The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the first one and devoted to the notion of p-adic manifold.
Total disconnectedness of p-adic numbers as the basic problem
The total dis-connectedness of p-adic topology and lacking correspondence with real manifolds could be seen as genuine problem in the purely formal construction of p-adic manifolds. Physical intuition suggests that path connected should be realized in some natural manner and that one should have a close connection with real topology which after all is the "lab topology".
In TGD framework one of the basic physical problems has been the connection between p-adic numbers and reals. Algebraic and topological approaches have been competing also here.
Canonical identification combined with the identification of common rationals in finite pinary resolution suggests also a manner of replacing p-adic topology with a path connected one. This topology would be essentially real topology induced to p-adic context by canonical identification used to build real chart leafs.
- Algebraic approach suggests the identification of reals and various p-adic numbers along common rationals but this correspondence is non-continuous. Above some resolution defined by power of p it must be replaced with a correspondence is continuous unless one uses pinary cutoff. Below this cutoff the pseudo-constants of p-adic differential equations would naturally relate to the identification of p-adics and reals along common rationals (plus common algebraics in the case of algebraic extensions).
- Topological approach relies on canonical identification and its variants mapping p-adic numbers to reals in a continuous manner. This correspondence is however problematic in the sense that does not commute with the basic symmetries as correspondence along common rationals would do for subgroups of the symmetries represented in terms of rational matrices. A further problematic aspect of canonical identification is that it does not commute with the field equations.
- The notion of finite measurement resolution allows to find a compromise between the symmetries and continuity (that is, algebra and topology). Canonical identification can be modified so that it maps rationals to themselves only up to some pinary digits but is still continuous in p-adic sense. Canonical identification could map only a skeleton formed by discrete point set - analogous to Bruhat-Tits building - from real to p-adic context and the preferred extremals on both sides would contain this skeleton.
What is very remarkable is that canonical identification can be seen as a continuous generalization of the p-adic norm defined as Np(x) == Ik,l(x) having the highly desired Archimedean property. Ik,l is the most natural variant of canonical identification.
- Canonical identification maps p-adic numbers ∑ xnpn to reals and is defined by the formula I(x) = ∑ xnp-n. I is a continuous map from p-adic numbers to reals. Its inverse is also continuous but two-valued for a finite number of pinary digits since the pinary expansion of real number is not unique (1=.999999.. is example of this in 10-adic case). For a real number with a finite number of pinary digits one can always choose the p-adic representative with a finite number of pinary digits.
- Canonical identification is used to map the predictions of p-adic mass calculations to map the p-adic value of the mass squared to its real counterpart. It makes also sense to map p-adic probabilities to their real counterparts by canonical identification. In TGD inspired theory of consciousness canonical identification is a good candidate for defining cognitive representations as representations mapping real preferred extremals to p-adic preferred extremals as also for the realization of intentional action as a quantum jump replacing p-adic preferred extremal representing intention with a real preferred extremal representing action. Could these cognitive representations and their inverses actually define real coordinate charts for the p-adic "mind stuff" and vice versa?
- Canonical identification has several variants. For instance, one can map p-adic rational number m/n regarded as a p-adic number to a real number I(m)/I(n). In this case canonical identification respects rationality but is ill-defined for p-adic irrationals. This is not a catastrophe if one has finite measurement resolution meaning that only rationals for which m<pl,n<pl are mapped to the reals (real rationals actually).
One can also express p-adic number as expansion of powers fo pk and define canonical identification Ik as ∑ xnpkn → ∑ xnp-kn. Also the variant Ik,l(m/n)=Ik,l(m)/Ik,l(n) with l defining pinary cutoff for m and l makes sense. One can say that Ik,l(m/n) identifies p-adic and real numbers along common rationals for p-adic numbers with a pinary cutoff defined by k and maps them to rationals for pinary cutoff defined by l. Discrete subset of rational points on p-adic side is mapped to a discrete subset of rational points on real side by this hybrid of canonical identification and identification along common rationals. This form of canonical identification is the one needed in TGD framework.
- Canonical identification does not commute with rational symmetries unless one uses the map Ik,l(m/n)=Ik,l(m)/Ik,l(n) and also now only in finite resolutions defined by k. For the large p-adic primes associated with elementary particles this is not a practical problem (electron corresponds to M127=2127-1!) The generalization to algebraic extensions makes also sense. Canonical identification breaks general coordinate invariance unless one uses group theoretically preferred coordinates for M4 and CP2 and subset of these for the space-time region considered.
To sum up, chart maps are constructed in two steps and works in both directions. For p-adic-to-real case a subset of rational points of the p-adic preferred extremal would be mapped using Ik,l to rational points of the real preferred extremal. Field equations for the preferred extremal would be then used to complete the resulting discrete skeleton to a full map leaf. Of course also algebraic extensions can and must be considered. This kind of completion performed in iterative manner has been also proposed assuming that space-time surfaces are quaternionic surfaces (tangent spaces are in well-defined sense quaterionic sub-space of octonionic space containing complex octonions as a preferred sub-space this).
- Canonical identification for the various coordinates defines a chart map mapping regions of p-adic manifold to Rn+. That each coordinate is mapped to a norm Np(x) means that the real coordinates are always non-negative. If real spaces Rn+ would provide only chart maps, it is not necessary to require approximate commutativity with symmetries. Also Berkovich considers norms but for a space of formal power series assigned with the p-adic disk: in this case however the norms have extremely low information content.
- Ik,l(x) indeed defines the analog of Archimedean norm in the sense that one has Np(x+y) ≤ Np(x)+Np(y). This follows immediately from the fact that the sum of pinary digits can vanish modulo p. The triangle inequality holds true also for the rational variant of I. Np(x) is however not multiplicative: only a milder condition Np(pnx)=N(pn)N(x)=p-n N(x) holds true.
- Archimedean property gives excellent hopes that p-adic space provided with chart maps for the coordinates defined by canonical identification inherits real topology and its path connectedness. A hierarchy of topologies would be obtained as induced real topologies and characterized by various norms defined by Ik,l labelled by a finite measurement resolution. This would give a very close connection with physics.
- The mapping of p-adic manifolds to real manifolds would make the construction of p-adic topologies very concrete. For instance, one can map real preferred subset of rationalp oints of a real extremal to a p-adic one by the inverse of canonical identification by mapping the real points with finite number of pinary digits to p-adic points with a finite number of pinary digits. This does not of course guarantee that the p-adic preferred extremal is unique. One could however hope that p-adic preferred extrremals can be said to possess the invariants of corresponding real topologies in finite measurement resolution.
- The maps between different real charts would be induced by the p-adically analytic maps between the inverse images of these charts. At the real side the maps would be consistent with the p-adic maps only in the discretization below pinary cutoff.
- As already mentioned, one must restrict the p-adic points mapped to reals to rationals since Ik,l(m/n) is not well-defined for p-adic irrationals (having non-periodic pinary expansion: note however that one can consider also p-adic integers). For the restriction to finite rationals the chart image on real side would consist of rational points. The cutoff would mean that these rationals are not dense in the set of reals. Preferred extremal property could however allow to identify the chart leaf as a piece of preferred extremal containing the rational points in the measurement resolution use. This would realize the dream of mapping p-adic p-adic preferred extremals to real ones playing a key role in number theoretical universality.
What about p-adic coordinate charts for a real preferred extremal?
What is remarkable that one can also build p-adic coordinate charts about real preferred extremal using the inverse of the canonical identification assuming that finite rationals are mapped to finite rationals. There are actually good reasons to expect that coordinate charts make sense in both directions.
Algebraic continuation from real to p-adic context is one such reason. At the real side one can calculate the values of various integrals like K ähler action. This would favor p-adic regions as map leafs. One can require that K ähler action for Minkowskian and Euclidian regions (or their appropriate exponents) make sense p-adically and define the values of these functions for the p-adic preferred extremals by algebraic continuation. This could be very powerful criterion allowing to assign only very few p-adic primes to a given real space-time surface. This would also allow to define p-adic boundaries as images of real boundaries in finite measurement resolution. p-Adic path connectedness would be induced from real path-connectedness.
p-Adic rationals include also the ratios of integers, which are infinite as real integers so that the pinary expansion of the rational is not periodic asymptotically. In principle one could imagine of mapping also these to real numbers but the resulting skeleton might be too dense and might not allow to satisfy the preferred extremal property. Furthermore, the representation of a p-adic number as a ratio of this kind of integers is not unique and can be always tranformed to an infinite p-adic integer multiplied by a power of p . In the same manner real points which can be regarded as images of ratios of p-adic integers infinite as real integers could be mapped to p-adic ones but same problem is encountered also now.
In the intersection of real and p-adic worlds the correspondence is certainly unique and means that one interprets the equations defining the p-adic space-time surface as real equations. The number of rational points (with cutoff) for the p-adic preferred extremal becomes a measure for how unique the chart map in the general case can be. For instance, for 2-D surfaces the surfaces xn+yn=zn allow no nontrivial rational solutions for n>2 for finite real integers. This criterion does not distinguish between different p-adic primes and algebraic continuation is needed to make this distinction.
Chart maps for p-adic manifolds
The real map leafs must be mutually consistent so that there must be maps relating coordinates used in the overlapping regions of coordinate charts on both real and p-adic side. On p-adic side chart maps between real map leafs are naturally induced by identifying the canonical image points of identified p-adic points on the real side. For discrete chart maps Ik,l with finite pinary cutoffs one one must complete the real chart map to - say diffeomorphism. That this completion is not unique reflects the finite measurement resolution.
In TGD framework the situation is dramatically simpler. For sub-manifolds the manifold structure is induced from that of imbedding space and it is enough to construct the manifold structure M4 × CP2 in a given measurement resolution (k,l). Due to the isometries of the factors of the imbedding space, the chart maps in both real and p-adic case are known in preferred imbedding space coordinates. As already discussed, this allows to achieve an almost complete general coordinate invariance by using subset of imbedding space coordinates for the space-time surface. The breaking of GCI has interpretation in terms of presence of cognition and selection of quantization axes.
For instance, in the case of Riemann sphere S2 the holomorphism relating the complex coordinates in which rotations act as M öbius tranformations and rotations around -call it z-axis- act as phase multiplications - the coordinates z and w at Norther and Southern hemispheres are identified as w=1/z restricted to rational points at both side. For CP2 one has three poles instead of two but the situation is otherwise essentially the same.
For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.