Mathematical Aspects of Consciousness Theory

ions about.... The ordinary second quantized quantum physics corresponds only to thelowest level infinite primes. 2. The ordinary primes appearing as building blocks of infinite primes at the first level of thehierarchy could be identified as coding for p-adic primes assignable to fermionic and bosonicpartons identified as 2-surfaces of a given space-time sheet. The hierarchy of infinite primeswould correspond to hierarchy of space-time sheets defined by the topological condensate. Thisleads also to a precise identification of p-adic and real variants of bosonic partonic 2-surfaces ascorrelates of intention and action and pairs of p-adic and real fermionic partons as correlates forcognitive representations. 3. The idea that infinite primes characterize quantum states of the entire Universe, perhaps groundstates of super-conformal representations, if not all states, could be taken further. Could 8-Dhyper-octonions correspond to 8-momenta in the description of TGD in terms of 8-D hyper-octonion space M? Could 4-D hyper-quaternions have an interpretation as four-momenta?The problems caused by non-associativity and non-commutativity however suggests that it isperhaps wiser to restrict the consideration to infinite primes associated with rationals and theiralgebraic extensions. Here however emerges the idea about the number theoretic analog of color confinement. Rational(infinite) primes allow not only a decomposition to (infinite) primes of algebraic extensions of rationalsbut also to algebraic extensions of quaternionic and octonionic (infinite) primes. The physical analogis the decomposition of a particle to its more elementary constituents. This fits nicely with theidea about number theoretic resolution represented as a hierarchy of Galois groups defined by theextensions of rationals and realized at the level of physics in terms of Jones inclusions [C6] defined bythese groups having a natural action on space-time surfaces, induced spinor fields, and on configurationspace spinor fields representing physical states [C1]. Infinite primes and physics as number theory The hierarchy of algebraic extensions of rationals implying corresponding extensions of p-adic numberssuggests that Galois groups, which are the basic symmetry groups of number theory, should haveconcrete physical representations using induced spinor fields and configuration space spinor fieldsand also infinite primes and real units formed as infinite rationals. These groups permute zeros ofpolynomials and thus have a concrete physical interpretation both at the level of partonic 2-surfacesdictated by algebraic equations and at the level of braid hierarchy. The vision about the role ofhyperfinite factors of II1 and of Jones inclusions as descriptions of quantum measurements with finitemeasurement resolution leads to concrete ideas about how these groups are realized. Space-time correlates of infinite primes One can assign to infinite primes at the n level of hierarchy rational functions of n arguments witharguments ordered in a hierarchical manner. It would be nice to assign some concrete interpretationto the polynomials of n arguments in the extension of field of rationals. 1. Do infinite primes code for space-time surfaces? Infinite primes code naturally for Fock states in a hierarchy of super-symmetric arithmetic quantumfield theories. Quantum classical correspondence leads to ask whether infinite primes could also codefor the space-time surfaces serving as symbolic representations of quantum states. This would ageneralization of algebraic geometry would emerge and could reduce the dynamics of Kähler actionto algebraic geometry and organize 4-surfaces to a physical hierarchy according to their algebraiccomplexity. Note that this conjecture which should be consistent with several conjectures aboutthe dynamics of space-time surfaces (space-time surfaces as preferred extrema of Kähler action, asKähler calibrations, as quaternionic or co-quaternionic (as associative or co-associative) 4-surfaces ofhyper-octonion space M.The most promising variant of this idea is based on the conjecture that hyper-octonion real-analytic maps define foliations of HO = M by hyper-quaternionic space-time surfaces providing inturn preferred extremals of Kähler action. This would mean that lowest level infinite primes would 392Chapter 7. Infinite Primes and Consciousness define hyper-analytic maps HO → HO as polynomials. The intuitive expectation is that higher levelsshould give rise to more complex HO analytic maps. The basic objections against the idea is the failure of associativity. The only manner to guaranteeassociativity is to assume that the arguments ohn in the polynomial are not independent but thatone has hi = fi(hi−1, i = 2, ..., n where fi is hyper-octonion real-analytic function. This assumptionmeans that one indeed obtains foliation of M by hyper-quaternionic surfaces also now and that thesefoliations become increasingly complex as n increases. One could of course consider also the possi-bility that the hierarchy of infinite primes is directly mapped to functions of single hyper-octonionicargument hn = ... = h1 = h. 2. What about the interpretation of zeros and poles of rational functions associated with infiniteprimes If one accepts this interpretation of infinite primes, one must reconsider the interpretation of thezeros and also poles of the functions f(o) defined by the infinite primes. The set of zeros and polesconsists of discrete points and this suggests an interpretation in terms of preferred points of M, whichappear naturally in the quantization of quantum TGD [C1] if one accepts the ideas about hyper-finitefactors of type II1 [C6] and the generalization of the notion of imbedding space and quantization ofPlanck constant [A9].The M projection of the preferred point would code for the position tip of future or past light-cone δM± whereas E 4 projection would choose preferred origin for coordinates transforming linearlyunder SO(4). At the level of CP2 the preferred point would correspond to the origin of coordinatestransforming linearly under U(2) ⊂ SU(3). These preferred points would have interpretation as argu-ments of n-point function in the construction of S-matrix and theory would assign to each argumentof n-point function (not necessarily so) ”big bang” or ”big crunch”. Also configuration space CH (the world of classical worlds) would decompose to a union CHh ofthe classical world consisting of 3-surfaces inside δM± × CP2 with CP2 possessing also a preferredpoint. The necessity of this decomposition in M degrees of freedom became clear long time ago. 3. Why effective 1-dimensionality in algebraic sense? The identification of arguments (via hyper-octonion real-analytic map in the most general case)means that one consider essentially functions of single variable in the algebraic sense of the word.Rational functions of single variable defined on curve define the simplest function fields having manyresemblances with ordinary number fields, and it is known that the dimension D = 1 is completelyexceptional in algebraic sense [51]. 1. Langlands program [50] is based on the idea that the representations of Galois groups can beconstructed in terms of so called automorphic functions to which zeta functions relate via Mellintransform. The zeta functions associated with 1-dimensional algebraic curve on finite field Fq,q = p, code the numbers of solutions to the equations defining algebraic curve in extensions ofFq which form a hierarchy of finite fields Fqm with m = kn [48]: these conjectures have beenproven. Algebraic 1-dimensionality is also responsible for the deep results related to the numbertheoretic Langlands program as far as 1-dimensional function fields on finite fields are considered[48, 50]. In fact, Langlands program is formulated only for algebraic extensions of 1-dimensionalfunction fields. 2. The exceptional character of algebraically 1-dimensional surfaces is responsible the successes ofconformal field theory inspired approach to the realization of the geometric Langlands program[51]. It is also responsible for the successes of string models. 3. Effective 1-dimensionality in the sense that the induced spinor fields anti-commute only along1-D curve of partonic 2-surface is also crucial for the stringy aspects of quantum TGD [C1]. 4. Associativity is a key axiom of conformal field theories and would dictate both classical andquantum dynamics of TGD in the approach based on hyper-finite factors of type II1[C6]. Henceit is rather satisfactory outcome that the mere associativity for octonionic polynomials forcesalgebraic 1-dimensionality. 7.2. Infinite primes, integers, and rationals393 7.1.4 About literature The reader not familiar with the basic algebra of quaternions and octonions is encouraged to studysome background material: the home page of Tony Smith provides among other things an excellentintroduction to quaternions and octonions [20]. String model builders are beginning to grasp thepotential importance of octonions and quaternions and the articles about possible applications ofoctonions [21, 22, 23] provide an introduction to octonions using the language of physicist.Personally I found quite frustrating to realize that I had neglected totally learning of the basic ideasof algebraic geometry, despite its obvious potential importance for TGD and its applications in stringmodels. This kind of losses are the price one must pay for working outside the scientific community.It is not easy for a physicist to find readable texts about algebraic geometry and algebraic numbertheory from the bookshelves of mathematical libraries. The book ”Algebraic Geometry for Scientistsand Engineers” by Abhyankar [24], which is not so elementary as the name would suggest, introducesin enjoyable manner the basic concepts of algebraic geometry and binds the basic ideas with the morerecent developments in the field. ”Problems in Algebraic Number Theory” by Esmonde and Murty[19] in turn teaches algebraic number theory through exercises which concretize the abstract ideas.The book ”Invitation to Algebraic Geometry” by K. E. Smith. L. Kahanpää, P. Kekäläinen and W.Traves is perhaps the easiest and most enjoyable introduction to the topic for a novice. It also containsreferences to the latest physics inspired work in the field. 7.2 Infinite primes, integers, and rationals By the arguments of introduction p-adic evolution leads to a gradual increase of the p-adic prime pand at the limit p→∞ Omega Point is reached in the sense that the negentropy gain associated withquantum jump can become arbitrarily large. There several interesting questions to be answered. Doesthe topology RP at the limit of infinite P indeed approximate real topology? Is it possible to generalizethe concept of prime number and p-adic number field to include infinite primes? This is is possible issuggested by the fact that sheets of 3-surface are expected to have infinite size and thus to correspondto infinite p-adic length scale. Do p-adic numbers RP for sufficiently large P give rise to reals bycanonical identification? Do the number fields RP provide an alternative formulation/generalizationof the non-standard analysis based on the hyper-real numbers of Robinson [40]? Is it possible togeneralize the adelic formula [E4] so that one could generalize quantum TGD so that it allows effectivep-adic topology for infinite values of p-adic prime? It must be emphasized that the consideration ofinfinite primes need not be a purely academic exercise: for infinite values of p p-adic perturbation seriescontains only two terms and this limit, when properly formulated, could give excellent approximationof the finite p theory for large p.It turns out that there is not any unique infinite prime nor even smallest infinite prime and thatthere is an entire hierarchy of infinite primes. Somewhat surprisingly, RP is not mapped to entireset of reals nor even rationals in canonical identification: the image however forms a dense subsetof reals. Furthermore, by introducing the corresponding p-adic number fields RP , one necessarilyobtains something more than reals: one might hope that for sufficiently large infinite values of P thissomething might be regarded as a generalization of real numbers to a number field containing bothinfinite numbers and infinitesimals.The pleasant surprise is that one can find a general construction recipe for infinite primes andthat this recipe can be characterized as a repeated second quantization procedure in which the manyboson states of the previous level become single boson states of the next level of the hierarchy:this realizes Cantor’s definition ’Set as Many allowing to regard itself as One’ in terms of the basicconcepts of quantum physics. Infinite prime allows decomposition to primes at lower level of infinityand these primes can be identified as primes labeling various space-time sheets which are in turngeometric correlates of selves in TGD inspired theory of consciousness. Furthermore, each infiniteprime defines decomposition of a fictive many particle state to a purely bosonic part and to part forwhich fermion number is one in each mode. This decomposition corresponds to the decompositionof the space-time surface to cognitive and material space-time sheets. Thus the concept of infiniteprime suggests completely unexpected connection between quantum field theory, TGD based theory ofconsciousness and number theory by providing in its structure nothing but a symbolic representationof mathematician and external world! 394Chapter 7. Infinite Primes and Consciousness The definition of the infinite integers and rationals is a straightforward procedure. Infinite primesalso allow generalization of the notion of ordinary number by allowing infinite-dimensional space ofreal units which are however non-equivalent in p-adic sense. This means that space-time points areinfinitely structured. The fact that this structure completely invisible at the level of real physicssuggests that it represents the space-time correlate for mathematical cognition. 7.2.1 The first level of hierarchy In the following the concept of infinite prime is developed gradually by stepwise procedure rather thangiving directly the basic definitions. The hope is that the development of the concept in the samemanner as it actually occurred would make it easier to understand it.