real and p-adic worlds. These observations suggest that all three views about evolution are closely related.

  2. The hierarchy of Planck constants suggests evolution as the gradual increase of the Planck constant characterizing p-adic space-time sheet (or partonic 2-surface for the minimal option). The original vision about this evolution was as a migration to the pages of the book like structure defined by the generalized imbedding space and has therefore quite concrete geometric meaning. It implies longer time scales of long term memory and planned action and macroscopic quantum coherence in longer scales.

    The new view is in terms of first quantum jumps to the opposite boundary of CD leading to the death of self and its re-incarnation at the opposite boundary.

  3. The vision about life as something in the intersection of real and p-adic words allows to see evolution information theoretically as the increase of number entanglement negentropy implying entanglement in increasing length scales. This option is equivalent with the second view and consistent with the first one if the effective p-adic topology characterizes the real partonic 2-surfaces in the intersection of p-adic and real worlds.

The third kind of evolution would mean also the evolution of spiritual consciousness if the proposed interpretation is correct. In each quantum jump U-process generates a superposition of states in which any sub-system can have both real and algebraic entanglement with the external world. If state function reduction process involves also the choice of the type of entanglement it could be interpreted as a choice between good and evil. The hedonistic complete freedom resulting as the entanglement entropy is reduced to zero on one hand, and the negentropic entanglement implying correlations with the external world and meaning giving up the maximal freedom on the other hand. The selfish option means separation and loneliness. The second option means expansion of consciousness - a fusion to the ocean of consciousness as described by spiritual practices.

In this framework one could understand the physics correlates of ethics and moral. The ethics is simple: evolution of consciousness to higher levels is a good thing. Anything which tends to reduce consciousness represents violence and is a bad thing. Moral rules are related to the relationship between individual and society and presumably develop via self-organization process and are by no means unique. Moral rules however tend to optimize evolution. As blind normative rules they can however become a source of violence identified as any action which reduces the level of consciousness.

There is an entire hierarchy of selves and every self has the selfish desire to survive and moral rules develop as a kind of compromise and evolve all the time. ZEO leads to the notion that I have christened cosmology of consciousness. It forces to extend the concept of society to four-dimensional society. The decisions of "me now" affect both my past and future and time like quantum entanglement makes possible conscious communication in time direction by sharing conscious experiences. One can therefore speak of genuinely four-dimensional society. Besides my next-door neighbors I had better to take into account also my nearest neighbors in past and future (the nearest ones being perhaps copies of me!). If I make wrong decisions those copies of me in future and past will suffer the most. Perhaps my personal hell and paradise are here and are created mostly by me.

What could the quantum correlates of moral be?

We make moral choices all the time. Some deeds are good, some deeds are bad. In the world of materialist there are no moral choices, the deeds are not good or bad, there are just physical events. I am not a materialist so that I cannot avoid questions such as how do the moral rules emerge and how some deeds become good and some deeds bad. Negentropic entanglement is the obvious first guess if one wants to understand emergence of moral.

  1. One can start from ordinary quantum entanglement. It corresponds to a superposition of pairs of states. Second state corresponds to the internal state of the self and second state to a state of external world or biological body of self. In negentropic quantum entanglement each is replaced with a pair of sub-spaces of state spaces of self and external world. The dimension of the sub-space depends on the which pair is in question. In state function reduction one of these pairs is selected and deed is done. How to make some of these deeds good and some bad?

  2. Obviously the value of heff/h=n gives the criterion in the case that weak form of NMP holds true. Recall that weak form of NMP allows only the possibility to generate negentropic entanglement but does not force it. NMP is like God allowing the possibility to do good but not forcing good deeds.

    Self can choose any sub-space of the subspace defined by n-dimensional projector and 1-D subspace corresponds to the standard quantum measurement. For n=1 the state function reduction leads to vanishing negentropy, and separation of self and the target of the action. Negentropy does not increase in this action and self is isolated from the target: kind of price for sin.

    For the maximal dimension of this sub-space the negentropy gain is maximal. This deed is good and by the proposed criterion the negentropic entanglement corresponds to love or more generally, positively colored conscious experience. Interestingly, there are 2n possible choices which is the dimension of Boolean algebra consisting of n independent bits. This could relate directly to fermionic oscillator operators defining basis of Boolean algebra. The deed in this sense would be a choice of how loving the attention towards system of external world is.

  3. Could the moral rules of society be represented as this kind of entanglement patterns between its members? Here one of course has entire fractal hierarchy of societies corresponding different length scales. Attention and magnetic flux tubes serving as its correlates is the basic element also in TGD inspired quantum biology already at the level of bio-molecules and even elementary particles. The value of heff/h=n associated with the magnetic flux tube connecting members of the pair, would serve as a measure for the ethical value of maximally good deed. Dark phases of matter would correspond to good: usually darkness is associated with bad!

  4. These moral rules seem to be universal. There are however also moral rules or should one talk about rules of survival, which are based on negative emotions such as fear. Moral rules as rules of desired behavior are often tailored for the purposes of power holder. How this kind of moral rules could develop? Maybe they cannot be realized in terms of negentropic entanglement. Maybe the superposition of the allowed alternatives for the deed contains only the alternatives allowed by the power holder and the superposition in question corresponds to ordinary entanglement for which the signature is simple: the probabilities of various options are different. This forces the self to choose just one option from the options that power holder accepts. These rules do not allow the generation of loving relationship.
Moral rules seem to be generated by society, up-bringing, culture, civilization. How the moral rules develop? One can try to formulate and answer in terms of quantum physical correlates.
  1. Basically the rules should be generated in the state function reductions which correspond to volitional action which corresponds to the first state function reduction to the earlier active boundary of CD. Old self dies and new self is born at the opposite boundary of CD and the arrow of time associated with CD changes.

  2. The repeated sequences of state function reductions can generate negentropic entanglement during the quantum evolutions between them. This time evolution would be the analog for the time evolution defined by Hamiltonian - that is energy - associated with ordinary time translation whereas the first state function reduction at the opposite boundary inducing scaling of heff and CD would be accompanied by time evolution defined by conformal scaling generator L0.

    Note that the state at passive boundary does not change during the sequence of repeated state function reductions. These repeated reductions however change the parts of zero energy states associated with the new active boundary and generate also negentropic entanglement. As the self dies the moral choices can made if the weak form of NMP is true.

  3. Who makes the moral choices? It looks of course very weird that self would apply free will only at the moment of its death or birth! The situation is saved by the fact that self has also sub-selves, which correspond to sub-CDs and represent mental images of self. We know that mental images die as also we do some day and are born again (as also we do some day) and these mental images can generate negentropic resources within CD of self.

    One can argue that these mental images do not decide about whether to do maximally ethical choice at the moment of death. The decision must be made by a self at higher level. It is me who decides about the fate of my mental images - to some degree also after their death! I can choose the how negentropic the quantum entanglement characterizing the relationship of my mental image and the world outside it. I realize, that the misused idea of positive thinking seems to unavoidably creep in! I have however no intention to make money with it!

There are still many questions that are waiting for more detailed answer. These questions are also a good manner to detect logical inconsistencies.
  1. What is the size of CD characterizing self? For electron it would be at least of the order of Earth size. During the lifetime of CD the size of CD increases and the order of magnitude is measured in light-life time for us. This would allow to understand our usual deeds affecting the environment in terms of our subselves and their entanglement with the external world which is actually our internal world, at least if magnetic bodies are considered.

  2. Can one assume that the dynamics inside CD is independent from what happens outside CD. Can one say that the boundaries of CD define the ends of space-time or does space-time continue outside them. Do the boundaries of CD define boundaries for 4-D spotlight of attention or for one particular reality? Does the answer to this question have any relevance if everything physically testable is formulated in term physics of string world sheets associated with space-time surfaces inside CD?

    Note that the (average) size of CDs (, which could be in superposition but need not if every repeated state function reduction is followed by a localization in the moduli space of CDs) increases during the life cycle of self. This makes possible generation of negentropic entanglement between more and more distant systems. I have written about the possibility that ZEO could make possible interaction with distant civilizations (see this. The possibility of having communications in both time directions would allow to circumvent the barrier due to the finite light-velocity, and gravitational quantum coherence in cosmic scales would make possible negentropic entanglement.

  3. How selves interact? CDs as spot-lights of attention should overlap in order that the interaction is possible. Formation of flux tubes makes possible quantum entanglement. The string world sheets carrying fermions also essential correlates of entanglement and the possibly entanglement is between fermions associated with partonic 2-surfaces. The string world sheets define the intersection of real and p-adic worlds, where cognition and life resides.

For details see the article Good and Evil, Life and Death.

Intentions, cognitions, time, and p-adic physics

Intentions involved time in an essential manner and this led to the idea that p-adic-to-real quantum jumps could correspond to a realization of intentions as actions. It however seems that this hypothesis posing strong additional mathematical challenges is not needed if one accepts adelic approach in which real space-time time and its p-adic variants are all present and quantum physics is adelic. I have developed the first formulation of p-adic space-time surface in and the ideas related to the adelic vision (see this, this, and this).

1. What intentions are?

One of the earlier ideas about the flow of subjective time was that it corresponds to a phase transition front representing a transformation of intentions to actions and propagating towards the geometric future quantum jump by quantum jump. The assumption about this front is un-necessary in the recent view inspired by ZEO.

Intentions should relate to active aspects of conscious experience. The question is what the quantum physical correlates of intentions are and what happens in the transformation of intention to action.

  1. The old proposal that p-adic-to-real transition could correspond to the realization of intention as action. One can even consider the possibility that the sequence of state function reductions decomposes to pairs real-to-padic and p-adic-to-real transitons. This picture does not explain why and how intention gradually evolves stronger and stronger, and is finally realized. The identification of p-adic space-time sheets as correlates of cognition is however natural.

  2. The newer proposal, which might be called adelic, is that real and p-adic space-time sheets form a larger sensory-cognitive structure: cognitive and sensory aspects would be simultaneously present. Real and p-adic space-time surfaces would form single coherent whole which could be called adelic space-time. All p-adic manifolds could be present and define kind of chart maps about real preferred extremals so that they would not be independent entities as for the first option. The first objection is that the assignment of fermions separately to the every factor of adelic space-time does not make sense. This objection is circumvented if fermions belong to the intersection of realities and p-adicities.

    This makes sense if string world sheets carrying the induced spinor fields define seats of cognitive representations in the intersection of reality and p-adicities. Cognition would be still associated with the p-adic space-time sheets and sensory experience with real ones. What can sensed and cognized would reside in the intersection.

    Intention would be however something different for the adelic option. The intention to perform quantum jump at the opposite boundary would develop during the sequence of state function reductions at fixed boundary and eventually NMP would force the transformation of intention to action as first state function reduction at opposite boundary. NMP would guarantee that the urge to do something develops so strong that eventually something is done.

    Intention involves two aspects. The plan for achieving something which corresponds to cognition and the will to achieve something which corresponds to emotional state. These aspects could correspond to p-adic and real aspects of intentionality.

2. p-Adic physics as physics of only cognition?

There are two views about p-adic-real correspondence corresponding to two views about p-adic physics. According to the first view p-adic physics defines correlates for both cognition and intentionality whereas second view states that it provides correlates for cognition only.

  1. Option A: The older view is that p-adic -to-real transitions realize intentions as actions and opposite transitions generate cognitive representations. Quantum state would be either real or p-adic. This option raises hard mathematical challenges since scattering amplitudes between different number fields are needed and the needed mathematics might not exist at all.

  2. Option B: Second view is that cognition and sensory aspects of experience are simultaneously present at all levels and means that real space-time surface and their real counterparts form a larger structure in the spirit of what might be called Adelic TGD. p-Adic space-time charts could be present for all primes. It is of course necessary to understand why it is possible to assign definite prime to a given elementary particle.

    This option could be developed by generalizing the existing mathematics of adeles by replacing number in given number field with a space-time surface in the imbedding space corresponding that number field. Therefore this option looks more promising. For this option also the development of intention can be also understood. The condition that the scattering amplitudes are in the intersection of reality and p-adicities is very powerful condition on the scattering amplitudes and would reduce the realization of number theoretical universality and p-adicization to that for string world sheets and partonic 2-surfaces.

    For instance, the difficult problem of defining p-adic analogs of topological invariant would trivialize since these invariants (say genus) have algebraic representation for 2-D geometries. 2-dimensionality of cognitive representation would be perhaps basically due to the close correspondence between algebra and topology in dimension D=2.

Most of the following considerations apply in both cases.

3. Some questions to ponder

The following questions are part of the list of question that one must ponder.

a) Do cognitive representations reside in the intersection of reality and p-adicities?

The idea that cognitive representation reside in the intersection of reality and various p-adicities is one of the key ideas of TGD inspired theory of consciousness.

  1. All quantum states have vanishing total quantum numbers in ZEO, which now forms the basis of quantum TGD (see this). In principle conservation laws do not pose any constraints on possibly occurring real--p-adic transitions (Option A) if they occur between zero energy states.

    On the other hand, there are good hopes about the definition of p-adic variants of conserved quantities by algebraic continuation since the stringy quantal Noether charges make sense in all number fields if string world sheets are in the real--p-adic intersection. This continuation is indeed needed if quantum states have adelic structure (Option B). In accordance with this quantum classical correspondence (QCC) demands that the classical conserved quantities in the Cartan algebra of symmetries are equal to the eigenvalues of the quantal charges.

  2. The starting point is the interpretation of fermions as correlates for Boolean cognition and p-adic space-time sheets space-time correlates for cognitions (see this). Induced spinor fields are localized at string world sheets, which suggests that string world sheets and partonic 2-surfaces define cognitive representations in the intersection of realities and p-adicities. The space-time adele would have a book-like structure with the back of the book defined by string world sheets.

  3. At the level of partonic 2-surfaces common rational points (or more generally common points in algebraic extension of rationals) correspond to the real--p-adic intersection. It is natural to identify the set of these points as the intersection of string world sheets and partonic 2-surfaces at the boundaries of CDs. These points would also correspond to the ends of strings connecting partonic 2-surfaces and the ends of fermion lines at the orbits of partonic 2-surfaces (at these surfaces the signature of the induced 4-metric changes). This would give a direct connection with fermions and Boolean cognition.

    1. For option A the interpretation is simple. The larger the number of points is, the higher the probability for the transitions to occur. This because the transition amplitude must involve the sum of amplitudes determined by data from the common points.

    2. For option B the number of common points measures the goodness of the particular cognitive representation but does not tell anything about the probability of any quantum transition. It however allows to discriminate between different p-adic primes using the precision of the cognitive representation as a criterion. For instance, the non-determinism of Kähler action could resemble p-adic non-determinism for some algebraic extension of p-adic number field for some value of p. Also the entanglement assignable to density matrix which is n-dimensional projector would be negentropic only if the p-adic prime defining the number theoretic entropy is divisor of n. Therefore also entangled quantum state would give a strong suggestion about the value of the optimal p-adic cognitive representation as that associated with the largest power of p appearing in n.

b) Could cognitive resolution fix the measurement resolution?

For p-adic numbers the algebraic extension used (roots of unity fix the resolution in angle degrees of freredom and pinary cutoffs fix the resolution in "radial" variables which are naturally positive. Could the character of quantum state or perhaps quantum transition fix measurement resolution uniquely?

  1. If transitions (state function reductions) can occur only between different number fields (Option A), discretization is un-avoidable and unique if maximal. For real-real transitions the discretization would be motivated only by finite measurement resolution and need be neither necessary nor unique. Discretization is required and unique also if one requires adelic structure for the state space (Option B). Therefore both options A and B are allowed by this criterion.

  2. For both options cognition and intention (if p-adic) would be one half of existence and sensory perception and motor actions would be second half of existence at fundamental level. The first half would correspond to sensory experience and motor action as time reversals of each other. This would be true even at the level of elementary particles, which would explain the amazing success of p-adic mass calculations.

  3. For option A the state function reduction sequence would correspond to a formation of p-adic maps about real maps and real maps about p-adic maps: real → p-adic → real →..... For option B it would correspond the sequence adelic → adelic → adelic →.....

  4. For both options p-adic and real physics would be unified to single coherent whole at the fundamental level but the adelic option would be much simpler. This kind of unification is highly suggestive - consider only the success of p-adic mass calculations - but I have not really seriously considered what it could mean.

c) What selects the preferred p-adic prime?

What determines the p-adic prime or preferred p-adic prime assignable to the system considered? Is it unique? Can it change?

  1. An attractive hypothesis is that the most favorable p-adic prime is a factor of the integer n defining the dimension of the n× n density matrix associated with the flux tubes/fermionic strings connecting partonic 2-surfaces: the presence of fermionic strings already implies at least two partonic 2-surfaces. During the sequence of reductions at same boundary of CD n receives additional factors so that p cannot change. If wormhole contacts behave as magnetic monopoles there must be at least two of them connected by monopole flux tubes. This would give a connection with negentropic entanglement and for heff/h=n to quantum criticality, dark matterm and hierarchy of inclusions of HFFs.

  2. Second possibility is that the classical non-determinism making itself visible via super-symplectic invariance acting as broken conformal gauge invariance has same character as p-adic non-determinism for some value of p-adic prime. This would mean that p-adic space-time surfaces would be especially good representations of real space-time sheets. At the lowest level of hierarchy this would mean large number of common points. At higher levels large number of common parameter values in the algebraic extension of rationals in question.

d) How finite measurement resolution relates to hyper-finite factors?

The connection with hyper-finite factors suggests itself.

  1. Negentropic entanglement can be said to be stabilized by finite cognitive resolution if hyper-finite factors are associated with the hierarchy of Planck constants and cognitive resolutions. For HFFs the projection to single ray of state space in state function reduction is replaced with a projection to an infinite-dimensional sub-space whose von Neumann dimension is not larger than one.

  2. This raises interesting question. Could infinite integers constructible from infinite primes correspond to these infinite dimensions so that prime p would appear as a factor of this kind of infinite integer? One can say that for inclusions of hyperfinite factors the ratio of dimensions for including and included factors is quantum dimension which is algebraic number expressible in terms of quantum phase q=exp(i2π/n). Could n correspond to the integer ratio n=nf/ni for the integers characterizing the sub-algebra of super-symplectic algebra acting as gauge transformations?

4. Generalizing the notion of p-adic space-time surface

The notion of p-adic manifold \citealb/picosahedron is an attempt to formulate p-adic space-time surfaces identified as preferred extremal of p-adic variants of p-adic field equations as cognitive charts of real space-time sheets. Here the essential point is that p-adic variants of field equations make sense: this is due to the fact that induced metric and induced gauge fields make sense (differential geometry exists p-adically unlike global geometry involving notions of lengths, area, etc does not exist: in particular the notion of angle and conformal invariance make sense).

The second key element is finite resolution so that p-adic chart map is not unique. Same applies to the real counterpart of p-adic extremal and having representation as space-time correlate for an intention realized as action.

The discretization of the entire space-time surface proposed in the formulation of p-adic manifold concept (see this) looks too naive an approach. It is plausible that one has an abstraction hierarchy for discretizations at various abstraction levels.

  1. The simplest discretization would occur at space-time level only at partonic 2-surfaces in terms of string ends identified as algebraic points in the extension of p-adics used. For the boundaries of string world sheets at the orbits of partonic 2-surface one would have discretization for the parameters defining the boundary curve. By field equations this curve is actually a segment of light-like geodesic line and characterized by initial light-like 8-velocity, which should be therefore a number in algebraic extension of rationals. The string world sheets should have similar parameterization in terms of algebraic numbers.

    By conformal invariance the finite-dimensional conformal moduli spaces and topological invariants would characterize string world sheets and partonic 2-surfaces. The p-adic variant of Teichmueller parameters was indeed introduced in p-adic mass calculations and corresponds to the dominating contribution to the particle mass (see this and this).

  2. What might be called co-dimension 2 rule for discretization suggests itself. Partonic 2-surface would be replaced with the ends of fermion lines at it or equivalently: with the ends of space-like strings connecting partonic 2-surfaces at it. 3-D partonic orbit would be replaced with the fermion lines at it. 4-D space-time surface would be replaced with 2-D string world sheets. Number theoretically this would mean that one has always commutative tangent space. Physically the condition that em charge is well-defined for the spinor modes would demand co-dimension 2 rule.

  3. This rule would reduce the real-p-adic correspondence at space-time level to construction of real and p-adic space-time surfaces as pairs to that for string world sheets and partonic 2-surfaces determining algebraically the corresponding space-time surfaces as preferred extremals of Kähler action. Strong form of holography indeed leads to the vision that these geometric objects can be extended to 4-D space-time surface representing preferred extremals.

  4. In accordance with the generalization of AdS/CFT correspondence to TGD framework cognitive representations for physics would involve only partonic 2-surfaces and string world sheets. This would tell more about cognition rather than Universe. The 2-D objects in question would be in the intersection of reality and p-adicities and define cognitive representations of 4-D physics. Both classical and quantum physics would be adelic.

  5. Space-time surfaces would not be unique but possess a degeneracy corresponding to a sub-algebra of the super-symplectic algebra isomorphic to it and and acting as conformal gauge symmetries giving rise to n conformal gauge invariance classes. The conformal weights for the sub-algebra would be n-multiples of those for the entire algebra and n would correspond to the effective Planck constant heff/h=n. The hierarchy of quantum criticalities labelled by n would correspond to a hierarchy of cognitive resolutions defining measurement resolutions.

Clearly, very many big ideas behind TGD and TGD inspired theory of consciousness would have this picture as a Boolean intersection.

5. Number theoretic universality for cognitive representations

  1. By number theoretic universality p-adic zero energy states should be formally similar to their real counterparts for option B. For option A the states between which real--p-adic transitions are highly probable would be similar. The states would have as basic building bricks the elements of the Yangian of the super-symplectic algebra associated with these strings which one can hope to be algebraically universal.

  2. Finite measurement resolution demands that all scattering amplitudes representing zero energy states involve discretization. In purely p-adic context this is unavoidable because the notion of integral is highly problematic. Residue integral is p-adically well-defined if one can deal with π.

    p-Adic integral can be defined as the algebraic continuation of real integral made possible by the notion of p-adic manifold and this works at least in the real--p-adic intersection. String world sheets would belong to the intersection if they are cognitive representations as the interpretation of fermions as correlates of Boolean cognition suggests. In this case there are excellent hopes that all real integrals can be continued to various p-adic sectors (which can involve algebraic extensions of p-adic number fields). Quantum TGD would be adelic. There are of course potential problems with transcendentals like powers of π.

  3. Discrete Fourier analysis allows to define integration in angle degrees of freedom represented in terms of algebraic extension involving roots of unity. In purely p-adic context the notion of angle does not make sense but trigonometric functions make sense: the reason is that only the local aspect of geometry generalize characterized by metric generalize. The global aspects such as line length involving integral do not. One can however introduce algebraic extensions of p-adic numbers containing roots of unity and this gives rise to a realistic notion of trigonometric function. One can also define the counterpart of integration as discrete Fourier analysis in discretized angle degrees of freedom.

  4. Maybe the 2-dimensionality of cognition has something to do with the fact that quaternions and octonions do not have p-adic counterpart (the p-adic norm squared of quaternion/octonion can vanish). I have earlier proposed that life and cognitive representations resides in real-p-adic intersection. Stringy description of TGD could be seen as number theoretically universal cognitive representation of 4-D physics. The best that the limitations of cognition allow to obtain. This hypothesis would also guarantee that various conserved quantal charges make sense both in real and p-adic sense as p-adic mass calculations demand.

Monday, April 13, 2015

Manifest unitarity and information loss in gravitational collapse

There was a guest posting in the blog of Lubos by Prof. Dejan Stojkovic from Buffalo University. The title of the post was Manifest unitarity and information loss in gravitational collapse. It explained the contents of the article Radiation from a collapsing object is manifestly unitary by Stojkovic and Saini.

The posting

The posting describes calculations carried out for a collapsing spherical mass shell, whose radius approaches its own Scwartschild radius. The metric outside the shell with radius larger than rS is assumed to be Schwartschild metric. In the interior of the shell the metric would be Minkowski metric. The system considered is second quantized massless scalar field. One can calculate the Hamiltonian of the radiation field in terms of eigenmodes of the kinetic and potential parts and by canonical quantization the Schrödinger equation for the eigenmodes reduces to that for a harmonic oscillator with time dependent frequency. Solutions can be developed in terms of solutions of time-independent harmonic oscillator. The average value of the photon number turns out to approach to that associated with a thermal distribution irrespective of initial values at the limit when the of the shell approaches its blackhole radius. The temperature is Hawking temperature. This is of course highly interesting result and should reflect the fact that Minkowski vacuum looks from the point of view of an accelerated system to be in thermal equilibrium. Manifest unitary is just what one expects.

The authors assign a density matrix to the state in the harmonic oscillator basis. Since the state is pure, the density matrix is just a projector to the quantum state since the components of the density matrix are products of the coefficients characterizing the state in the oscillator basis (there are a couple of typos in the formulas, reader certainly notices them). In Hawking's original argument the non-diagonal cross terms are neglected and one obtains a non-pure density matrix. The approach of authors is of course correct since they consider only the situation before the formation of horizon. Hawking consider the situation after the formation of horizon and assumes some un-specified process taking the non-diagonal components of the density matrix to zero. This decoherence hypothesis is one of the strange figments of insane theoretical imagination which plagues recent day theoretical physics.

Authors mention as a criterion for purity of the state the condition that the square of the density matrix has trace equal to one. This states that the density matrix is N-dimensional projector. The criterion alone does not however guarantee the purity of the state for N> 1. This is clear from the fact that the entropy is in this case non-vanishing and equal to log(N). I notice this because negentropic entanglement in TGD framework corresponds to the situation in entanglement matrix is proportional to unit matrix (that is projector). For this kind of states number theoretic counterpart of Shannon entropy makes sense and gives negative entropy meaning that entanglement carries information. Note that unitary 2-body entanglement gives rise to negentropic entanglement.

Authors inform that Hawkins used Bogoliubov transformations between initial Minkowski vacuum and final Schwartschild vacum at the end of collapse which looks like thermal distribution with Hawking temperature in terms from Minkowski space point of view. I think that here comes an essential physical point. The question is about the relationship between two observers - one might call them the observer falling into blackhole and the observer far away approximating space-time with Minkowski space. If the latter observer traces out the degrees of freedom associated with the region below horizon, the outcome is genuine density matrix and information loss. This point is not discussed in the article and authors inform that their next project is to look at the situation after the spherical shell has reached Schwartschild radius and horizon is born. One might say that all that is done concerns the system before the formation of blackhole (if it is formed at all!).

Several poorly defined notions arise when one tries to interpret the results of the calculation.

  1. What do we mean with observer? What do we mean with information? For instance, authors define information as difference between maximum entropy and real entropy. Is this definition just an ad hoc manner to get sum well-defined number christened as an information? Can we really reduce the notion of information to thermodynamics? Shouldn't we be very careful in distinguishing between thermodynamical entropy and entanglement entropy? A sub-system possessing entanglement entropy with its complement can be purified by seeing it as a part of the entire system. This entropy relates to pair of systems. Thermal entropy can be naturally assigned to an average representative of ensemble and is single particle observable.

  2. Second list of questions relates to quantum gravitation. Is blackhole really a relevant notion or just a singular outcome of a theory exceeding its limits? Does something deserving to be called blackhole collapse really occur? Is quantum theory in its recent form enough to describe what happens in this process or its analog? Do we really understand the quantal description of gravitational binding?

What TGD can say about blackholes?

The usual objection of string theory hegemony is that there are no competing scenarios so that superstring is the only "known" interesting approach to quantum gravitation (knowing in academic sense is not at all the same thing as knowing in the naive layman sense and involves a lot of sociological factors transforming actual knowing to sociological unknowing: in some situations these sociological factors can make a scientist practically blind, deaf, and as it looks - brainless!) . I dare however claim that TGD represents an approach, which leads to a new vision challenging a long list of cherished notions assigned with blackholes.

To my view blackhole science crystallizes huge amount of conceptual sloppiness. People can calculate but are not so good in concetualizing. Therefore one must start the conceptual cleaning from fundamental notions such as information, notions of time (experienced and geometric), observer, etc... In attempt to develop TGD from a bundle of ideas to a real theory I have been forced to carry out this kind of distillation and the following tries to summarize the outcome.

  1. TGD provides a fundamental description for the notions of observer and information. Observer is replaced with "self" identified in ZEO by a sequences of quantum jumps occurring at same boundary of CD and leaving it and the part of the zero energy state at it fixed whereas the second boundary of CD is delocalized and superposition for which the average distance between the tips of CDs involve increases: this gives to the experience flow of time and its correlation with the flow of geometric time. The average size of CDs simply increases and this means that the experiences geometric time increases. Self "dies" as the first state function reduction to the opposite boundary takes place and new self assignable it is born.

  2. Negentropy Maximizaton Principle favors the generation of entanglement negentropy. For states with projection operator as density matrix the number theoretic negentropy is possible for primes dividing the dimension of the projection and is maximum for the largest power of prime factor of N. Second law is replaced with its opposite but for negentropy which is two-particle observable rather than single particle observable as thermodynamical entropy. Second law follows at ensemble level from the non-determinism of the state function reduction alone.

The notions related to blackhole are also in need of profound reconsideration.

  1. Blackhole disappears in TGD framework as a fundamental object and is replaced by a space-time region having Euclidian signature of the induced metric identifiable as wormhole contact, and defining a line of generalized Feynman diagram (here "Feynmann" could be replaced with " twistor" or "Yangian" something even more appropriate). Blackhole horizon is replaced the 3-D light-like region defining the orbit of wormhole throat having degenerate metric in 4-D sense with signature (0,-1,-1,-1). The orbits of wormhole throats are carries of various quantum numbers and the sizes of M4 projections are of order CP2 size in elementary particle scales. This is why I refer to these regions also as light-like parton orbits. The wormhole contacts involved connect to space-time sheets with Minkowskian signature and stability requires that the wormhole contacts carry monopole magnetic flux. This demands at least two wormhole contacts to get closed flux lines. Elementary particles are this kind of pairs but also multiples are possible and valence quarks in baryons could be one example.

  2. The connection with GRT picture could emerge as follows. The radial component of Schwartschild-Nordström metric associated with electric charge can be deformed slightly at horizon to transform horizon to light-like surface. In the deep interior CP2 would provide gravitational instanton solution to Maxwell-Einstein system with cosmological constant and having thus Euclidian metric. This is the nearest to TGD description that one can get within GRT framework obtained from TGD at asymptotic regions by replacing many-sheeted space-time with slightly deformed region of Minkowski space and summing the gravitational fields of sheets to get the the gravitational field of M4 region.

    All physical systems have space-time sheets with Euclidian signature analogous to blackhole. The analog of blackhole horizon provides a very general definition of "elementary particle".

  3. Strong form of general coordinate invariance is central piece of TGD and implies strong form of holography stating that partonic 2-surfaces and their 4-D tangent space data should be enough to code for quantum physics. The magnetic flux tubes and fermionic strings assignable to them are however essential. The localization of induced spinor fields to string world sheets follows from the well-definedness of em charge and also from number theoretical arguments as well as generalization of twistorialization from D=4 to D=8.

    One also ends up with the analog of AdS/CFT duality applying to the generalization of conformal invariance in TGD framework. This duality states that one can describe the physics in terms of Kähler action and related bosonic data or in terms of Kähler-Dirac action and related data. In particular, Kähler action is expressible as string world sheet area in effective metric defined by Kähler-Dirac gamma matrices. Furthermore, gravitational binding is describable by strings connecting partonic 2-surfaces. The hierarchy of Planck constants is absolutely essential for the description of gravitationally bound states in thems of gravitational quantum coherence in macroscopic scales. The proportionality of the string area in effective metric to 1/heff2, heff=n× h=hgr=GMm/v0 is absolutely essential for achieving this.

    If the stringy action were the ordinary area of string world sheet as in string models, only gravitational bound states with size of order Planck length would be possible. Hence TGD forces to say that superstring models are at completely wrong track concerning the quantum description of gravitation. Even the standard quantum theory lacks something fundamental required by this goal. This something fundamental relates directly to the mathematics of extended super-conformal invariance: these algebras allow infinite number of fractal inclusion hierarchies in which algebras are isomorphic with each other. This allows to realize infinite hierarchies of quantum criticalities. As heff increases, some degrees are reduced from critical gauge degrees of freedom to genuine dynamical degrees of freedom but the system is still critical, albeit in longer scale.

  4. A naive model for the TGD analog of blackhole is as a macroscopic wormhole contact surrounded by particle wormhole contacts with throats connected to the large wormhole throats by flux tubes and strings to the large wormhole contact. The macroscopic wormhole contact would carry magnetic charge equal to the sum of those associated with elemenentary particle wormhole throats.

  5. What about black hole collapse and blackhole evaporation if blackholes are replaced with wormhole contacts with Euclidian signature of metric? Do they have any counterparts in TGD? Maybe! Any phase transition increasing heff=hgr would occur spontaneously as transitions to lower criticality and could be interpreted as analog of blackhole evaporation. The gravitationally bound object would just increase in size. I have proposed that this phase transition has happened for Earth (Cambrian explosion) and increases its radius by factor 2. This would explain the strange finding that the continents seem to fit nicely together if the radius of Earth is one half of the recent value. These phase transitions would be the quantum counterpart of smooth classical cosmic expansion.

    The phase transition reducing heff would not occur spontaneusly and in living systems metabolic energy would be needed to drive them. Indeed, from the condition that heff=hgr= GMm/v0 increases as M and v0 change also gravitational Compton length Lgr=hgr/m= GM/v0 defining the size scale of the gravitational object increases so that the spontaneous increase of hgr means increase of size.

    Does TGD predict any process resembling blackhole collapse? In Zero Energy Ontology (ZEO) state function reductions occurring at the same boundary of causal diamond (CD) define the notion of self possessing arrow of time. The first quantum state function reduction at opposite boundary is eventually forced by Negentropy Maximization Principle (NMP) and induces a reversal of geometric time. The expansion of object with a reversed arrow of geometric time with respect to observer looks like collapse. This is indeed what the geometry of causal diamond suggests.

  6. The role of strings (and magnetic flux tubes with which they are associated) in the description of gravitational binding (and possibly also other kinds of binding) is crucial in TGD framework. They are present in arbitrary long length scales since the value of gravitational Planck constant heff = hgr = GMm/v0, v0 (v0/c<1) has dimensions of velocity can have huge values as compared with those of ordinary Planck constant. This implies macroscopic quantum gravitational coherence and the fountain effect of superfluidity could be seen as an example of this.

    The presence of flux tubes and strings serves as a correlate for quantum entanglement present in all scales is highly suggestive. This entanglement could be negentropic and by NMP and could be transferred but not destroyed. The information would be coded to the relationship between two gravitationally bound systems and instead of entropy one would have enormous negentropy resources. Whether this information can be made conscious is a fascinating problem. Could one generalize the interaction free quantum measurement so that it would give information about this entanglement? Or could the transfer of this information make it conscious?

    Also super string camp has become aware about possibility of geometric and topological correlates of entanglement. The GRT based proposal relies on wormhole connections. Much older TGD based proposal applied systematically in quantum biology and TGD inspired theory of consciousness identifies magnetic flux tubes and associated fermionic string world sheets as correlates of negentropic entanglement.