pitkanen - tgd general theory. early review article (2001)

7/28/2019 Pitkanen - TGD General Theory. Early Review Article (2001)

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Topological Geometrodynamics: General Theory

M. PitkanenDept. of Physics, University of Helsinki, Helsinki, Finland.

Email: [email protected]://www.physics.helsinki.fi/matpitka/.

Abstract

The recent status of Topological Geometrodynamics (TGD) is reviewed. One canend up with TGD either by starting from the energy problem of General Relativityor from the need to generalize hadronic or super string models. The basic principle ofthe theory is Do not quantize! meaning that quantum physics is reduced to Kahlergeometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H= M4+CP2 with physical states represented by classical spinorfields. General coordinate invariance implies that classical theory becomes an exactpart of the quantum theory and configuration space geometry and that spacetimesurfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional K ahlergeometric existence fixes imbedding space and the dimension of the spacetime highlyuniquely and implies that superconformal and supercanonical symmetries acting onthe lightcone boundary M4+ CP2 are cosmologies symmetries.

The work with the p-adic aspects of TGD, the realization of the possible role of

quaternions and octonions in the formulation of quantum TGD, the discovery of infiniteprimes, and TGD inspired theory of consciousness encouraged the vision about TGDas a generalized number theory. The vision leads to a considerable generalization ofTGD and to an extension of the symmetries of the theory to include super-conformaland super-Kac-Moody symmetries associated with the group P SU(3) U(2)ew(P denotes Poincare group) acting as the local symmetries of the theory. Quantumcriticality, which can be seen as a prediction of the theory, fixes the value spectrumfor the coupling constants of the theory.

The proper mathematical and physical interpretation of the p-adic numbers hasremained a long-lasting challenge. Both TGD inspired theory of consciousness andthe vision about physics as a generalized number theory suggest that p-adic spacetimeregions obeying p-adic counterparts of the field equations are geometric correlates ofmind in the sense that they provide cognitive representations for the physics in thereal spacetime regions representing matter. Evolution identified as a gradual increaseof the infinite p-adic prime characterizing the entire Universe is basic prediction of thetheory.

S-matrix elements can be identified as Glebsch-Gordan coefficients between inter-

acting and free Super-Kac-Moody Algebra representations and it is now possible to

give Feynmann rules for the S-matrix in the approximation that elementary particles

correspond to the so called CP2 type extremals.

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Contents1 Introduction 2

2 How to end up with TGD? 4

3 Quantum TGD as configuration space spinor geometry 5

4 TGD as a generalized number theory 10

5 About the construction of S-matrix 17

1 Introduction

Topological Geometrodynamics [20, 21, 22] was born for more than twenty yearsago as an attempt to construct a Poincare invariant theory of gravitation byassuming that physically allowed spacetimes are representable as surfaces in thespace H = M4+CP2, where M4+ denotes the future lightcone of the Minkowskispace and CP2 is complex projective space having real dimension four (Appendixof [20]). Poincare group was identified as the isometry group of M4+ ratherthan of the spacetime surface itself. The isometries of CP2 were identified ascolor group and the geometrization of electroweak gauge fields and elementaryparticle quantum numbers was achieved in terms of the spinor structure ofCP2. Twenty years after this discovery one can still say that CP2 codes theknown elementary particle quantum numbers and interactions in its geometry.An alternative manner to end up with TGD is as a generalization of hadronicor super string models and the consistency of the two approaches forces tointroduce of the notion of the many-sheeted spacetime topologically nontrivialin all length scales.

The construction of a mathematical theory around these physically very at-tractive ideas became the basic challenge and I have devoted the last twentythree years to the realization of this dream. The great idea is that quantumphysics reduces to the construction of Kahler metric and spinor structure for theinfinite-dimensional space CH of all possible 3-surfaces of H. One can regardthis space as a union of infinite-dimensional symmetric spaces having constantcurvature and possessing infinite-dimensional group of isometries: symmetricspace property reduces to the conformal invariance implied by the special prop-erties of the 4-dimensional lightcone boundary. Physical states correspond to

classical spinor fields in this space and a natural geometrization of the fermionicstatistics in terms of the gamma matrices emerges (Chs Definition of KahlerFunction and Construction of Configuration Space Kahler Geometry fromSymmetry Principles of [20]). One of the most dramatic (and quite not cor-rect) heuristic predictions of the original framework was the uniqueness of the

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space H: quantum TGD exists mathematically (cancellation of various infini-ties occurs) only for the space M4+CP2, the choice which is forced also by thecosmological and symmetry considerations. TGD as a generalized number the-ory vision however led to the realization that spacetime resp. embedding spacedimensions can be multiples of four resp. eight in accordance with quaternion-resp. octonion-dimensional democracy. One can say that infinite-dimensionalKahler geometric existence, and thus also physics, is unique. Note that also theinfinite-dimensional generalization of cosmological principle implies that config-uration space is a union of symmetric spaces labelled by zero modes. Infinite-dimensional Kahler geometry is studied also in string model context, now loopspaces are in question [1]. It was the thesis of D. Freed on Kahler geometrizationof the loop spaces [6], which stimulated the idea that configuration space is aunion of symmetric spaces with a unique Kahler geometry.

p-Adic number fields Rp [2] (one number field for each prime obtained asa completion of the rational numbers) emerged for about ten years ago as aseparate thread only loosely related to quantum TGD. The application of p-adicnumbers to physics had been suggested already earlier by various authors [29,11, 7], for further references see [2]. What made p-adics so attractive was that,with certain additional assumptions about physically favoured p-adic primes, itbecame possible to understand the basic elementary particle mass scales numbertheoretically. This led to a successful calculation of the elementary particlemasses using p-adic thermodynamics assuming that Super Virasoro algebra andrelated Kac Moody algebras, which are also basic algebraic structures of stringmodels, act as symmetries of TGD (Chs p-Adic Particle Massivation:.. of[21]).

p-Adic fractality is the basic physical implication of the p-adicity. Fractalityof the spacetime at the fundamental level has been suggested also by other au-thors: namely Cantorian fractal spacetime model of G. Ord and of MohammedEl Naschie [15, 18] and Scale Relativity [17, 3]. The physics in the infinite-dimensional configuration space of TGD is in some respects analogous to thatassociated with the infinite-dimensional Cantorian fractal spacetime in the the-ory of El Naschie. In particular, the notion of resolution is fundamental in bothapproaches. The configuration space of TGD is infinite-dimensional symmetricspace whereas Cantorian fractal spacetime has infinite formal dimension and anexpectation value for the topological and Hausdorff dimension equal to 4 + 3,where = (

5 1)/2 is the Golden Mean. Pseudo-Riemannian spacetime is

assumed to emerge by a phase transition in certain resolution.This review article is devoted to the basic quantum TGD, in particular the

role of the p-adic numbers in quantum TGD. Second article summarizes vari-ous physical consequences of the new spacetime concept, which are nontrivialin all length scales. There are three on-line books on TGD which are warmlyrecommended for a reader interested in details: Topological Geometrodynam-ics [20], Topological Geometrodynamics and p-Adic Numbers [21] and TGDInspired Theory of Consciousness with Applications to Biosystems [22].

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2 How to end up with TGD?The basic physical picture behind TGD was formed as a fusion of two rather dis-parate approaches: namely TGD is as a Poincare invariant theory of gravitationand TGD as a generalization of the old-fashioned string model.

The first approach was born as an attempt to construct a Poincare invari-ant theory of gravitation. Spacetime, rather than being an abstract manifoldendowed with a pseudo-Riemannian structure, is regarded as a surface in the 8-dimensional space H = M4+CP2, where M4+ denotes the interior of the futurelight cone of the Minkowski space (to be referred as light cone in the sequel) andCP2 = SU(3)/U(2) is the complex projective space of two complex dimensions[4, 9, 8, 23]. The identification of the spacetime as a submanifold [5, 27] ofM4+

CP2 leads to an exact Poincare invariance at classical level and solves

the conceptual difficulties related to the definition of the energy-momentum inGeneral Relativity [Misner-Thorne-Wheeler]. It soon however turned out thatsubmanifold geometry, being considerably richer in structure than the abstractmanifold geometry, leads to a geometrization of all basic interactions and thegeometrization of the elementary particle quantum numbers of the classical fieldsresults.

The second approach was based on the generalization of the mesonic stringmodel describing mesons as strings with quarks attached to the ends of thestring. In the 3-dimensional generalization 3-surfaces correspond to free par-ticles and the boundaries of the 3- surface correspond to partons in the sensethat the quantum numbers of the elementary particles reside on the boundaries.Various boundary topologies (number of handles) correspond to various fermion

families so that one obtains an explanation for the known elementary particlequantum numbers. This approach leads also to a natural topological descriptionof the particle reactions as topology changes: for instance, two-particle decaycorresponds to a decay of a 3-surface to two disjoint 3-surfaces.

The only manner to unify these two versions of TGD is by a generalizationof the spacetime concept. The macroscopic spacetime with matter is identifiedas a many-sheeted surface with hierarchical structure. There are sheets gluedon larger sheets glued on larger sheets..... Each sheet has an outer boundaryand finite size and material objects are identified as spacetime sheets. Gluing isperformed by so called topological sum operation connecting different spacetimesheets by very tiny wormhole contacts with size of order CP2 radius R about104 Planck lengths (1030 meters). Wormhole contacts, whose function is tofeed gauge fluxes from smaller to larger spacetime sheet, naturally reside near

the boundaries of the smaller spacetime sheet.many-sheeted spacetime leads to the geometrization of structures and matter

in terms of the macroscopic topology of the spacetime surface and means thatwe live in the middle of the wildest science fiction. The larger spacetime sheetrepresents the external world from the view point of the smaller spacetime sheet.For instance, atomic spacetime sheet represents external world of an atomic

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nucleus, molecular spacetime sheet the external world of an atom, ..., organ theexternal world of a cell, etc..

3 Quantum TGD as configuration space spinor

geometry

The program of quantum TGD is to reduce quantum theory (apart from theconcept of quantum jump) to classical spinor geometry of the configurationspace.

a) Quantum theory for extended particles is free, classical field theory forthe classical spinor fields defined in the configuration space C(H) consisting ofall possible 3-surfaces in H.

b) Configuration space is endowed with the metric and spinor structureso that one can define various metric related differential operators, say Diracoperator, appearing in the field equations of the theory. It turns out that con-figuration space spinors at given point of CH correspond physically to the Fockspace of second quantized free fermions so that a very close connection to theordinary quantum field theory emerges. Configuration space spinor fields areDiff4 invariant so that quantum states correspond to entire histories rather thanconstant time snapshots of a single history in TGD.

The essential elements of configuration space geometry are Kahler prop-erty, four-dimensional general coordinate invariance and the decomposition ofthe configuration space to a union of infinite-dimensional symmetric spacesG/H, where G is infinite-dimensional group of the canonical transformations

of H = M4

+ CP2 (M4

+ denotes light cone boundary, the moment of bigbang) acting as isometries of CH. Configuration space metric exists only forH = M4+ S. The construction of the configuration space spinor structureleads to a supersymmetrization of the various symmetry structures and in caseof a four-dimensional spacetime exists only for the dimension D = 8 for H andimplies S = CP2. More generally, 4n-dimensional spacetimes in 8n-dimensionalspace M4+ CP2n1 are possible.

1. How to realize four-dimensional general coordinate invariance?

In order to realize 4-dimensional general coordinate invariance in the spaceof 3-surfaces, the definition of the configuration space metric should somehowassociate to a given 3-surface X3 a unique spacetime surface X4(X3) for Diff4 toact on. Physical considerations require that the metric should be, not only Diff4

invariant, but also Diff4 degenerate so that infinitesimal Diff4 transformationsshould correspond to zero norm vector fields of the configuration space.

Since Kahler function determines Kahler geometry, the definition of theKahler function should associate a unique spacetime surface X4(X3) to a given3-surface X3. The natural physical interpretation for this spacetime surface isas the classical spacetime X4(X3) associated with X3 so that in TGD classical

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physics becomes a part of the configuration space geometry and of the quantumtheory.2. Symmetric space structure and light cone boundary

In the finite-dimensional context, globally symmetric spaces are of form G/Hand connection and curvature are independent of the metric, provided it is leftinvariant under G. If configuration space is a union of infinite-dimensionalsymmetric spaces, an enormous calculational simplification results and giveshopes of a calculable theory. The task is to identify the infinite-dimensionalgroups G and H and at the infinitesimal level the Cartan decomposition g = thof the infinite-dimensional Lie-algebra g ofG to the direct sum of the Lie-algebrah of H and its complement t in g satisfying the crucial conditions [t, t] h,[h, t] t and [h, h] h.

The crux of the matter is Diff4 degeneracy: all 3-surfaces on the orbit of

3-surface X3 must be physically equivalent so that one can effectively replaceall 3-surfaces Z3 on the orbit of X3 with a suitably chosen surface Y3 on theorbit of X3. The Lorentz and Diff4 invariant choice of Y3 is as the intersectionof the 4-surface with the set M4+ CP2, where M4+ denotes the boundaryof the light cone: effectively the imbedding space can be replaced with theproduct M4+ CP2 as far as vibrational degrees of freedom are considered.More precisely: configuration space has a fiber structure: the 3-surfaces Y3 M4+ CP2 correspond to the base space and the 3-surfaces on the orbit ofgiven Y3 and diffeomorphic with Y3 correspond to the fiber and are separatedby a zero distance from each other in the configuration space metric.

These observations lead to the identification of the isometry group as somesubgroup G of the group of the diffeomorphisms of H = M4+ CP2. Thesediffeomorphisms indeed act in a natural manner in C H, the space of the 3-surfaces in H. Therefore one can identify the configuration space as the unionof the coset spaces G/H, where H corresponds to the subgroup of G acting asdiffeomorphisms for a given X3. H depends on the topology of X3 and since Gdoes not change the topology of the 3-surface, each 3-topology defines a separateorbit ofG. Therefore, the union involves the sum over all topologies of X3 pluspossibly other zero modes.

The task is to identify correctly G as a sub-algebra of the diffeomorphisms ofH. The only possibility seems to be that the canonical transformations of Hgenerated by the function algebra of H act as isometries of the configurationspace. The canonical transformations act nontrivially also in M4+ since M

4+

allows Kahler structure and thus also symplectic structure.The solution to the problem of identifying the Cartan decomposition turned

out to be amazingly simple. t corresponds naturally to the canonical generatorswith odd conformal weight whereas h corresponds to the generators with evenconformal weight. The requirement that h corresponds to zero modes statesnothing but the invariance of the theory with respect to supercanonical andand super-conformal symmetries of the lightcone boundary. Note that thesesymmetries are cosmological.

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4. Complexification and the magic properties of the light cone boundaryIn case of the Kahler metric, G- and H Lie-algebras must allow a complex-ification so that the isometries can act as holomorphic transformations. Theunique feature of M4+ is its metric degeneracy: the boundary of the light coneis metrically 2-dimensional sphere although it is topologically 3-dimensional.This implies that lightcone boundary allows an infinite-dimensional group ofconformal symmetries generated by an algebra, which is a generalization of theordinary Virasoro algebra. There is actually also an infinite-dimensional groupof isometries (!) isomorphic with the group of the conformal transformations.Even more, in case of H the groups of the conformal symmetries and isome-tries are local with respect to CP2. Furthermore, light cone boundary allows aninfinite-dimensional group of canonical transformations as the symmetries of thesymplectic structure automatically associated with the Kahler structure. There-

fore 4-dimensional Minkowski space is in a unique position in TGD approach.M4+ allows also complexification and Kahler structure unlike the boundariesof the higher-dimensional light cones so that it becomes possible to define acomplexification in the tangent space of the configuration space, too.

The space of the vector fields on H = M4+ CP2 inherits the complexstructure of the light cone boundary and CP2. The complexification can beinduced from the complex conjugation for the complexified vector fields of H.The crucial element is the complex conjugation for the scalar function basisassociated with the the light cone boundary. In M4+ degrees of freedom com-plexification works only provided that the radial vector fields posses zero normas configuration space vector fields (they have also zero norm as vector fields).

The effective two-dimensionality of the light cone boundary allows also to

circumvent the no-go theorems associated with the higher-dimensional Abelianextensions. First, in the dimensions D > 2 Abelian extensions of the gauge alge-bra are extensions by an infinite dimensional Abelian group rather than centralextensions by the group U(1). In the present case the extension is a symplecticextension analogous to the extension defined by the Poisson bracket {p, q} = 1rather than the standard central extension but is indeed 1- dimensional andwell defined provided that the configuration space metric is Kahler. Secondly,D > 2 extensions possess no unitary faithfull representations (satisfying certainwell motivated physical constraints) [12]. The point is that light cone boundaryis metrically and conformally 2-sphere and therefore the gauge algebra is effec-tively the algebra associated with the 2-sphere and, as a consequence, also theconfiguration space metric is Kahler.

5. Definition of Kahler function

In principle the construction of the Kahler geometry reduces to the com-plexification of the tangent space of CH and to that of finding Kahler functionK(X3) with the property that it associates a unique spacetime surface X4(X3)to a given 3-surface X3 and possesses mathematically and physically acceptableproperties. The guess for the Kahler function is the following one.

The value of the Kahler function K for a given 3-surface Y3 on lightcone

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boundary is obtained in the following manner (Ch. Definition of Kahler func-tion of [20]).a) Consider all possible 4-surfaces X4 M4+ CP2 having Y3 as its sub-

manifold: Y3 X4. If Y3 has boundary then it belongs to the boundary ofX4: Y3 X4. In soap film analogy Y3 is analogous to the frame spanningminimal area soap film.

b) Associate to each four surface Kahler action as the Maxwell action forthe Abelian gauge field defined by the projection of the CP2 Kahler form to thefour-surface. For a Minkowskian signature of the induced metric Kahler electricfield gives a negative contribution to the action density whereas for an Euclidiansignature the action density is always non-positive.

c) Define the value of the Kahler function K for Y3 as the absolute mini-mum of the Kahler action SK over all possible four-surfaces having Y

3 as its

submanifold: K(Y3) = M in{SK(X4)|X4 Y3}.d) IfX3 is arbitrary 3-surface, not necessarily located on lightcone boundary,

and if it belongs to the orbit of X4(Y3) and is diffeomorphic with the intersectionof the orbit with a=constant hyperboloid of M4+ CP2, define K(X3) to beequal to K(Y3). In this manner one achieves automatically, not only generalcoordinate invariance, but also Diff4 degeneracy.

e) Kahler action turns out to be non-determistic in the sense that there areseveral degenerate absolute minima associated with Y3 M4+. In this kind ofsituation one must generalize the notion of 3-surface allowing also associationsequencies consisting of unions of a minimal number of 3-surfaces with timelike separations fixing uniquely single degenerate absolute minimum. The fixingof the additional three-surfaces inside lightcone is like fixing auxiliary frame

in case that there are several minimal area soap films through a given frame.This non-determinism is absolutely essential for the physicality of the theory:without it the theory would reduce to a theory defined in the space M4+CP2and time would be lost.

6. The value of Kahler coupling strength from the requirement of quantum

criticality

Since the Theory of Everything should be unique it would be highly desir-able to find arguments fixing the normalization or equivalently the value of theKahler coupling strength K. The exponent exp(K) of Kahler function appearsas the vacuum functional of the theory and is completely analogous to the ex-ponent exp(H/T) defining partition function in thermodynamics. Hence Kis mathematically in the same role as temperature in thermodynamics. Thissuggests that the physical value of K is analogous to critical temperature and

hence more or less unique. One can wonder whether the value of Kahler couplingstrength is unique from the criticality requirement. The answer is very probablyNo. p-Adic considerations suggest strongly Kahler coupling depends on the p-adic length scale Lp in a logarithmic manner just like the gauge coupling ofU(1)gauge theory (Ch. Configuration Space Spinor Structure of [20]). Thus fixedpoint property is not in conflict with the renormalization group evolution since

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this evolution is discretized. Even quantum criticality follows as a prediction ifone assumes that the fundamental action principle is second quantized modifiedDirac action fixed by the requirement that the variation of the c-number termresulting from the normal ordering of the action with respect to the imbeddingspace coordinates gives rise to the field equations associated with the Kahleraction (Ch. TGD as a Generalized Number Theory of [20]).

Quantum criticality implies the existence of macroscopic quantum systemsin all length scales: this is especially important implication for understandingbrain as macroscopic quantum system. The explanation of 1/f noise (see [26]and Ch. Quantum Control and Coordination in Biosystems of [22]), whichhas been found to be a universal phenomenon, in terms of quantum criticalitycircumvents the basic objection against self-organized criticality related to theinherent unstability of criticality. Quantum criticality is stable because the

critical parameter is constant of Nature so that small perturbations cannot thuschange it. This view leads also to a model of 1/f noise in terms of thermalarithmetic quantum field theory and the theory predicts deviations from 1/fnoise reflecting directly the distribution of primes.

7. Group theoretic construction of the configuration space geometry

The gigantic size of the isometry group suggests that it might be possible todeduce very detailed information about the metric of the configuration space bygroup theoretical arguments. This turns out to be the case (Ch. Constructionof configuration space Kahler geometry from symmetry principles of [20]). Inorder to have a Kahler structure, one must define a complexification of theconfiguration space. Also one should identify the Lie algebra of the isometrygroup and try to derive explicit form of the Kahler metric using this information.

One can indeed construct the metric in this manner but a rigorous proof thatthe corresponding Kahler function is the one defined by Kahler action doesnot exist yet although both approaches predict the same general qualitativeproperties for the metric. The argument stating the equivalence of the twoapproaches reduces to the hypothesis stating electric-magnetic duality of thetheory. For the absolute minima of Kahler action magnetic configuration spaceHamiltonians derivable from group theoretical approach are essentially identicalwith electric configuration space Hamiltonians derivable from Kahler action.

8. Configuration space spinor structure

The construction of the spinor structure (Ch. Configuration Space SpinorStructure of [20]) relies on the idea that configuration space gamma matricesprovide a geometrization for the second quantized free induced spinor fields onspacetime surface. This process decomposes into following steps. Perform a

second quantization for the free induced spinor field in X4; express configura-tion space gamma matrices and canonical super generators as superpositions ofthe fermionic oscillator operators; deduce the anticommutation relations of thespinor fields from the requirement of super canonical invariance; generalize theflux representation for the configuration space Hamiltonians to a spinorial fluxrepresentation for their super partners.

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Anticommutation relations must be written for super currents with a vectorindex rather than for the spinor components itself. This is consistent only pro-vided the number of the vector indices given by the dimension D of imbeddingspace is not larger the number of the spinor indices associated with a givenspinor chirality: this number equals to 2D/21. Thus the procedure works onlyprovided one has

D 2D/21 .Contrary to the earlier erraneous belief, D = 8 is the minimal imbedding spacedimension but not the only possible one. Together with the other constraintsthis implies that the minimal choice for the imbedding space is H = M4+CP2.TGD as a number theory vision to be discussed later forces spacetimes to be

n-quaternion-dimensional surfaces in n-octonion-dimensional imbedding spacesH = M4+ CP2n12 . Macroscopically spacetime surfaces are predicted to beeffectively at most four-dimensional.

4 TGD as a generalized number theory

The vision about the fundamental formulation of quantum TGD as a generalizednumber theory is based on the gradual accumulation of wisdom coming fromdifferent sources. The attempts to find a formulation allowing to understandreal and p-adic physics as aspects of some more general scenario have been animportant stimulus and generated a lot of, not necessarily mutually consistent,ideas. In the following one possible vision trying to build a consistent narrative

on basis of these ideas is summarized briefly.1. p-Adic aspects of quantum TGDThe hunch that p-adic numbers might be of relevance for TGD led rather

rapidly to p-adic mass calculations and to other other applications of p-adicnumbers discussed in second part of the article. The understanding of how p-adic numbers should be imbedded into the basic mathematical structure TGDhas developed with a much slower pace and the theory is still in a rather spec-ulative stage.

A longheld hypothesis has been that the spin glass analogy implied by thehuge vacuum degeneracy of the Kahler action could imply p-adic topology asan effective spacetime topology. TGD inspired theory of consciousness howeversuggests that p-adic topology is genuine rather than effective topology, and thatboth p-adic and real physics are needed in order to have complete description

of reality. Real topology is the topology of reality and various p-adic topologiesare topologies of possible experiences about reality. More concretely, p-adicspacetime regions (mind) provide cognitive representations about real regionsrepresenting matter. p-Adic length scale hypothesis, to be discussed in thearticle about applications of TGD, quantifies the notion of the many-sheetedspacetime.

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2. Quaternions, octonions, and TGDThe choice of the imbedding space seems to be fixed uniquely both by thespectrum of elementary particle quantum numbers and by the mathematicalarguments related to the construction of quantum TGD. The dimensions fourand eight are also p-adically very special. Imbedding space dimension is amultiple of eight from the requirement that the tangent space of imbedding spaceallows local octonionic structure. The discussions with Tony Smith [Smith]revealed that one can endow octonions and quaternions with Minkowski metricby defining the metric as Re(xy) instead of Re(xy) as done usually, so thatmetric signature is not a problem. M4+ has very natural quaternion structureconsistent with the Lorentz invariance and color group SU(3) in turn is theunique simple Lie-group possessing local octonionic structure so that also CP2emerges very naturally.

The concept of induction procedure can be generalized to the inductionof octonionic structure and it turns that any 4-surface is either associative orco-associative (tangent space or normal space is associative subalgebra withrespect to the induced algebraic structure). This suggests that 4n-dimensionalspacetimes could be regarded as maximal associative, perhaps quaternionic,submanifolds of 8n-dimensional imbedding space, which itself has octonionicstructure in some sense. This idea appears in Tony Smiths own unified theory[Smith] relying heavily on global octonionic structures and special properties ofthe 8-dimensional spacetime as well as lattice structures in 4 and 8 dimensions.The concept of calibration closely related to the associative structure is describedin the book Spinors and Calibrations of Reese Harvey [Harvey].

The second observation due to Tony Smith, suggests a partial reduction of

the p-adic length scale hypothesis to the number theory of integer quaternionsforming a 4-dimensional lattice D4: the physically interesting p-adic lengthscales are in one-one correspondence with quaternionic primes and their powers.Lattice structures indeed emerge naturally in p-adic QFT and below the socalled elementary particle horizon, the signature of the metric is Euclidian sothat quaternionic lattice structure is natural. In fact, this idea together withblackhole-elementary particle analogy gives a beatiful number theoretic contentto the p-adic length scale hypothesis (Ch. Quaternions, Octonions, and InfinitePrimes of [20]).

3. The painting is the landscape

The work with TGD inspired theory of consciousness has led to a visionabout the relationship of mathematics and physics. Physics is not in this view amodel of reality but objective reality itself: painting is the landscape. One can

equate mathematics and physics in a well defined sense and the often implic-itly assumed Cartesian theory-world division disappears. Physical realities aremathematical ideas represented by configuration space spinor fields (quantumhistories) and quantum jumps between quantum histories give rise to conscious-ness and to the subjective existence of a mathematician.

In this view evolution becomes also evolution of mathematical structures,

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which become more and more self-conscious quantum jump by quantum jump.The notion of p-adic evolution is indeed a basic prediction of quantum TGDbut it seems that even this vision might be generalized by allowing rational-adictopologies for which topology is defined by a norm associated with an algebrarather than number field.

Painting is the landscape view is not in conflict with the map is not terri-tory vision since p-adic spacetime regions are the geometric correlates of mindproviding cognitive representations are the map about territory identifiable asreal regions serving as the geometric correlates of matter.

4. Infinite primes and infinite hierarchy of second quantizations

There are several arguments leading to the notion of infinite primes (Ch.Quaternions, Octonions, and Infinite Primes of [20]). Simple arguments showthat the p-adic prime characterizing the 3-surface representing the entire uni-

verse increases in a statistical sense. This leads to a peculiar paradox: if thenumber of quantum jumps already occurred is infinite, this prime is most nat-urally infinite. On the other hand, if one assumes that only finite number ofquantum jumps have occurred, one encounters the problem of understandingwhy the initial quantum history was what it was. Furthermore, since the sizeof the 3-surface representing the entire Universe is infinite, p-adic length scalehypothesis suggest also that the p-adic prime associated with the entire universeis infinite.

These arguments motivate the attempt to construct a theory of infiniteprimes and to extend quantum TGD so that also infinite primes are possible.Rather surprisingly, one can construct infinite primes by repeating a procedureanalogous to a quantization of a super symmetric quantum field theory. The

simplest infinite primes at the lowest level of the hierarchy are of the form

P(m,n,s) = mXs ns , X =ipi , s =

piS

pi ,

m =piS

pkii , n =piS

pkii .(1)

Each finite prime pi represents one mode of an arithmetic quantum field theory.X is product of all finite primes and analogous to Dirac sea. s representsthe presence of holes in modes belonging to S and m and n represent manyboson states with ki bosons in mode pi. These primes will be referred to asgenerating infinite primes. On can formally assign opposite fermion numbers1 to the primes P+ and P: these primes are analogous to many particlestates constructed from Fock vacua obtained by filling all positive resp. negative

energy states.One can construct more general infinite primes as sums of the productsn+

i=1 P+,in

j=1 P,i of primes of type P(..) with rational coefficients. Internalconsistency requires that the net fermion number n = n+ n is same for allproducts appearing in the sum. There are also additional constraints from theprimeness requirement. These states are analogous to bound states of quantum

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field theories. The construction of infinite primes can be repeated again andagain and corresponds to a repeated second quantization of an arithmetic quan-tum field theory such that the many particle states of the previous level becomesingle particle states at the next level.

This and other observations suggest that the Universe of quantum TGDmight basically provide a physical representation of number theory allowingalso infinite primes. The proposal suggests also a possible generalization of realnumbers to a number system akin to hyper-reals introduced by Robinson inhis non-standard calculus [Robinson] providing rigorous mathematical basis forcalculus. In fact, some rather natural requirements lead to a unique general-ization for the concepts of integer, rational and real. Somewhat surprisingly,infinite integers and reals can be regarded as infinite-dimensional vector spaceswith integer and real valued coefficients respectively and one cannot exclude the

possibility that the tangent space for the configuration space of 3-surfaces couldbe regarded as the space of generalized octonions!

The construction generalizes so that it applies even in case of quaternionicand octonionic primes. This hierarchy of second quantizations means a profoundgeneralization of physics to what might be regarded a physical counterpart fora hierarchy of abstractions about abstractions about.... The ordinary secondquantized quantum physics corresponds only to the lowest level infinite primes.Also the TGD inspired model for 1/f noise (Ch. Quantum Control and Co-ordination in Biosystems of [22]) based on thermal arithmetic quantum fieldtheory encouraged to consider the idea about quaternionic arithmetic quantumfield theory as an essential element of quantum TGD.

5. Infinite primes as a bridge between quantum and classical

The final stimulus came from the observation stimulated by algebraic numbertheory [Esmonde and Murty]. Infinite primes can be mapped to polynomialprimes and this observation allows to identify completely generally the spectrumof infinite primes whereas hitherto it was possible to construct explicitely onlywhat might be called generating infinite primes.

The mapping of infinite primes to polynomial primes is constructed as fol-lows. The map of general infinite prime to a polynomial P(p, q) of quaternionicvariables p and q is induced by the replacements P+(m,n,s) = mX/s + ns p + m/ns. P(m,n,s) = mX/s ns q m/ns in the representation of theinfinite prime as a superposition of products of generating infinite primes withrational coefficients. If P(p, q) does not reduce to a product of rational polyno-mials, P(p, q) represents polynomial prime and also an infinite prime. Thereforethe construction of infinite primes reduces to the construction of polynomial

primes of special type. The construction generalizes in a straightforward man-ner to the higher levels of the hierarchy. For instance, at the second level oneobtains polyonials P(p1, q1|p2, q2) of p1, q1 with coefficients which are rationalfunctions of p2, q2.

This in turn led to the observation that one can represent infinite primes(integers) geometrically as surfaces related to the polynomials associated with

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infinite primes (integers). For instance, lowest infinite primes correspond to thesurfaces defined by the condition qP(p, q) = 0 implying that p is quaternionanalytic function of q (Taylor series with real coefficients). Thus infinite primeswould serve as a bridge between Fock-space descriptions and geometric descrip-tions of physics: quantum and classical. Geometric objects could be seen asconcrete representations of infinite numbers providing an amplification of infin-itesimals to macroscopic deformations of spacetime surface. We perhaps see theinfinitesimals as concrete geometric shapes!

6. Physics as a generalized number theory and Universe as algebraic holo-

gram

The third stimulus encouraging to think that TGD might be reduced to al-gebraic number theory and algebraic geometry in some generalized sense, camefrom the work with Riemann hypothesis (Ch. TGD and Number Theory: Rie-

mann Hypothesis of [20]). One can assign to Riemann Zeta a superconformalquantum field theory and identify Zeta as a Hermitian form in the state spacepossibly defining a Hilbert space metric. The proposed form of the Riemannhypothesis implies that the zeros of code for infinite primes which in turn haveinterpretation as Fock states of a supersymmetric quantum field theory if theproposed vision is correct.

A further stimulus came from the realization that algebraic extensions ofrationals, which make possible a generalization of the notion of prime, couldprovide an enormous representative and information storage power in arith-metic quantum field theory. Algebraic symmetries defined as transformationspreserving the algebraic norm represent new kind of symmetries commutingwith the ordinary symmetries (Ch. TGD as a Generalized Number Theory

of [20]). Fractal scalings and discrete symmetries are in question so that thenotion of fractality emerges to the fundamental physics in this manner.The basic observation, completely consistent with fractality, is that these

symmetries make possible what might be called algebraic hologram. The al-gebraic quantum numbers associated with elementary particle depend on theenvironment of the particle. The only possible conclusion seems to be thatthese fractal quantum numbers provide some kind of cognitive representationabout external world. This kind of an algebraic hologram would be in completeaccordance with fractality and would provide first principle realization for frac-tality observed everywhere in Nature but not properly understood in standardphysics framework. A further basic idea which emerged was the principle of al-gebraic democracy: all possible algebraic extensions of rational quaternions andoctonions are possible and emerge dynamically as properties of physical systems

in algebraic physics.In this framework Golden Mean, which is a fundamental constant in the

approach of M. El Naschie [15], could be understood as being related with thealgebraic extensions of rationals emerging dynamically for the p-adic solutionsp = p(q) to the equations qP(p, q) = 0 defining spacetime surfaces as geometricrepresentations of quaternionic polynomials representing infinite primes. For

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instance, the cyclic extension of dimension four of rationals generated by thefifth root of unity and having the form Z = t + xU + yU2 + zU3, U5 = 1,characterizes the most symmetric algebraic extension of rationals. The phaseangle ofU is 2/5 so that that the units Uk are expressible in terms of GoldenMean.

7. Maximal algebraic, quaternion- and octonion-dimensional, and topological

democracy

One can assign with quaternionic infinite primes at the level n of the hier-archy n-quaternion dimensional surfaces in n-octonion-dimensional space Hn =M4+ CP2n12 , which are selected by the property of being associative. Thismeans dimensional hierarchy of spacetime surfaces and a maximal dimensionaland algebraic democracy: quite a considerable generalization of quantum TGDfrom its original formulation. Since quantum field theories are based on the

notion of a point like particle, the hierarchy of the arithmetic quantum field the-ories associated with infinite primes cannot code entire quantum TGD but onlythe ground states of the superconformal representations. This might howeverbe the crucial element needed to understand the construction of the S-matrixat the general level.

There is also topological democracy and an evolution of algebraic topologicalstructures. At the lowest, primordial level there are just algebraic surfacesallowing no completion to smooth ...-adic or real surfaces, and defined only inthe algebraic extensions of rationals by algebraic field equations. At higherlevels rational-adic, p-adic, and even infinite-p p-adic completions appear andprovide natural completions of function spaces. Note that maximal topologicaland dimensional democracy conforms with the basic philosophy of the Cantorian

space time approach [15].9. Real and p-adic regions of the spacetime as geometric correlates of matterand mind

The solutions of the equations determining spacetime surfaces are restrictedby the requirement that the components of quaternions are real. When this isnot the case, one might apply instead of a real completion some rational-adic orp-adic completion: this is how rational-adic or p-adic physics could emerge fromthe basic equations of the theory. Of course, real and p-adic solutions mightexist simultaneously. One can interpret the resulting rational-adic or p-adic re-gions as geometrical correlates for mind stuff. p-Adic non-determinism impliesextreme flexibility so that the identification of the p-adic spacetime regions asseats of cognitive representations is very natural. Unlike real completion, p-adiccompletions preserve the information about the algebraic extension of ratio-

nals and algebraic quantum numbers must be associated with the mindlikespacetime regions. p-Adics and reals are in the same relationship as map andterritory.

The implications are far-reaching and consistent with TGD inspired theoryof consciousness: p-adic regions are present even at the elementary particlelevel and could provide some kind of model of self and external world. In

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fact, p-adic physics must model the p-adic cognitive regions representing realelementary particle regions rather than elementary particles themselves! p-Adicreal phase transitions can be induced in quantum jumps and could beinterpreted either as transformations of a thought into an action or of a sensoryinput into a cognitive image. TGD inspired theory of consciousness suggeststhat this occurs already at the elementary particle level (the so called cognitiveneutrinos are in a central role in the TGD based model of cognition, see Ch.Biosystems as Superconductors of [22]).

10. Various equivalent characterizations of spacetimes as surfaces

The idea about spacetimes as associative, quaternion-analytic surfaces ofan octonionic imbedding space and the notion of infinite prime serving as abridge between classical and quantum are the two basic tenets of the algebraicapproach. This vision leads to a hypothesis about an equivalence of several

quite different views about spacetime: a) spacetime as an associative surface ofan octonionic imbedding space; b) spacetime as a Lagrange manifold for octo-symplectic structure; c) spacetime as a quaternion-analytic surface; d) space-time as a geometric counterpart of an infinite prime representing also Fock stateidentifiable as a particular ground state of a quaternionic superconformal rep-resentation; and finally, e) spacetime surface as an absolute minimum of theKahler action. The basic challenge is to prove that the last characterization isequivalent with the others in the real context.

11. Transition from string models to TGD as replacement of reals-complex

numbers with quaternions-octonions

This vision allows to characterize very precisely the general solutions of thefield equations in both bosonic and fermionic sectors in terms of supersymmetry.

One can fairly say, that quantum TGD results from the string model withthe pair reals-complex numbers replaced with the pair quaternions-octonions.It is also possible to understand how p-adicity and its generalizations emergefrom the theory. Also a much more detailed first principle definition of U-matrix emerges if the ground states of superconformal representations for a givenspacetime surface are coded by the infinite prime associated with the spacetimesurface. What is especially fascinating is the possibility of a purely number-theoretical description of interactions: interactions are automatically generatedwhen many-partice state is constructed by multiplying infinite primes to giverise to an infinite integer since the spacetime surface associated with an infiniteinteger is not the union of the spacetime surfaces associated with the primefactors.

12. Some implications of the number-theoretical vision

Number-theoretical vision has led to quite impressive progress and several in-consistencies and weak points in the earlier construction have been detected andeliminated. Mention only the rather precise proposal for the characterization ofthe absolute minima of Kahler action; a modification of the Dirac action to al-low maximal supersymmetry; the exact solution of the modified Dirac equationbased on quaternion-conformal invariance; the identification of the quaternion-

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conformal invariance associated with P SU(3)U(2)ew (P denotes Poincaregroup) super Kac-Moody algebra as a symmetry of the fundamental interac-tions; the realization that the local quaternion-conformal symmetries and thecosmological supercanonical symmetries associated with the lightcone boundaryare practically independent of each other; the identification of the TGD counter-part of the Higgs field and the realization that ordinary Higgs mechanism andp-adic thermodynamics based particle massivation combine to a theory consis-tent with standard model in the bosonic sector; the formulation of the preciseFeynmann rules for elementary particle physics in the approximation that theso called CP2 extremals do not couple to other particle like 3-surfaces; the re-alization that the spin-glass nature of the TGD Universe is realized already atthe elementary particle level; that even elementary particles are adaptive sys-tems as far as the infrared behaviour of the propagators is considered; that also

color confinement might be coded into the geometry of the CP2 type extremalsrepresenting quarks and gluons and determines the behaviour of propagators:quarks and gluons cannot simply propagate too far from their fellow partons.

Of course, there are myriads of open technical questions, even the proof thatabsolute minimization of Kahler action is equivalent with associativity propertyis lacking, and the TGD as a generalized number theory is a work in progress.

5 About the construction of S-matrix

The enormous symmetries of quantum are bound to lead to a highly unique S-matrix but the practical construction of S-matrix is a formidable challenge andnecessitates deep grasp about the physics involved so that one can make the

needed approximations. The evolution of the ideas related to S-matrix involvesseveral side-tracks and strange twists characteristic for a mathematical problemsolving when a direct contact with the experimental reality is lacking. The workwith S-matrix has taught that principles are more important than formulas andthat the only manner to proceed is from top to bottom by gradually solvingthe philosophical problems, identifying all the relevant symmetries and under-standing the horribly nonlinear dynamics defined by the absolute minimizationof Kahler action.

The lack of explicit formulas for S-matrix elements have been the basic weak-ness of quantum TGD approach as compared to the concrete perturbative for-mulas provided by superstring approach. Fortunately, the new number theoreticvision leads to concrete Feynmann rules for S-matrix in the approximation that

elementary particles can be regarded as CP2 type extremals (Ch. Constructionof S-matrix of [20]). Of course, this is only small piece of quantum TGD butcertainly the most important one as far as the empirical testing of the theory isconsidered.

1. The fundamental identification of U- and S-matrices

Single quantum jump corresponds to the sequence

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i Ui f .U does not certainly correspond to a genuine time-development and is morefundamental ob ject than S-matrix understood as a unitary time translationoperator. A good guess is that U has interpretation as Glebsch-Gordan coef-ficients between free an interacting representations of Super Virasoro algebraassociated with the surfaces iX4(Y3i ) and X4(iY3i ) representing free and in-teracting states geometrically. For a given unentangled subsystem (subsystemin self-organizing self-state) the eigenstates of the density matrix of the sub-system becoming unentangled in quantum jump determines what are the finalstates of the quantum jump. Negentropy Maximization Principle (Ch. Negen-tropy Maximization Principle of [22]) states that the subsystem of unentangled

subsystems whose measurement gives rise to maximal entanglement negentropygain, is quantum measured. If subsystem is entangled then in a reasonableapproximation nothing happens to it during quantum jump sequence and thesub-quantum history remains unchanged.

2. Quaternion conformal invariance and S-matrix

Super conformal invariance should dictate also the dynamics of the theory.As far as elementary particle physics is considered, one can restrict the atten-tion to a tiny tensor factor of U-matrix describing local physics and only thequaternion-conformal symmetries are interesting. The construction of S-matrixshould reduce more or less to that encountered in super string models sincecomplex-one-dimensional objects are replaced with one-quaternion-dimensionalobjects. It is even possible that the two-dimensional commutative submanifoldsof spacetime surface might effectively represent spacetime surfaces so that at

this limit TGD would reduce to superstring model type theory.CP2 type extremals are basic classical vacuum extremals and excellent can-

didates for the absolute minima of Kahler action and for the geometric coun-terparts of elementary particles. Of utmost importance is the classical non-determinism of the CP2 type extremals. The M4+ projection of the CP2 typeextremal is lightlike random curve expressible as a superposition of a geodesiccm motion and zitterbewegung: the condition for the ligtlikeness boils downto the classical Virasoro conditions. The quantized 4-momentum associatedwith the modified Dirac operator is in the direction of the cm motion. Masssquared operator can have all possible values for internal lines but for on-massshell states mass squared operator is integer valued by quaternion-conformalinvariance.

These observations suggest that stringy Feynmann diagrams constructedfrom CP2 type extremals could provide excellent approximation of the S-matrixat the high energy limit of the theory by reproducing the Feynmann diagram-matics of quantum field theories purely topologically (to achieve this formulationrequired 23 years of hard work!). A very rough description of Feynmann rulesgoes as follows.

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a) Assign to the internal lines of the generalized Feynmann graph the expo-nent of a translation operator for the modified Dirac action along the directionof the line. Since the momentum operator in question is quadratic in the os-cillator operators, the theory is free in the standard sense of the word, and itis only the absolute minimization of Kahler action which induces interactionsand makes the theory nontrivial. Translation operator gives only a phase fac-tor for momentum eigenstates. Besides this lines contain 1/(L0 + i) stringypropagator and the exponent of the Kahler action for the CP2 type extremalrepresenting the line. The latter factor is characteristic for TGD and impliesinfrared supression of the diagrams.

b) The 3-vertices where the lines join give rise to vertex operators whichcan be regarded as Glebch-Gordan coefficients for super-Kac-Moody represen-tations. In QFT approximation diagrams are obtained by gluing together the

incoming lines along their ends assumed to correspond to identical 3-surfacesin CP2. Vertex factors are determined purely combinatorially by the innerproducts of the states obtained by applying the Super-Kac-Moody operatorscreating quantum states appearing in the lines of the vertex: generalized Zweigrule summarizes the situation. In this approximation four-surfaces are homolog-ically equivalent to QFT Feynmann diagrams and singular as 4-surfaces so thatUV divergences are unavoidable. The requirement that Feynmann diagrams arenonsingular as 4-surfaces forces to deform CP2 type extremals near the verticesand this smoothening introduces effective UV cutoff at CP2 length scale sincevertices cannot be too close to each other.

c) In QFT approximation CP2 type extremals representing virtual particlesare far from full ones and the exponents of the Kahler action associated with

the internal lines give only moderately small supression factors determining thevalues of the effective coupling constants. The weakness of the gravitational cou-pling can be understood if virtual gravitons are emitted and absorbed as almostpointlike 3-surfaces and thus correspond to almost full virtual CP2 type ex-tremals not possible in QFT approximation. This picture is in accordance withthe string model view that nonpointlike vertices are essential for gravitation.The extremely small value for the exponential of the gravitonic Kahler actionexplains the small value of the gravitational coupling and is consistent with theproposed formula for the gravitational coupling in terms of the Kahler actionfor CP2 type extremal and p-adic length scale (Ch. TGD as a GeneralizedNumber Theory of [20]).

d) There are some important deviations from the standard physics basi-cally due to the non-determinism of CP2 type extremals and implying that

self-organization, is present already in elementary particle length scales. In-frared cutoff is coded automatically as an intrinsic characteristic of the CP2type extremal representing virtual particle: particle stays in a box, not becauseof the boundary conditions but because the command of staying in box is codedinto the structure of the particle. One could understand color confinement asbeing due to this kind of a learned infrared cutoff. It seems that even ele-

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mentary particles resemble biomatter in the sense that they can adapt to thesurrounding world and p-adic length scale hierarchy represents a hierarchy ofthis kind of adaptations. Spin glass analogy at the quantum level means thatthe predictions for the scattering rates must be averaged over an ensemble ofquantum field theories with propagators and coupling constants having a milddependence on an arbitrary function of CP2 coordinates. In the lowest orderapproximation this dependence is trivial.

References

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[3] C. Castro (2000), The Status and Programs of the New Relativity Theory,arXiv:physics/0011040.

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[12] J. Mickelson (1989), Current Algebras and Groups. Plenum Press, NewYork.

[13] J. Milnor (1965), Topology form Differential Point of View. The UniversityPress of Virginia.

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[20] M. Pitkanen (1990), Topological Geometrodynamics Internal Report HU-TFT-IR-90-4 (Helsinki University). This online-book can be found athttp://www.physics.helsinki.fi/matpitka/tgd.html.

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TFT-IR-95-5 (Helsinki University). This online book can be found as pdffiles at http://www.physics.helsinki.fi/ matpitka/padtgd.html.

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[Smith] T. Smith (1997), D4-D5-E6 Physics. Homepage of Tony Smith.http : //galaxy.cau.edu/tsmith/d4d5e6hist.html. The homepage containsa lot of information and ideas about the possible relationship of octonionsand quaternions to physics.

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