Thursday, May 21, 2015

p-Adic physics as physics of cognition and imagination

The vision about p-adic physics as physics of cognition and imagination has gradually established itself as one of the key idea of TGD inspired theory of consciousness. There are several motivations for this idea.

The strongest motivation is the vision about living matter as something residing in the intersection of real and p-adic worlds. One of the earliest motivations was p-adic non-determinism identified tentatively as a space-time correlate for the non-determinism of imagination. p-Adic non-determinism follows from the fact that functions with vanishing derivatives are piecewise constant functions in the p-adic context.

More precisely, p-adic pseudo constants depend on the pinary cutoff of their arguments and replace integration constants in p-adic differential equations. In the case of field equations this means roughly that the initial data are replaced with initial data given for a discrete set of time values chosen in such a manner that unique solution of field equations results. Solution can be fixed also in a discrete subset of rational points of the imbedding space. Presumably the uniqueness requirement implies some unique pinary cutoff. Thus the space-time surfaces representing solutions of p-adic field equations are analogous to space-time surfaces consisting of pieces of solutions of the real field equations. p-Adic reality is much like the dream reality consisting of rational fragments glued together in illogical manner or pieces of child's drawing of body containing body parts in more or less chaotic order.

The obvious looking interpretation for the solutions of the p-adic field equations would be as a geometric correlate of imagination. Plans, intentions, expectations, dreams, and cognition in general could have p-adic space-time sheets as their geometric correlates. A deep principle could be involved: incompleteness is characteristic feature of p-adic physics but the flexibility made possible by this incompleteness is absolutely essential for imagination and cognitive consciousness in general.

The original idea was that p-adic space-time regions can suffer topological phase transitions to real topology and vice versa in quantum jumps replacing space-time surface with a new one is given up as mathematically awkward: quantum jumps between different number fields do not make sense. The new adelic view states that both real and p-adic space-time sheets are obtained by continuation of string world sheets and partonic 2-surfaces to various number fields by strong form of holography.

The idea about p-adic pseudo constants as correlates of imagination is however too nice to be thrown away without trying to find an alternative interpretation consistent with strong form of holography. Could the following argument allow to save p-adic view about imagination in a mathematically respectable manner?

  1. Construction of preferred extremals from data at 2-surfaces is like boundary value problem. Integration constants are replaced with pseudo-constants depending on finite number pinary digits of variables depending on coordinates normal to string world sheets and partonic 2-surfaces.

  2. Preferred extremal property in real context implies strong correlations between string world sheets and partonic 2-surfaces by boundary conditions a them. One cannot choose these 2- surfaces completely independently. Pseudo-constant could allow a large number of p-adic configurations involving string world sheets and partonic 2-surfaces not allowed in real context and realizing imagination.

  3. Could imagination be realized as a larger size of the p-adic sectors of WCW? Could the realizable intentional actions belong to the intersection of real and p-adic WCWs? Could the modes of WCW spinor fields for which 2-surfaces are extandable to space-time surfaces only in some p-adic sectors make sense? The real space-time surface for them be somehow degenerate, for instance, consisting of string world sheets only.

    Could imagination be search for those collections of string world sheets and partonic 2-surfaces, which allow extension to (realization as) real preferred extremals? p-Adic physics would be there as an independent aspect of existence and this is just the original idea. Imagination could be realized in state function reduction, which always selects only those 2-surfaces which allow continuation to real space-time surfaces. The distinction between only imaginable and also realizable would be the extendability by using strong form of holography.

I have the feeling that this view allows respectable mathematical realization of imagination in terms of adelic quantum physics. It is remarkable that strong form of holography derivable from - you can guess, strong form of General Coordinate Invariance (the Big E again!), plays an absolutely central role in it.

How time reversed mental images differ from mental images?

The basic predictions of ZEO based quantum measurement theory is that self corresponds to a sequence of state function reductions to a fixed boundary of CD (passive boundary) and that the first reduction to the opposite boundary means death of self and re-incarnation at the opposite boundary. The re-incarnated self has reversed arrow of geometric time. This applies also to sub-selves of self giving rise to mental images. One can raise several questions.

Do we indeed have both mental images and time-reversed mental images? How the time-reversed mental image differs from the original one? Does the time flow in opposite direction for it? The roles of boundaries of CD have changed. The passive boundary of CD define the static back-ground the observed whereas the non-static boundary defines kind of dynamic figure. Does the change of the arrow of time change the roles of figure and background?

I have also proposed that motor action and sensory perception are time reversals of each other. Could one interpret this by saying that sensory perception is motor action affecting the body of self (say emotional expression) and motor action sensory perception of the environment about self.

In the sequel reverse speech and figure-background illusion is represented as examples of what time reversal for mental images could mean.

Time reversed cognition

Time reflection yields time reversed and spatially reflected sensory-cognitive representations. When mental image dies it is replaced with its time-reversal at opposite boundary of its CD. The observation of these representations could serve as a test of the theory.

There is indeed some evidence for this rather weird looking time and spatially reversed cognition.

  1. I have a personal experience supporting the idea about time reversed cognition. During the last psychotic episodes of my "great experience" I was fighting to establish the normal direction of the experienced time flow. Could this mean that for some sub-CDs the standard arrow of time had reversed as some very high level mental images representing bodily me died and was re-incarnated?

  2. The passive boundary of CD corresponds to static observing self - kind of background - and active boundary the dynamical - kind of figure. Figure-background division of mental image in this sense would change as sub-self dies and re-incarnates since figure and background change their roles. Figure-background illusion could be understood in this manner.

  3. The occurrence of mirror writing is well known phemonenon (my younger daughter was reverse writer when she was young). Spatial reflections of MEs are also possible and might be involved with mirror writing. The time reversal would change the direction of writing from right to left.

  4. Reverse speech would be also a possible form of reversed cognition. Time reversed speech has the same power spectrum as ordinary speech and the fact that it sounds usually gibberish means that phase information is crucial for storing the meaning of speech. Therefore the hypothesis is testable.

Reverse speech

Interestingly, the Australian David Oates claims that so called reverse speech is a real phenomenon, and he has developed entire technology and therapy (and business) around this phenomenon. What is frustrating that it seems impossible to find comments of professional linguistics or neuro-scientits about the claims of Oates. I managed only to find comments by a person calling himself a skeptic believer but it became clear that the comments of this highly rhetoric and highly arrogant commentator did not contain any information. This skeptic even taught poor Mr. Oates in an aggressive tone that serious scientists are not so naive that they would even consider the possibility of taking seriously what some Mr. Oates is saying. The development of science can often depend on ridiculously small things: in this case one should find a shielded place (no ridiculing skeptics around) to wind tape recorder backwards and spend few weeks or months to learn to recognize reverse speech if it really is there! Also computerized pattern recognition could be used to make speech recognition attempts objective since it is a well-known fact that brain does feature recognition by completing the data into something which is familiar.

The basic claims of Oates are following.

  1. Reverse speech contains temporal mirror images of ordinary words and even metaphorical statements, that these words can be also identified from Fourier spectrum, that brain responds in unconscious manner to these words and that this response can be detected in EEG. Oates classifies these worlds to several categories. These claims could be tested and pity that no professional linguist nor neuroscientist (as suggested by web search) has not seen the trouble of finding whether the basic claims of Oates are correct or not.

  2. Reverse speech is complementary communication mode to ordinary speech and gives rise to a unconscious (to us) communication mechanism making lying very difficult. If person consciously lies, the honest alter ego can tell the truth to a sub-self understanding the reverse speech. Reverse speech relies on metaphors and Oates claims that there is general vocabulary. Could this taken to suggest that reverse speech is communication of right brain whereas left brain uses ordinary speech? The notion of semitrance used to model bicameral mind suggests that reverse speech could be communication of higher levels of self hierarchy dispersed inside the ordinary speech. There are also other claims relating the therapy using reverse speech, which sound rather far-fetched but one should not confuse these claims to those which are directly testable.

Physically reverse speech could correspond to phase conjugate sound waves which together with their electromagnetic counterparts can be produced in laboratory . Phase conjugate waves have rather weird properties due the fact that second law applies in a reversed direction of geometric time. For this reason phase conjugate waves are applied in error correction. ZEO predicts this phenomenon.

Negative energy topological light rays are in a fundamental role in the TGD based model for living matter and brain. The basic mechanism of intentional action would rely on time mirror mechanism utilizing the TGD counterparts of phase conjugate waves producing also the nerve pulse patterns generating ordinary speech. If the language regions of brain contain regions in which the the arrow of psychological time is not always the standard one, they would induce phase conjugates of the sound wave patterns associated with the ordinary speech and thus reverse speech.

ZEO based quantum measurement theory, which is behind the recent form of TGD inspired theory of consciousness, provides a rigorous basis for this picture. Negative energy signals can be assigned with sub-CDs representing selves with non-standard direction of geometric time and every time when mental image dies, a mental images with opposite arrow of time is generated. It would be not surprising if the reverse speech would be associated with these time reversed mental images.

Figure-background rivalry and time reversed mental images

The classical demonstration of figure-background rivalry is is a pattern experienced either as a vase or two opposite faces. This phenomenon is not the same thing as bi-ocular rivalry in which the percepts associated with left and right eyes produced by different sensory inputs are rivalling. There is also an illusion in which one perceices the dancer to make a pirouette in either counter-clockwise or clockwise direction althought the figure is static. The direction of pirouette can change. In this case time-reversal would naturally change the direction of rotation.

Figure-background rivalry gives a direct support for the TGD based of self relying on ZEO if the following argument is accepted.

  1. In ZEO the state function reduction to the opposite boundary of CD means the death of the sensory mental image and birth of new one, possibly the rivalling mental image. During the sequence of state function reductions to the passive boundary of CD defining the mental image a boundary quantum superposition of rivalling mental images associated with the active boundary of CD is generated.

    In the state function reduction to the opposite boundary the previous mental image dies and is replaced with new one. In the case of bin-ocular rivalry this might be the either of the sensory mental images generated by the sensory inputs to eyes. This might happen also now but also different interpretation is possible.

  2. The basic questions concern the time reversed mental image. Does the subject person as a higher level self experience also the time reversed sensory mental image as sensory mental image as one might expect. If so, how the time reversed mental image differs from the mental image? Passive boundary of CD defines quite generally the background - the static observer - and active boundary the figure so that their roles should change in the reduction to the opposite boundary.In sensory rivalry situation this happens at least in the example considered (vase and two faces).

    I have also identified motor action as time reversal of sensory percept. What this identification could mean in the case of sensory percepts? Could sensory and motor be interpreted as an exchange of experiencer (or sub-self) and environment as figure and background?

If this interpretation is correct, figure-background rivalry would tell something very important about consciousness and would also support ZEO. Time reversal would permute figure and background. This might happen at very abstract level. Even subjective-objective duality and first - and third person aspects of conscious experience might relate to the time reversal of mental images. In near death experiences person sees himself as an outsider: could this be interpreted as the change of the roles of figure and background indentified as first and third person perspectives? Could the first moments of the next life be seeing the world from the third person perspective?

An interesting question is whether right- and left hemispheres tend to have opposite directions of geometric time. This would make possible metabolic energy transfer between them making possible kind of flip-flop mechanism. The time-reversed hemisphere would receive negative energy serving as metabolic energy resource for it and the hemisphere sending negative energy would get in this manner positive metabolic energy. Deeper interpretation would be in terms of periodic transfer of negentropic entanglement. This would also mean that hemispheres would provide two views about the world in which figure and background would be permuted.

Tuesday, May 19, 2015

Voevoedski's univalent foundations of mathematics

I found a very nice article about the work of mathematician Voevoedski: he talks about univalent foundations of mathematics. To put in nutshell: the world deals with mathematical proofs and the deep idea is that proofs are like paths in some abstract space. One deals with paths also in homotopy theory. What is remarkable that Voevoedski's work leads to computer programs allowing to check that proof does not contain an error. Something very badly needed when the proofs are getting increasingly complex. I dare to guess that the article is understandable by laymen too.

Russell's problem

The article tells about type theory originated already by Russell, who was led to his paradox with the set consisting of sets which do not contain itself as an element. The fatal question was "Does this set contain itself?".

Russell proposed a solution of the problems by introducing a hierarchy of types. Sets are at the bottom and defined so that they do not contain as set collection of sets. In usual applications of set theory - say in manifold theory - this is always true. Type hierarchy guarantees that you do not put apples and oranges in the same basket.

Voevoedski's idea

Voevoedski's work relates to proof theory and to formalising what mathematical proof does mean.

Consider demonstration that two sets A and B are equivalent. This means simple thing: construct a one-to-one map between them. Usually one is only interested in the existence of this map but one can also get interested on all manners to perform this map. All manners to make this map define in rather abstract sense a collection of paths between A and B regarded as objects. Single path consists of a collection of the arrows connecting element in A with element in B.

More generally, in mathematical proof theory all proofs of theorem define this kind of collection. In topology all paths connecting two points defined this kind of collection. In this framework Goedel's theorem becomes obvious: given axioms define rules for constructing paths and cannot give the paths connecting arbitrarily chosen two truths.

One can again abstract this process. Just as one can make statements about statements about..., one can build paths between paths, and paths between paths between paths.... Voevoedsky studied this problem in his attempts to formalise mathematics and with a practical goal to develop eventually tools for checking by computer that mathematical proofs are correct.

A more rigorous manner to define path is in terms of groupoid. It is more general notion that that of group since the inverse of groupoid element need not exist. For intense open paths between two points form a group but only closed paths can be given a structure of group.

Voevoedski introduced the notion of infinite groupoid containing paths, paths between paths, .... ad infinitum. Voevoedski talks about univalent foundations. The great idea is that homotopy theory becomes a foundation of mathematics: proofs are paths in some abstract structure. This suggests in my non-professional mind that one can talk about continuous deformations of proofs and that one can classify them to homotopy types with proofs of same homotopy type deformable to each other.

How this relates to quantum TGD

What made me fascinated was how closely the basic hierarchies of quantum TGD relate to the objects studies in type theory and Voevoedski's approach.

Let us start with set theory and type theory. TGD provides a non-trivial example about types, which by the way distinguishes TGD from super string models. Imbedding space is set, space-time surface are its subsets. "World of classical worlds" (WCW) is the key abstraction distinguishing TGD from super string models where one still tries to deal by working at space-time level.

What has surprised me against and again that super string modellers have spend decades in the landscape instead of making super string models a real theory by introducing loop space as a key notion although it has very nice mathematics: just the existence of Kähler geometry fixes it uniquely: this observation actually led to the realisation that quantum TGD might be unique just from its mathematical existence.

The points of WCW are 3-surfaces and its sets are collections of 3-surfaces. They are of higher type than the sets of imbedding space. There would be no sense in putting points of WCW and of imbedding space in the same basket. But in the set theory before Russell you could in principle do this. We have got as as birth gift the ability to not put cows and tooth brushes into same set. But the ability to take seriously the existence of more abstract types does not seem to be a birth gift.

Voevoedski and others deal with statements about statements about statements…. What is amusing that this vision has direct counterparts in TGD based quantum physics where various hierarchies have taken key role. Some deep ideas seem to burst out simultaneously in totally different contexts! Voevoedski noticed the same thing in his work but with within the realm of mathematics.

Just a list of examples should be enough. Consider first type theory.

  1. The hierarchy of infinite primes (integers, rationals) was the first mathematical discovery inspired by TGD inspired theory of consciousness. Infinite primes are constructed by a process analogous to a repeated second quantisation of arithmetic quantum field theory having interpretation as making statements about statements about..... up to arbitrary high order. Hierarchy of space-time sheets of many-sheeted space-time is the classical counterpart. Physics prediction is that higher level of quantisation are part of generalised quantum physics and allow quantum description of macroscopic and even astrophysical objects. The map of the sheets of many-sheeted space-time to single region of Minkowski space defines the contraction of TGD to GRT and is approximate operation: it maps a hierarchy of types to single type and is a violent procedure meaning a loss of information.

    Infinite integers could provide a generalisation of Goedel number numbering in a quantum mathematics based on the replacement of axiomatics with anti-axiomatics: specify what you cannot do instead of what you can do! I wrote about this in earlier posting.

    Infinite rationals of unit real norm lead also to a generalisation of the real number. Each real point becomes infinite dimensional space consisting of all infinite rationals of unit norm but well-defined number theoretic anatomy.

  2. There are several very closely related infinite hierarchies. Fractal hierarchy of quantum criticalities ( ball at the top of a hill at the top...) and isomorphic super-symplectic sub-algebras with conformal structure. There is infinite fractal hierarchy of conformal gauge symmetry breakings. This defines infinite hierarchy of dark matters with Planck constant heff=n× h. The algebraic extensions of rationals giving rise to evolutionary hierarchy for physics and perhaps explaining biological evolution define a hierarchy. The inclusions of hyper-finite factors realizing finite measurement resolution define a hierarchy. Hierarchy of infinite integers and rationals relates also closely to these hierarchies.

  3. In TGD inspired theory of conscioussness hierarchy of selves having sub-selves (experienced as mental images) having.... This hierarchy relates also very closely to the above hierarchies.

The notion of mathematical operation sequences as path is second key idea in Voevoedski's work. The idea about paths representing mathematical computations, proofs, etc.. is realised quite concretely in TGD quantum physics. Scattering amplitudes are identified as representations for sequences of algebraic operations of Yangian leading from an initial collection of elements of super-symplectic Yangian (physical states) to a final one. The duality symmetry of old fashioned string models generalises to a statement that any two sequences connecting same collections are equivalent and correspond to same amplitudes. This means extremely powerful predictions and it seems that in twistor programs analogous results are obtained too: very many twistor Grassmann diagrams represent the same scattering amplitude.

More about physical interpretation of algebraic extensions of rationals

The number theoretic vision has begun to show its power. The basic hierarchies of quantum TGD would reduce to a hierarchy of algebraic extensions of rationals and the parameters - such as the degrees of the irreducible polynomials characterizing the extension and the set of ramified primes - would characterize quantum criticality and the physics of dark matter as large heff phases. The identification of preferred p-adic primes as ramified primes of the extension and generalization of p-adic length scale hypothesis as prediction of NMP are basic victories of this vision (see this and this).

By strong form of holography the parameters characterizing string world sheets and partonic 2-surfaces serve as WCW coordinates. By various conformal invariances, one expects that the parameters correspond to conformal moduli, which means a huge simplification of quantum TGD since the mathematical apparatus of superstring theories becomes available and number theoretical vision can be realized. Scattering amplitudes can be constructed for a given algebraic extension and continued to various number fields by continuing the parameters which are conformal moduli and group invariants characterizing incoming particles.

There are many un-answered and even un-asked questions.

  1. How the new degrees of freedom assigned to the n-fold covering defined by the space-time surface pop up in the number theoretic picture? How the connection with preferred primes emerges?

  2. What are the precise physical correlates of the parameters characterizing the algebraic extension of rationals? Note that the most important extension parameters are the degree of the defining polynomial and ramified primes.

1. Some basic notions

Some basic facts about extensions are in order. I emphasize that I am not a specialist.

1.1. Basic facts

The algebraic extensions of rationals are determined by roots of polynomials. Polynomials be decomposed to products of irreducible polynomials, which by definition do not contain factors which are polynomials with rational coefficients. These polynomials are characterized by their degree n, which is the most important parameter characterizing the algebraic extension.

One can assign to the extension primes and integers - or more precisely, prime and integer ideals. Integer ideals correspond to roots of monic polynomials Pn(x)=xn+..a0 in the extension with integer coefficients. Clearly, for n=0 (trivial extension) one obtains ordinary integers. Primes as such are not a useful concept since roots of unity are possible and primes which differ by a multiplication by a root of unity are equivalent. It is better to speak about prime ideals rather than primes.

Rational prime p can be decomposed to product of powers of primes of extension and if some power is higher than one, the prime is said to be ramified and the exponent is called ramification index. Eisenstein's criterion states that any polynomial Pn(x)= anxn+an-1xn-1+...a1x+ a0 for which the coefficients ai, i<n are divisible by p and a0 is not divisible by p2 allows p as a maximally ramified prime. mThe corresponding prime ideal is n:th power of the prime ideal of the extensions (roughly n:th root of p). This allows to construct endless variety of algebraic extensions having given primes as ramified primes.

Ramification is analogous to criticality. When the gradient potential function V(x) depending on parameters has multiple roots, the potential function becomes proportional a higher power of x-x0. The appearance of power is analogous to appearance of higher power of prime of extension in ramification. This gives rise to cusp catastrophe. In fact, ramification is expected to be number theoretical correlate for the quantum criticality in TGD framework. What this precisely means at the level of space-time surfaces, is the question.

1.2 Galois group as symmetry group of algebraic physics

I have proposed long time ago that Galois group acts as fundamental symmetry group of quantum TGD and even made clumsy attempt to make this idea more precise in terms of the notion of number theoretic braid. It seems that this notion is too primitive: the action of Galois group must be realized at more abstract level and WCW provides this level.

First some facts (I am not a number theory professional, as the professional reader might have already noticed!).

  1. Galois group acting as automorphisms of the field extension (mapping products to products and sums to sums and preserves norm) characterizes the extension and its elements have maximal order equal to n by algebraic n-dimensionality. For instance, for complex numbers Galois group acs as complex conjugation. Galois group has natural action on prime ideals of extension mapping them to each other and preserving the norm determined by the determinant of the linear map defined by the multiplication with the prime of extension. For instance, for the quadratic extension Q(51/2) the norm is N(x+51/2y)=x2-5y2: not that number theory leads to Minkowkian metric signatures naturally. Prime ideals combine to form orbits of Galois group.

  2. Since Galois group leaves the rational prime p invariant, the action must permute the primes of extension in the product representation of p. For ramified primes the points of the orbit of ideal degenerate to single ideal. This means that primes and quite generally, the numbers of extension, define orbits of the Galois group.

Galois group acts in the space of integers or prime ideals of the algebraic extension of rationals and it is also physically attractive to consider the orbits defined by ideals as preferred geometric structures. If the numbers of the extension serve as parameters characterizing string world sheets and partonic 2-surfaces, then the ideals would naturally define subsets of the parameter space in which Galois group would act.

The action of Galois group would leave the space-time surface invariant if the sheets co-incide at ends but permute the sheets. Of course, the space-time sheets permuted by Galois group need not co-incide at ends. In this case the action need not be gauge action and one could have non-trivial representations of the Galois group. In Langlands correspondence these representation relate to the representations of Lie group and something similar might take place in TGD as I have indeed proposed.

Remark: Strong form of holography supports also the vision about quaternionic generalization of conformal invariance implying that the adelic space-time surface can be constructed from the data associated with functions of two complex variables, which in turn reduce to functions of single variable.

If this picture is correct, it is possible to talk about quantum amplitudes in the space defined by the numbers of extension and restrict the consideration to prime ideals or more general integer ideals.

  1. These number theoretical wave functions are physical if the parameters characterizing the 2-surface belong to this space. One could have purely number theoretical quantal degrees of freedom assignable to the hierarchy of algebraic extensions and these discrete degrees of freedom could be fundamental for living matter and understanding of consciousness.

  2. The simplest assumption that Galois group acts as a gauge group when the ends of sheets co-incide at boundaries of CD seems however to destroy hopes about non-trivial number theoretical physics but this need not be the case. Physical intuition suggests that ramification somehow saves the situation and that the non-trivial number theoretic physics could be associated with ramified primes assumed to define preferred p-adic primes.

2. How new degrees of freedom emerge for ramified primes?

How the new discrete degrees of freedom appear for ramified primes?

  1. The space-time surfaces defining singular coverings are n-sheeted in the interior. At the ends of the space-time surface at boundaries of CD however the ends co-incide. This looks very much like a critical phenomenon.

    Hence the idea would be that the end collapse can occur only for the ramified prime ideals of the parameter space - ramification is also a critical phenomenon - and means that some of the sheets or all of them co-incide. Thus the sheets would co-incide at ends only for the preferred p-adic primes and give rise to the singular covering and large heff. End-collapse would be the essence of criticality! This would occur, when the parameters defining the 2-surfaces are in a ramified prime ideal.

  2. Even for the ramified primes there would be n distinct space-time sheets, which are regarded as physically distinct. This would support the view that besides the space-like 3-surfaces at the ends the full 3-surface must include also the light-like portions connecting them so that one obtains a closed 3-surface. The conformal gauge equivalence classes of the light-like portions would give rise to additional degrees of freedom. In space-time interior and for string world sheets they would become visible.

    For ramified primes n distint 3-surfaces would collapse to single one but the n discrete degrees of freedom would be present and particle would obtain them. I have indeed proposed number theoretical second quantization assigning fermionic Clifford algebra to the sheets with n oscillator operators. Note that this option does not require Galois group to act as gauge group in the general case. This number theoretical second quantization might relate to the realization of Boolean algebra suggested by weak form of NMP (see this).

3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework

It seems that Galois group is naturally associated with the hierarchy heff/h=n of effective Planck constants defined by the hierarchy of quantum criticalities. n would naturally define the maximal order for the element of Galois group. The analog of singular covering with that of z1/n would suggest that Galois group is very closely related to the conformal symmetries and its action induces permutations of the sheets of the covering of space-time surface.

Without any additional assumptions the values of n and ramified primes are completely independent so that the conjecture that the magnetic flux tube connecting the wormhole contacts associated with elementary particles would not correspond to very large n having the p-adic prime p characterizing particle as factor (p=M127=2127-1 for electron). This would not induce any catastrophic changes.

TGD based physics could however change the situation and reduce number theoretical degrees of freedom: the intuitive hypothesis that p divides n might hold true after all.

  1. The strong form of GCI implies strong form of holography. One implication is that the WCW Kähler metric can be expressed either in terms of Kähler function or as anti-commutators of super-symplectic Noether super-charges defining WCW gamma matrices. This realizes what can be seen as an analog of Ads/CFT correspondence. This duality is much more general. The following argument supports this view.

    1. Since fermions are localized at string world sheets having ends at partonic 2-surfaces, one expects that also Kähler action can be expressed as an effective stringy action. It is natural to assume that string area action is replaced with the area defined by the effective metric of string world sheet expressible as anti-commutators of Kähler-Dirac gamma matrices defined by contractions of canonical momentum currents with imbedding space gamma matrices. It string tension is proportional to heff2, string length scales as heff.

    2. AdS/CFT analogy inspires the view that strings connecting partonic 2-surfaces serve as correlates for the formation of - at least gravitational - bound states. The distances between string ends would be of the order of Planck length in string models and one can argue that gravitational bound states are not possible in string models and this is the basic reason why one has ended to landscape and multiverse non-sense.

  2. In order to obtain reasonable sizes for astrophysical objects (that is sizes larger than Schwartschild radius rs=2GM) For heff=hgr=GMm/v0 one obtains reasonable sizes for astrophysical objects. Gravitation would mean quantum coherence in astrophysical length scales.

  3. In elementary particle length scales the value of heff must be such that the geometric size of elementary particle identified as the Minkowski distance between the wormhole contacts defining the length of the magnetic flux tube is of order Compton length - that is p-adic length scale proportional to p1/2. Note that dark physics would be an essential element already at elementary particle level if one accepts this picture also in elementary particle mass scales. This requires more precise specification of what darkness in TGD sense really means.

    One must however distinguish between two options.

    1. If one assumes n≈ p1/2, one obtains a large contribution to classical string energy as Δ ∼ mCP22Lp/hbar2eff ∼ mCP2/p1/2, which is of order particle mass. Dark mass of this size looks un-feasible since p-adic mass calculations assign the mass with the ends wormhole contacts. One must be however very cautious since the interpretations can change.

    2. Second option allows to understand why the minimal size scale associated with CD characterizing particle correspond to secondary p-adic length scale. The idea is that the string can be thought of as being obtained by a random walk so that the distance between its ends is proportional to the square root of the actual length of the string in the induced metric. This would give that the actual length of string is proportional to p and n is also proportional to p and defines minimal size scale of the CD associated with the particle. The dark contribution to the particle mass would be Δ m ∼ mCP22Lp/hbar2eff∼ mCP2/p, and completely negligible suggesting that it is not easy to make the dark side of elementary visible.

  4. If the latter interpretation is correct, elementary particles would have huge number of hidden degrees of freedom assignable to their CDs. For instance, electron would have p=n=2127-1 ≈ 1038 hidden discrete degrees of freedom and would be rather intelligent system - 127 bits is the estimate- and thus far from a point-like idiot of standard physics. Is it a mere accident that the secondary p-adic time scale of electron is .1 seconds - the fundamental biorhythm - and the size scale of the minimal CD is slightly large than the circumference of Earth?

    Note however, that the conservation option assuming that the magnetic flux tubes connecting the wormhole contacts representing elementary particle are in heff/h=1 phase can be considered as conservative option.

Thursday, May 14, 2015

Quantum Mathematics in TGD Universe

Some comments about quantum mathematics, quantum Boolean thinking and computation as they might happen at fundamental level.

  1. One should understand how Boolean statements A→B are represented. Or more generally: How a computation like procedure leading from a collection A of math objects collection B of math objects takes place? Recall that in computations the objects coming in and out are bit sequences. Now one have computation like process. → is expected to correspond to the arrow of time.

    If fermionic oscillator operators generate Boolean basis, zero energy ontology is necessary to realize rules as rules connecting statements realized as bit sequences. Positive energy ontology would allow only statements A,B but not statements A→B about them. ZEO allows also to avoid restrictions due to fermion number conservation and its well-definedness.

    Collection A is at the passive boundary of CD and not changed in state function reduction sequence defining self and B is at the active one. As a matter fact, it is not single statement but a quantum superpositions of statements B, which resides there! In the quantum jump selecting single B at the active boundary, A is replaced with a superposition of A:s: self dies and re-incarnates as more negentropic entity. Q-computation halts.

    That both a and b cannot be known precisely is a quantal limitation to what can be known: philosopher would talk about epistemology here. The different pairs (a,b) in superposition over b:s are analogous to different implications of a. Thinker is doomed to always live in a quantum cognitive dust and never be quite sure of.

  2. What is the computer like structure now? Turing computer is discretized 1-D time-like line. This quantum computer is superposition of 4-D space-time surfaces with the basic computational operations located along it as partonic 2-surfaces defining the algebraic operations and connected by fermion lines representing signals. Also string world sheets are involved. In some key aspects this is very similar to ordinary computer. By strong form of holography computations use only data at string world sheets and partonic 2-surfaces.

  3. What is the computation? It is sequence of repeated state function reduction leaving the passive boundary of CD
    intact but affecting the position (moduli) of upper boundary of CD and also the parts of zero energy states there.
    It is a sequence of unitary processes delocalizing the active boundary of CD followed by localization but no reduction. This the counterpart for a sequence of reductions leaving quantum state invariant in ordinary measurement theory (Zeno etc). Commutation halts as the first reduction to the opposite boundary occurs. Self dies and re-incarnates at the opposite boundary. Negentropy gain results in general and can be see as the information gained in the computation. One might hope that the new self (maybe something at higher level of dark matter hierarchy) is a little bit wiser - at least statistically speaking this seems to be true by weak form of NMP!

  4. One should understand the quantum counterparts for the basic rules of manipulation. ×,/,+, and - are the most familiar example.

    1. The basic rules correspond physically to generalized Feynman/twistor diagrams representing sequences of algebraic manipulations in the Yangian of super-symplectic algebra. Sequences correspond now to collections of partonic 2-surfaces defining vertices of generalized twistor diagrams.

    2. 3- vertices correspond to product and co-product for quantal stringy Noether charges. Geometrically the vertex - analog of algebraic operation - is a partonic 2-surface at with incoming and outgoing light-like 3-surfaces meet - like vertex of Feynman diagram. Co-product vertex is not encountered in simple algebraic systems, and is time reversed variant of vertex. Fusion instead of annihilation.

    3. This diagrammatics has a huge symmetry just like ordinary computations have. All computation sequences (note that the corresponding space-time surfaces are different!) connecting same collections A and B of objects produce the same scattering amplitude. This generalises the duality symmetry of hadronic string models. This is really gigantic simplification and the results of twistor Grassmann approach suggest that something similar is obtained there. This implication was so gigantic that I gave up the idea for years.

  5. One should understand the analogs for the mathematical axioms. What are the fundamental rules of manipulation?

    1. The classical computation/deduction would obey deterministic rules at vertices. The quantal formulation cannot be deterministic for the simple reason that one has quantum non-determinism (weak form of NMP allowing also good and evil) . The quantum rules obey the format that God used when communicating with Adam and Eve: do anything else but do not the break the conservation laws. Classical rules would list all the allowed possibilities and this leads to difficulties as Goedel demonstrated. I think that chess players follow the "anti-axiomatics".

    2. I have the feeling that anti-axiomatics - not any well-established idea, it occurred to me as I wrote this - could provide a more natural approach to quantum computation and even allow a new manner to approach to the problematics of axiomatisations. It is also interesting to notice a second TGD inspired notion - the infinite hierarchy of mostly infinite integers (generated from infinite primes obtained by a repeated second quantization of an arithmetic QFT) - could make possible a generalisation of Gödel numbering for statements/computations. This view has at least one virtue: it makes clear how extremely primitive conscious entities we are in a bigger picture!

  6. The laws of physics take care that the anti-axioms are obeyed. Quite concretely:

    1. Preferred extremal property of Kähler action and Käler-Dirac action plus conservation laws for charges associated with super-symplectic and other generalised conformal symmetries would define the rules not broken in vertices.

    2. At the fermion lines connecting the vertices the propagator would be determined by the boundary part of Kahler-Dirac action. K-D equation for spinors and consistency consistency conditions from Kahler action (strong form of holography) would dictate what happens to fermionic oscillator operators defining the analog of quantum Boolean algebra as super-symplectic algebra.

Thursday, May 07, 2015

Breakthroughs in the number theoretic vision about TGD

Number theoretic universality states that besides reals and complex numbers also p-adic number fields are involved (they would provide the physical correlates of cognition). Furthermore, scattering amplitudes should be well-defined in all number fields be obtained by a kind of algebraic continuation. I have introduced the notion of intersection of realities and p-adicities which corresponds to some algebraic extension of rationals inducing an extension of p-adic numbers for any prime p. Adelic physics is a strong candidate for the realization of fusion of real and p-adic physics and would mean the replacement of real numbers with adeles. Field equations would hold true for all numer fields and the space-time surfaces would relate very closely to each other: one could say that p-adic space-time surfaces are cognitive representations of the real ones.

I have had also a stronger vision which is now dead. This sad event however led to a discovery of several important results.

  1. The idea has been that p-adic space-time sheets would be not only "thought bubbles" representing real ones but also correlates for intentions and the transformation of intention to action would would correspond to a quantum jump in which p-adic space-time sheet is transformed to a real one. Alternatively, there would be a kind of leakage between p-adic and real sectors. Cognitive act would be the reversal of this process. It did not require much critical thought to realize that taking this idea seriously leads to horrible mathematical challenges. The leakage takes sense only in the intersection, which is number theoretically universal so that there is no point in talking about leakage. The safest assumption is that the scattering amplitudes are defined separately for each sector of the adelic space-time. This means enormous relief, since there exists mathematics for defining adelic space-time.

  2. This realization allows to clarify thoughts about what the intersection must be. Intersection corresponds by strong form of holography to string world sheets and partonic 2-surfaces at which spinor modes are localized for several reasons: the most important reasons are that em charge must be well-defined for the modes and octonionic and real spinor structures can be equivalent at them to make possible twistorialization both at the level of imbedding space and its tangent space.

    The parameters characterizing the objects of WCW are discretized - that is belong to an appropriate algebraic extension of rationals so that surfaces are continuous and make sense in real number field and p-adic number fields. By conformal invariance they might be just conformal moduli. Teichmueller parameters, positions of punctures for partonic 2-surfaces, and corners and angles at them for string world sheets. These can be continued to real and p-adic sectors and

  3. Fermions are correlates for Boolean cognition and anti-commutation relations for them are number theoretically universal, even their quantum variants when algebraic extension allows quantum phase. Fermions and Boolean cognition would reside in the number theoretically universal intersection. Of course they must do so since Boolean thought and cognition in general is behind all mathematics!

  4. I have proposed this in p-adic mass calculations for two decades ago. This would be wonderful simplification of the theory: by conformal invariance WCW would reduce to finite-dimensional moduli space as far as calculations of scattering amplitudes are considered. The testing of the theory requires classical theory and 4-D space-time. This holography would not mean that one gives up space-time: it is necessary. Only cognitive and as it seems also fundamental sensory representations are 2-dimensional. All that one can mathematically say about reality is by using data at these 2-surfaces. The rest is needed but it require mathematical thinking and transcendence! This view is totally different from the sloppy and primitive philosophical idea that space-time could somehow emerge from discrete space-time.

This has led also to modify the ideas about the relation of real and p-adic physics.
  1. The notion of p-adic manifolds was hoped to provide a possible realization of the correspondence between real and p-adic numbers at space-time level. It relies on the notion canonical identification mapping p-adic numbers to real in continuous manner and realizes finite measurement resolution at space-time level. p-Adic length scale hypothesis emerges from the application of p-adic thermodynamics to the calculation of particle masses but generalizes to all scales.

  2. The problem with p-adic manifolds is that the canonical identification map is not general coordinate invariant notion. The hope was that one could overcome the problem by finding preferred coordinates for imbedding space. Linear Minkowski coordinates or Robertson-Walker coordinates could be the choice for M4. For CP2 coordinates transforming linearly under U(2) suggest themselves. The non-uniqueness however persists but one could argue that there is no problem if the breaking of symmetries is below measurement resolution. The discretization is however also non-unique and makes the approach to look ugly to me although the idea about p-adic manifold as cognitive chargt looks still nice.

  3. The solution of problems came with the discovery of an entirely different approach. First of all, realized discretization at the level of WCW, which is more abstract: the parameters characterizing the objects of WCW are discretized - that is assumed to belong to an appropriate algebraic extension of rationals so that surfaces are continuous and make sense in real number field and p-adic number fields.

    Secondly, one can use strong form of holography stating that string world sheets and partonic 2-surfaces define the "genes of space-time". The only thing needed is to algebraically extend by algebraic continuation these 2-surfaces to 4-surfaces defining preferred extremals of Kähler action - real or p-adic. Space-time surface have vanishing Noether charges for a sub-algebra of super-symplectic algebra with conformal weights coming as n-ples of those for the full algebra- hierarchy of quantum criticalities and Planck constants and dark matters!

    One does not try to map real space-time surfaces to p-adic ones to get cognitive charts but 2-surfaces defining the space-time genes to both real and p-adic sectors to get adelic space-time! The problem with general coordinate invariance at space-time level disappears totally since one can assume that these 2-surfaces have rational parameters. One has discretization in WCW, rather than at space-time level. As a matter fact this discretization selects punctures of partonic surfaces (corners of string world sheets) to be algebraic points in some coordinatization but in general coordinate invariant manner

  4. The vision about evolutionary hierarchy as a hierarchy of algebraic extensions of rationals inducing those of p-adic number fields become clear. The algebraic extension associated with the 2-surfaces in the intersection is in question. The algebraic extension associated with them become more and more complex in evolution. Of course, NMP, negentropic entanglement (NE) and hierarchy of Planck constants are involved in an essential manner too. Also the measurement resolution characterized by the number of space-time sheets connecting average partonic 2-surface to others is a measure for "social" evolution since it defines measurement resolution.

There are two questions, which I have tried to answer during these two decades.
  1. What makes some p-adic primes preferred so that one can say that they characterizes elementary particles and presumably any system?

  2. What is behind p-adic length scale hypothesis emerging from p-adic mass calculations and stating that primes near but sligthly below two are favored physically, Mersenne primes in particular. There is support for a generalization of this hypothesis: also primes near powers of 3 or powers of 3 might be favored as length sand time scales which suggests that powers of prime quite generally are favored.

The adelic view led to answers to these questions. The answer to the first question has been staring directly to my eyes for more than decade.
  1. The algebraic extension of rationals allow so called ramified primes. Rational primes decompose to product of primes of extension but it can happen that some primes of extension appear as higher than first power. In this case one talks about ramification. The product of ramified primes for rationals defines an integer characterizing the ramification. Also for extension allows similar characteristic. Ramified primes are an extremely natural candidate for preferred primes of an extension (I know that I should talk about prime ideals, sorry for a sloppy language): that preferred primes could follow from number theory itself I had not though earlier and tried to deduce them from physics. One can assign the characterizing integers to the string world sheets to characterize their evolutionary level. Note that the earlier heuristic idea that space-time surface represents a decomposition of integer is indeed realized in terms of holography!

  2. Also infinite primes seem to find finally the place in the big picture. Infinite primes are constructed as an infinite hierarchy of second quantization of an arithmetic quantum field theory. The infinite primes of the previous level label the single fermion - and boson states of the new level but also bound states appear. Bound states can be mapped to irreducible polynomials of n-variables at n:th level of infinite obeying some restrictions. It seems that they are polynomials of a new variable with coefficients which are infinite integers at the previous level.

    At the first level bound state infinite primes correspond to irreducible polynomials: these define irreducible extensions of rationals and as a special case one obtains those satisfying so called Eistenstein criterion: in this case the ramified primes can be read directly from the form of the polynomial. Therefore the hierarchy of infinite primes seems to define algebraic extension of rationals, that of polynomials of one variables, etc.. What this means from the point of physics is a fascinating question. Maybe physicist must eventually start to iterate second quantization to describe systems in many-sheeted space-time! The marvellous thing would be the reduction of the construction of bound states - the really problematic part of quantum field theories - to number theory!

The answer to the second question requires what I call weak form of NMP.
  1. Strong form of NMP states that negentropy gain in quantum jump is maximal: density matrix decompose into sum of terms proportional to projection operators: choose the sub-space for which number theoretic negentropy is maximal. The projection operator containing the largest power of prime is selected. The problem is that this does not allow free will in the sense as we tend to use: to make wrong choices!

  2. Weak NMP allows to chose any projection operator and sub-space which is any sub-space of the sub-space defined by the projection operator. Even 1-dimensional in which case standard state function reduction occurs and the system is isolated from the environment as a prize for sin! Weak form of NMP is not at all so weak as one might think. Suppose that the maximal projector operator has dimension nmax which is product of large number of different but rather small primes. The negentropy gain is small. If it is possible to choose n=nmax-k, which is power of prime, negentropy gain is much larger!

    It is largest for powers of prime defining n-ary p-adic length scales. Even more, large primes correspond to more refined p-adic topology: p=1 (one could call it prime) defines discrete topology, p=2 defines the roughest p-adic topology, the limit p→ ∞ is identified by many mathematicians in terms of reals. Hence large primes p<nmax are favored. In particular primes near but below powers of prime are favored: this is nothing but a generalization of p-adic length scale hypothesis from p=2 to any prime p.

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

Tuesday, May 05, 2015

Updated Negentropy Maximization Principle

Quantum TGD involves "holy trinity" of time developments. There is the geometric time development dictated by the preferred extremal of Kähler action crucial for the realization of General Coordinate Invariance and analogous to Bohr orbit. There is what I originally called unitary "time development" U: Ψi→ UΨi→ Ψf, associated with each quantum jump. This would be the counterpart of the Schrödinger time evolution U(-t,t→ ∞). Quantum jump sequence itself defines what might be called subjective time development.

Concerning U, there is certainly no actual Schrödinger equation involved: situation is in practice same also in quantum field theories. It is now clear that in Zero Energy Ontology (ZEO) U can be actually identified as a sequence of basic steps such that single step involves a unitary evolution inducing delocalization in the moduli space of causal diamonds CDs) followed by a localization in this moduli space selecting from a superposition of CDs single CD. This sequence replaces a sequence of repeated state function reductions leaving state invariant in ordinary QM. Now it leaves in variant second boundary of CD (to be called passive boundary) and also the parts of zero energy states at this boundary. There is now a very attractive vision about the construction of transition amplitudes for a given CD, and it remains to be see whether it allows an extension so that also transitions involving change of the CD moduli characterizing the non-fixed boundary of CD.

A dynamical principle governing subjective time evolution should exist and explain state function reduction with the characteristic one-one correlation between macroscopic measurement variables and quantum degrees of freedom and state preparation process. Negentropy Maximization Principle is the candidate for this principle. In its recent form it brings in only a single little but overall important modification: state function reductions occurs also now to an eigen-space of projector but the projector can now have dimension which is larger than one. Self has free will to choose beides the maximal possible dimension for this sub-space also lower dimension so that one can speak of weak form of NMP so that negentropy gain can be also below the maximal possible: we do not live in the best possible world. Second important ingredient is the notion of negentropic entanglement relying on p-adic norm.

The evolution of ideas related to NMP has been slow and tortuous process characterized by misinterpretations, over-generalizations, and unnecessarily strong assumptions, and has been basically evolution of ideas related to the anatomy of quantum jump and of quantum TGD itself.

Quantum measurement theory is generalized to theory of consciousness in TGD framework by replacing the notion of observer as outsider of the physical world with the notion of self. Hence it is not surprising that several new key notions are involved.

  1. ZEO is in central role and brings in a completely new element: the arrow of time changes in the counterpart of standard quantum jump involving the change of the passive boundary of CD to active and vice versa. In living matter the changes of the of time are inn central role: for instance, motor action as volitional action involves it at some level of self hierarchy.

  2. The fusion of real physics and various p-adic physics identified as physics of cognition to single adelic physics is second key element. The notion of intersection of real and p-adic worlds (intersection of sensory and cognitive worlds) is central and corresponds in recent view about TGD to string world sheets and partonic 2-surfaces whose parameters are in an algebraic extension of rationals. By strong form of of holography it is possible to continue the string world sheets and partonic 2-surfaces to various real and p-adic surfaces so that what can be said about quantum physics is coded by them. The physics in algebraic extension can be continued to real and various p-adic sectors by algebraic continuation meaning continuation of various parameters appearing in the amplitudes to reals and various p-adics.

    An entire hierarchy of physics labeled by the extensions of rationals inducing also those of p-adic numbers is predicted and evolution corresponds to the increase of the complexity of these extensions. Fermions defining correlates of Boolean cognition can be said so reside at these 2-dimensional surfaces emerging from strong form of holography implied by strong form of general coordinate invariance (GCI).

    An important outcome of adelic physics is the notion of number theoretic entanglement entropy: in the defining formula for Shannon entropy logarithm of probability is replaced with that of p-adic norm of probability and one assumes that the p-adic prime is that which produces minimum entropy. What is new that the minimum entropy is negative and one can speak of negentropic entanglement (NE). Consistency with standard measurement theory allows only NE for which density matrix is n-dimensional projector.

  3. Strong form of NMP states that state function reduction corresponds to maximal negentropy gain. NE is stable under strong NMP and it even favors its generation. Strong form of NMP would mean that we live in the best possible world, which does not seem to be the case. The weak form of NMP allows self to choose whether it performs state function reduction yielding the maximum possible negentropy gain. If n-dimensional projector corresponds to the maximal negentropy gain, also reductions to sub-spaces with n-k-dimensional projectors down to 1-dimensional projector are possible. Weak form has powerful implications: for instance, one can understand how primes near powers of prime are selected in evolution identified at basic level as increase of the complexity of algebraic extension of rationals defining the intersection of realities and p-adicities.

  4. NMP gives rise to evolution. NE defines information resources, which I have called Akashic records - kind of Universal library. The simplest possibility is that under the repeated sequence of state function reductions at fixed boundary of CD NE at that boundary becomes conscious and gives rise to experiences with positive emotional coloring: experience of love, compassion, understanding, etc... One cannot exclude the possibility that NE generates a conscious experience only via the analog of interaction free measurement but this option looks un-necessary in the recent formulation.

  5. Dark matter hierarchy labelled by the values of Planck constant heff=n× h is also in central role and interpreted as a hierarchy of criticalities in which sub-algebra of super-symplectic algebra having structure of conformal algebra allows sub-algebra acting as gauge conformal algebra and having conformal weights coming as n-ples of those for the entire algebra. The phase transition increasing heff reduces criticality and takes place spontaneously. This implies a spontaneous generation of macroscopic quantum phases interpreted in terms of dark matter. The hierarchies of conformal symmetry breakings with n(i) dividing n(i+1) define sequences of inclusions of HFFs and the conformal sub-algebra acting as gauge algebra could be interpreted in terms of measurement resolution.

    n-dimensional NE is assigned with heff=n× h and is interpreted in terms of the n-fold degeneracy of the conformal gauge equivalence classes of space-time surfaces connecting two fixed 3-surfaces at the opposite boundaries of CD: this reflects the non-determinism accompanying quantum criticality. NE would be between two dark matter system with same heff and could be assigned to the pairs formed by the n sheets. This identification is important but not well enough understood yet. The assumption that p-adic primes p divide n gives deep connections between the notion of preferred p-adic prime, negentropic entanglement, hierarchy of Planck constants, and hyper-finite factors of type II1.

  6. Quantum classical correspondence (QCC) is an important constraint in ordinary measurement theory. In TGD QCC is coded by the strong form of holography assigning to the quantum states assigned to the string world sheets and partonic 2-surfaces represented in terms of super-symplectic Yangian algebra space-time surfaces as preferred extremals of Kähler action, which by quantum criticality have vanishing super-symplectic Noether charges in the sub-algebra characterized by integer n. Zero modes, which by definition do not contribute to the metric of "world of classical worlds" (WCW) code for non-fluctuacting classical degrees of freedom correlating with the quantal ones. One can speak about entanglement between quantum and classical degrees of freedom since the quantum numbers of fermions make themselves visible in the boundary conditions for string world sheets and their also in the structure of space-time surfaces.

NMP has a wide range of important implications.
  1. In particular, one must give up the standard view about second law and replace it with NMP taking into account the hierarchy of CDs assigned with ZEO and dark matter hierarchy labelled by the values of Planck constants, as well as the effects due to NE. The breaking of second law in standard sense is expected to take place and be crucial for the understanding of evolution.

  2. Self hierarchy having the hierarchy of CDs as imbedding space correlate leads naturally to a description of the contents of consciousness analogous to thermodynamics except that the entropy is replaced with negentropy.

  3. In the case of living matter NMP allows to understand the origin of metabolism. NMP demands that self generates somehow negentropy: otherwise a state function reduction to tjhe opposite boundary of CD takes place and means death and re-incarnation of self. Metabolism as gathering of nutrients, which by definition carry NE is the manner to avoid this fate. This leads to a vision about the role of NE in the generation of sensory qualia and a connection with metabolism. Metabolites would carry NE and each metabolite would correspond to a particular qualia (not only energy but also other quantum numbers would correspond to metabolites). That primary qualia would be associated with nutrient flow is not actually surprising!

  4. NE leads to a vision about cognition. Negentropically entangled state consisting of a superposition of pairs can be interpreted as a conscious abstraction or rule: negentropically entangled Schrödinger cat knows that it is better to keep the bottle closed.

  5. NMP implies continual generation of NE. One might refer to this ever expanding universal library as "Akaschic records". NE could be experienced directly during the repeated state function reductions to the passive boundary of CD - that is during the life cycle of sub-self defining the mental image. Another, less feasible option is that interaction free measurement is required to assign to NE conscious experience. As mentioned, qualia characterizing the metabolite carrying the NE could characterize this conscious experience.

  6. A connection with fuzzy qubits and quantum groups with NE is highly suggestive. The implications are highly non-trivial also for quantum computation allowed by weak form of NMP since NE is by definition stable and lasts the lifetime of self in question.

For details see the chapter Negentropy Maximization Principleof "TGD Inspired Theory of Consciousness".

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

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.