Still about non-planar twistor diagrams

A question about how non-planar Feynman diagrams could be represented in twistor Grassmannian approach inspired a re-reading of the recent article by recent article by Nima Arkani-Hamed et al.

This inspired the conjecture that non-planar twistor diagrams correspond to non-planar Feynman diagrams and a concrete proposal for realizing the earlier proposal that the contribution of non-planar diagrams could be calculated by transforming them to planar ones by using the procedure applied in knot theories to eliminate crossings by reducing the knot diagram with crossing to a combination of two diagrams for which the crossing is replaced with reconnection. The Wikipedia article about magnetic reconnection explains what reconnection means. More explicitly, the two reconnections for crossing line pair (AB,CD) correspond to the non-crossing line pairs (AD,BC) and (AC,BD).

I do not bother to type the 5 pages of text here. Instead I give a link to the article Still about non-planar twistor diagrams at my homepage. For background see the chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-matrix".

Comments on the recent experiments by the group of Michael Persinger

Michael Persinger's group reports three very interesting experimental findings related to EEG, magnetic fields, photon emissions from brain, and macroscopic quantum coherence. The findings provide also support for the proposal of Hu and Wu that nerve pulse activity could induce spin flips of spin networks assignable to cell membrane.

In this article (see also the new chapter of "TGD based view about living matter and remote mental interactions") I analyze the experiments from TGD point of view. It turns out that the experiments provide support for several TGD inspired ideas about living matter. Magnetic flux quanta as generators of macroscopic quantum entanglement, dark matter as a hierarchy of macroscopic quantum phases with large effective Planck constant, DNA-cell membrane system as a topological quantum computer with nucleotides and lipids connected by magnetic flux tubes with ends assignable to phosphate containing molecules, and the proposal that "dark" nuclei consisting of dark proton strings could provide a representation of the genetic code. The proposal of Hu and Wu translates to the assumption that lipids of the two layers of the cell membrane are accompanied by dark protons which arrange themselves to dark protonic strings defining a dark analog of DNA double strand (see this ).

Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD

The generation of matter-antimatter asymmetry is still poorly understood. There exists a multitude of models but no convincing one. In TGD framework the generation of matter-antimatter asymmetry can be explained in terms of cosmic strings carrying dark energy identified as Kähler magnetic energy (see this). Their decay to ordinary and dark matter would be the analog for the decay of the inflaton field to matter and the asymmetry would be generated in this process. The details of the process have not been considered hitherto.

The stimulus for constructing a general model for this process came from attempt to understand the notion of sphaleron claimed to allow a non-perturbative description for a separate non-conservation of baryon and lepton numbers in standard model. The separate non-conservation of B and L would make possible models of baryo-genesis and even lepto-genesis assuming that in the primordial situation only right-handed inert neutrinos are present. To my opinion these models however fail mathematically because they equate the non-conservation of axial fermion numbers - which is on a mathematically sound basis - with the non-conservation of fermion numbers. This kind of assumption is unjustified and to my opinion is misuse of the attribute "non-perturbative".

The basic vision about lepto-genesis followed by baryo-genesis is however very attractive. This even more so because right-handed neutrino is in a completely unique role in TGD Universe. The obvious question therefore is whether this vision could make sense also in TGD framework. It would be wonderful if cosmic strings - infinitely thin Kähler magnetic flux tubes carrying magnetic monopole field, which later develop finite sized and expanding M⁴ projection - carrying only right-handed neutrinos were the fundamental objects from which matter would have emerged in a manner analogous to the decay of vacuum expectations of instanton fields (see this). Even better, Kähler magnetic energy has interpretation as dark energy and magnetic tension gives rise to the negative "pressure" inducing accelerated expansion of the Universe.

The basic question is whether B and L are conserved separately or not. In TGD Universe one can consider two options depending on the answer to this question. For option I - the "official" version of TGD - quarks and leptons correspond to opposite 8-D chiralities of the induced spinor fields and B and L are conserved separately. For option II (see this) only leptonic spinor fields would be fundamental, and the idea is that quarks could be fractionally charged leptons. This option could lead to genuine baryo-genesis, and in the simplest model baryons would be generated from 3-leptons as 3-sheeted structures for which fractionization of color hyper-charge occurs. Leptonic imbedding space spinors moving in triality zero color partial waves would be replaced with triality +/- 1 partial waves assigned with quarks. Whether this replacement is on a mathematically sound basis, is far from obvious since induced spinor fields at space-time level would couple to induced spinor fields with leptonic couplings.

In any case, one can check whether leptogenesis, baryogenesis, and matter antimatter asymmetry are possible for either of both of these options. It turns out that for both option I and II one can construct simple model in terms for the generation of quarks from leptons via emission of lepto-quarks analogous to gauge bosons but differing from their counterparts in GUTs. Option II allows also genuine baryogenesis from leptons. The conclusion is that the "official" version of TGD predicting separate conservation of B and L allows an elegant vision about the generation of matter from cosmic strings containing only right-handed neutrinos in the initial states.

For details see the article Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD of the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time".

p-Adic symmetries

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 third one and devoted to p-adic symmetries.

A further objection relates to symmetries. It has become already clear that discrete subgroups of Lie-groups of symmetries cannot be realized p-adically without introducing algebraic extensions of p-adics making it possible to represent the p-adic counterparts of real group elements. Therefore symmetry breaking is unavoidable in p-adic context: one can speak only about realization of discrete sub-groups for the direct generalizations of real symmetry groups. The interpretation for the symmetry breaking is in terms of discretization serving as a correlate for finite measurement resolution reflecting itself also at the level of symmetries.

This observation has led to TGD inspired proposal for the realization of the p-adic counterparts symmetric spaces resembling the construction of P¹(K) in many respects but also differing from it.

1. For TGD option one considers a discrete subgroup G₀ of the isometry group G making sense both in real context and for extension of p-adic numbers. One combines G₀ with a p-adic counterpart of Lie group G_p obtained by exponentiating the Lie algebra by using p-adic parameters t_i in the exponentiation exp(t_iT_i).
   
   
2. One obtains actually an inclusion hierarchy of p-adic Lie groups. The levels of the hierarchy are labelled by the maximum p-adic norms |t_i|_p= p^-n_i, n_i ≥ 1 and in the special case n_i=n - strongly suggested by group invariance - one can write G_p,1 ⊃ G_p,2 ⊃ ...G_p,n .... G_p,i defines the p-adic counterpart of the continuous group which gets the smaller the larger the value of n is. The discrete group cannot be obtained as a p-adic exponential (although it can be obtained as real exponential), and one can say that group decomposes to a union of disconnected parts corresponding to the products of discrete group elements with G_p,n.
   

   This decomposition to totally uncorrelated disjoint parts is of course worrying from the point of view of algebraic continuation. The construction of p-adic manifolds by using canonical identification to define coordinate charts as real ones allows a correspondence between p-adic and real groups and also allows to glue together the images of the disjoint regions at real side: this induces gluing at p-adic side. The procedure will be discussed later in more detail.
   

3. There is a little technicality is needed. The usual Lie-algebra exponential in the matrix representation contains an imaginary unit. For p mod 4 =3 this imaginary unit can be introduced as a unit in the algebraic extension. For p mod 4 =1 it can be realized as an algebraic number. It however seems that imaginary unit or its p-adic analog should belong to an algebraic extension of p-adic numbers. The group parameters for algebraic extension of p-adic numbers belong to the algebraic extension. If the algebraic extension contains non-trivial roots of unity U_m,n= exp(i2 π m/n), the differences U_m,n-U^*_m,n are proportional to imaginary unit as real numbers and one can replace imaginary unit in the exponential with U_m,n-U^*_m,n. In real context this means only a rescaling of the Lie algebra generator and Planck constant by a factor (2sin(2 π m/n))^-1. A natural imaginary unit is defined in terms of U_1,pⁿ.
   
   
4. This construction is expected to generalize to the case of coset spaces and give rise to a coset space G/H identified as the union of discrete coset spaces associated with the elements of the coset G₀/H₀ making sense also in the real context. These are obtained by multiplying the element of G₀/H₀ by the p-adic factor space G_p,n/H_p,n.
   

One has two hierarchies corresponding to the hierarchy of discrete subgroups of G₀ requiring each some minimal algebraic extension of p-adic numbers and to the hierarchy of G_p:s defined by the powers of p. These two hierarchies can be assigned to angles (actually phases coming as roots of unity) and p-adic length scales in the space of group parameters.

The Lie algebra of the rotation group spanned by the generators L_x,L_y,L_z provides a good example of the situation and leads to the question whether the hierarchy of Planck constants kenociteallb/Planck could be understood p-adically.

1. Ordinary commutation relations are [L_x,L_y]= i hbar L_z. For the hierarchy of Lie groups it is convenient to extend the algebra by introducing the generators L_iⁿ⁾= pⁿL_i and one obtains [L_x^m),L_yⁿ⁾]= i hbar L_z^m+n). This resembles the commutation relations of Kac-Moody algebra structurally.
   
   
2. For the generators of Lie-algebra generated by L_i^m) one has [L_x^m),L_y^m)]= ip^m hbar L_z^m). One can say that Planck constant is scaled from hbar to p^m hbar. Could the effective hierarchy of Planck constants assigned to the multi-furcations of space-time sheets correspond in p-adic context to this hierarchy of Lie-algebras?
   
   
3. The values of the Planck constants would come as powers of primes: the hypothesis has been that they comes as positive integers. The integer n defining the number of sheets for n-furcation would come as powers n=p^m. The connection between p-adic length scale hierarchy and hierarchy of Planck constants has been conjectured already earlier but the recent conjecture is the most natural one found hitherto. Of course, the question whether the number sheets of furcation correlates with the power of p characterizing "small" continuous symmetries remains an open question. Note that also n-adic and even q=m/n-adic topology is possible with norms given by powers of integer or rational. Number field is however obtained only for primes. This suggests that if also integer - and perhaps even rational valued scales are allowed for causal diamonds, they correspond to effective n-adic or q-adic topologies and that powers of p are favored.
   

The difficult questions concern again integration. The integrals reduce to sums over the discrete subgroup of G multiplied with an integral over the p-adic variant G_p,n of the continuous Lie group. The first integral - that is summation - is number theoretically universal. The latter integral is the problematic one.

1. The easy way to solve the problem is to interpret the hierarchy of continuous p-adic Lie groups G_p,n as analogs of gauge groups. But if the wave functions are invariant under G_p,n, what is the situation with respect to G_p,m for m<n? Infinitesimally one obtains that the commutator algebras [G_p,k,G_p,l] ⊂ G_p,k+l must annihilate the functions for k+l ≥ n. Does also G_p,m, m<n annihilate the functions for as a direct calculation demonstrates in the real case. If this is the case also p-adically the hierarchy of groups G_p,n would have no physical implications. This would be disappointing.
   
   
2. One must however be very cautious here. Lie algebra consists of first order differential operators and in p-adic context the functions annihilated by these operators are pseudo-constants. It could be that the wave functions annihilated by G_p,n are pseudo-constants depending on finite number of pinary digits only so that one can imagine of defining an integral as a sum. In the recent case the digits would naturally correspond to powers p^m, m<n. The presence of these
   functions could be purely p-adic phenomenon having no real counterpart and emerge when one
   leaves the intersections of real and p-adic worlds. This would be just the non-determinism of imagination assigned to p-adic physics in TGD inspired theory of consciousness.
   

Is there any hope that one could define harmonic analysis in G_p,n in a number theoretically universal manner? Could one think of identifying discrete subgroups of G_p,n allowing also an interpretation as real groups?

1. Exponentiation implies that in matrix representation the elements of G_p,n are of form g= Id+ pⁿg₁: here Id represents real unit matrix. For compact groups like SU(2) or CP₂ the group elements in real context are bounded above by unity so that this kind of sub-groups do not exist as real groups. For non-compact groups like SL(2,C) and T⁴ this kind of subgroups make sense also in real context.
   
   
2. Zero energy ontology suggests that discrete but infinite sub-groups Γ of SL(2,C) satisfying certain additional conditions define hyperbolic spaces as factor spaces H³/ Γ (H³ is hyperboloid of M⁴ lightcone). These spaces have constant sectional curvature and very many 3-manifolds allow a hyperbolic metric with hyperbolic volume defining a topological invariant. The moduli space of CDs contains the groups Γ defining lattices of H³ replacing it in finite measurement resolution. One could imagine hierarchies of wave functions restricted to these subgroups or H³ lattices associated with them. These wave functions would have the same form in both real and p-adic context so that number theoretical universality would make sense and one could perhaps define the inner products in terms of "integrals" reducing to sums.
   
   
3. The inclusion hierarchy G_p,n ⊃ G_p,n+1 would in the case of SL(2,C) have interpretation in terms of finite measurement resolution for four-momentum. If G_p,n annihilate the physical states or creates zero norm states, this inclusion hierarchy corresponds to increasing IR cutoff (note that short length scale in p-adic sense corresponds to long scale in real sense!). The hierarchy of groups G_p,n makes sense also in the case of translation group T⁴ and also now the interpretation in terms of increasing IR cutoff makes sense. This picture would provide a group theoretic realization for with the vision that p-adic length scale hierarchies correspond to hierarchies of length scale measurement resolutions in M⁴ degrees of freedom.
   

Canonical identification and the definition of p-adic counterparts of Lie groups

For Lie groups for which matrix elements satisfy algebraic equations, algebraic subgroups with rational matrix elements could regarded as belonging to the intersection of real and p-adic worlds, and algebraic continuation by replacing rationals by reals or p-adics defines the real and p-adic counterparts of these algebraic groups. The challenge is to construct the canonical identification map between these groups: this map would identify the common rationals and possible common algebraic points on both sides and could be seen also a projection induced by finite measurement resolution.

A proposal for a construction of the p-adic variants of Lie groups was discussed in previous section. It was found that the p-adic variant of Lie group decomposes to a union of disjoint sets defined by a discrete subgroup G₀ multiplied by the p-adic counterpart G_p,n of the continuous Lie group G. The representability of the discrete group requires an algebraic extension of p-adic numbers. The disturbing feature of the construction is that the p-adic cosets are disjoint. Canonical identification I_k,l suggests a natural solution to the problem. The following is a rough sketch leaving a lot of details open.

1. Discrete p-adic subgroup G₀ corresponds as such to its real counterpart represented by matrices in algebraic extension of rationals. G_p,n can be coordinatized separately by Lie algebra parameters for each element of G₀ and canonical identification maps each G_p,n to a subset of real G. These subsets intersect and the chart-to-chart identification maps between Lie algebra coordinates associated with different elements of G₀ are defined by these intersections. This correspondence induces the correspondence in p-adic context by the inverse of canonical identification.
   
   
2. One should map the p-adic exponentials of Lie-group elements of G_p,n to their real counterparts by some form of canonical identification.
   
   1. Consider first the basic form I=I_{0, ∞} of canonical identification mapping all p-adics to their real counterparts and maps only the p-adic integers 0 ≤ k<p to themselves.
      

      The gluing maps between groups G_p,n associated with elements g_m and g_n of G₀ would be defined by the condition g_m I(exp(it_aT^a)= g_n I(exp(iv_aT^a). Here t_a and v_a are Lie-algebra coordinates for the groups at g_m and g_n. The delicacies related to the identification of p-adic analog of imaginary unit have been discussed in the previous section. It is important that Lie-algebra coordinates belong to the algebraic extension of p-adic numbers containing also the roots of unity needed to represent g_n. This condition allows to solve v_a in terms of t_a and v_a= v_a(t_b) defines the chart map relating the two coordinate patches on the real side. The inverse of the canonical identification in turn defines the p-adic variant of the chart map in p-adic context. For I this map is not p-adically analytic as one might have guessed.
      
      

   2. The use of I_k,n instead of I gives hopes about analytic chart-to chart maps on both sides. One must however restrict I_k,n to a subset of rational points (or generalized points in algebraic extension with generalized rational defined as ratio of generalized integers in the extension). Canonical identification respects group multiplication only if the integers defining the rationals m/n appearing in the matrix elements of group representation are below the cutoff p^k. The points satisfying this condition do not in general form a rational subgroup. The real images of rational points however generate a rational sub-group of the full Lie-group having a manifold completion to the real Lie-group.
      

      One can define the real chart-to chart maps between the real images of G_p,k at different points of G₀ using I_k,l(exp(iv_aT^a)= g_n^-1g_m × I_k,l(exp(it_aT^a). When real charts intersect, this correspondence should allow solutions v_a,t_b belonging to the algebraic extension and satisfying the cutoff condition. If the rational point at the other side does not correspond to a rational point it might be possible to perform pinary cutoff at the other side.
      

      Real chart-to-chart maps induce via common rational points discrete p-adic chart-to-chart maps between G_p,k. This discrete correspondence should allow extension to a unique chart-to-chart map the p-adic side. The idea about algebraic continuation suggests that an analytic form for real chart-to-chart maps using rational functions makes sense also in the p-adic context.
      

3. p-Adic Lie-groups G_p,k for an inclusion hierarchy with size characterized by p^-k. For large values of k the canonical image of G_p,k for given point of G₀ can therefore intersect its copies only for a small number of neighboring points in G₀, whose size correlates with the size of the algebraic extension. If the algebraic extension has small dimension or if k becomes large for a given algebraic extension, the number of intersection points can vanish. Therefore it seems that in the situations, where chart-to-chart maps are possible, the power p^k and the dimension of algebraic extension must correlate. Very roughly, the order of magnitude for the minimum distance between elements of G₀ cannot be larger than p^-k+1. The interesting outcome is that the dimension of algebraic extension would correlate with the pinary cutoff analogous to the IR cutoff defining measurement resolution for four-momenta.
   

For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.

Could canonical identification make possible definition of integrals in p-adic context?

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 second one and devoted to p-adic integration.

The notion of p-adic manifold using using real chart maps instead of p-adic ones allows an attractive approach also to p-adic integration and to the problem of defining p-adic version of differential forms and their integrals.

1. If one accepts the simplest form of canonical identification I(x): ∑_n x_npⁿ → ∑ x_np^-n, the image of the p-adic surface is continuous but not differentiable and only integers n<p are mapped to themselvs. One can define integrals of real functions along images of the p-adically analytic curves and define the values of their p-adic counterparts as their algebraic continuation when it exists.
   

   In TGD framework this does not however work. If one wants to define induced quantities - such as metric and K ähler form - on the real side one encounters a problem since the image surface is not smooth and the presence of edges implies that these quantities containing derivatives of imbedding space coordinates possess delta function singularities. These singularities could be even dense in the integration region so that one would have no-where differentiable continuous functions and the real integrals would reduce to a sum which do not make sense.
   
   

2. In TGD framework finite measurement resolution realized in terms of pinary cutoff however saves the situation. The canonical identification I_k,l(m/n) = I_k,l(m)/I_kl(n) maps rationals to themselves for m<p^k,n<p^k. The second pinary cutoff m<p^l,n<p^l, l>k implies that the chart map takes a discrete subset of p-adic rationals to a discrete set of real rationals. The completion of the discrete image of p-adic preferred extremal under I_k,l to a real preferred extremal is very natural. This preferred extremal can be said to be unique apart from a finite measurement resolution represented by the pinary cutoffs k and l. All induced quantities are well defined on both sides.
   

   p-Adic integrals can be defined as pullbacks of real integrals by algebraic continuation when this is possible. The inverse image of the real integration region in canonical identification defines the p-adic integration region.
   
   

3. The integrals of p-adic differential forms can be defined as pullbacks of the real integrals. The integrals of closed forms, which are typically integers, would be the same integers but interpreted as p-adic integers.
   

It is interesting to study the algebraic continuation of K ähler action from real sector to p-adic sectors.

1. K ähler action for both Euclidian and Minkowskian regions reduces to the algebraic continuation of the integral of Chern-Simons-K ähler form over preferred 3-surfaces. The contributions from Euclidian and Minkowskian regions reduce to integrals of Chern-Simons form over 3-surfaces. I have somewhere considered the possibility that the 3-surfaces for Minkowskian and Euclidian contribution might be identical: this cannot be the case since the space-like 3-surfaces at the boundaries of CD for Minkowskian and Euclidian regions are disjoint.
   

   The contribution from Euclidian regions defines K ähler function of WCW and the contribution from Minkowskian regions giving imaginary exponential of K ähler action has interpretation as Morse function whose stationary points are expected to select special preferred extremals. One would expect that both functions have a continuous spectrum of values. In the case of K ähler function this is necessary since K ähler function defines the K ähler metric of WCW via its second derivatives in complex coordinates by the well-known formula. Note that by the above observation K ähler and Morse functions are not in general proportional to each other.
   
   

2. The algebraic continuation of the exponent of K ähler function for a given p-adic prime is expected to require the proportionality to pⁿ so that not all preferred extremals are expected to allow a continuation to a given p-adic number field. This kind of assumption has been indeed made in the case of deformations of CP₂ type extremals in order to derive formula for the gravitational constant in terms of basic parameters of TGD but without real justification (see this).
   
   
3. The condition that the action exponential in the Minkowskian regions is a genuine phase factor implies that it reduces to a root of unity (one must have an algebraic extension of p-adic numbers). Therefore the contribution to the imaginary exponent K ähler action from these regions for the p-adicizable preferred extremals should be of form 2 π (k+m/n).
   

   If all preferred real extremals allow p-adic counterpart, the value spectrum of the Morse function on the real side is discrete and could be forced by the preferred extremal property. If this were the case the stationary phase approximation around extrema of K ähler function on the real side would be replaced by sum with varying phase factors weighted by K ähler function.
   

   An alternative conclusion is that the algebraic continuation of K ähler action to any p-adic field is possible only for a subset of preferred extremals with a quantized spectrum of Morse function. One the real side stationary phase approximation would make sense. It however seems that the stationary phases must obey the above discussed quantization rule.
   

Also holomorphic forms allow algebraic continuation and one can require that also their integrals over cycles do so. An important example is provided by the holomorphic one-forms integrals over cycles of partonic 2-surface defining the Teichmueller parameters characterizing the conformal equivalence class of the partonic 2-surfaces as Riemann surface. The p-adic variants exist of these parameters exist if they allow an algebraic continuation to a p-adic number. The algebraic continuation from the real side to the p-adic side would be possible on for certain p-adic primes p if any: this would allow to assign p-adic prime or primes to a given real preferred extremal. This justifies the assumptions of p-adic mass calculations concerning the contribution of conformal modular degrees of freedom to mass squared (see this).

For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.

Could canonical identification allow construction of path connected topologies for p-adic manifolds?

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.

1. 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).
   
   
2. 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.
   
   
3. 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.
   

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.

1. Canonical identification maps p-adic numbers ∑ x_npⁿ to reals and is defined by the formula I(x) = ∑ x_np^-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.
   
   
2. 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?
   
   
3. 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<p^l,n<p^l are mapped to the reals (real rationals actually).
   

   One can also express p-adic number as expansion of powers fo p^k and define canonical identification I_k as ∑ x_np^kn → ∑ x_np^-kn. Also the variant I_k,l(m/n)=I_k,l(m)/I_k,l(n) with l defining pinary cutoff for m and l makes sense. One can say that I_k,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.
   
   

4. Canonical identification does not commute with rational symmetries unless one uses the map I_k,l(m/n)=I_k,l(m)/I_k,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.
   

What is very remarkable is that canonical identification can be seen as a continuous generalization of the p-adic norm defined as N_p(x) == I_k,l(x) having the highly desired Archimedean property. I_k,l is the most natural variant of canonical identification.

1. 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 N_p(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.
   
   
2. I_k,l(x) indeed defines the analog of Archimedean norm in the sense that one has N_p(x+y) ≤ N_p(x)+N_p(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. N_p(x) is however not multiplicative: only a milder condition N_p(pⁿx)=N(pⁿ)N(x)=p^-n N(x) holds true.
   
   
3. 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 I_k,l labelled by a finite measurement resolution. This would give a very close connection with physics.
   
   
4. 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.
   
   
5. 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.
   
   
6. As already mentioned, one must restrict the p-adic points mapped to reals to rationals since I_k,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.
   

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 I_k,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).

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 I_k,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 and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.