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.
- 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.
- 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
- 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!
- 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.
- 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.
- 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.
- 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
- 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.
- What makes some p-adic primes preferred so that one can say that they characterizes elementary particles and presumably any system?
- 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 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!
- 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!
- 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!
- 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.