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.
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 Gp: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.
- 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 Gp obtained by exponentiating the Lie algebra by using p-adic parameters ti in the exponentiation exp(tiTi).
- One obtains actually an inclusion hierarchy of p-adic Lie groups. The levels of the hierarchy are labelled by the maximum p-adic norms |ti|p= p-ni, ni ≥ 1 and in the special case ni=n - strongly suggested by group invariance - one can write Gp,1 ⊃ Gp,2 ⊃ ...Gp,n .... Gp,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 Gp,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.
- 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 Um,n= exp(i2 π m/n), the differences Um,n-U*m,n are proportional to imaginary unit as real numbers and one can replace imaginary unit in the exponential with Um,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 U1,pn.
- 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 Gp,n/Hp,n.
The Lie algebra of the rotation group spanned by the generators Lx,Ly,Lz 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.
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 Gp,n of the continuous Lie group. The first integral - that is summation - is number theoretically universal. The latter integral is the problematic one.
- Ordinary commutation relations are [Lx,Ly]= i hbar Lz. For the hierarchy of Lie groups it is convenient to extend the algebra by introducing the generators Lin)= pnLi and one obtains [Lxm),Lyn)]= i hbar Lzm+n). This resembles the commutation relations of Kac-Moody algebra structurally.
- For the generators of Lie-algebra generated by Lim) one has [Lxm),Lym)]= ipm hbar Lzm). One can say that Planck constant is scaled from hbar to pm 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?
- 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=pm. 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.
Is there any hope that one could define harmonic analysis in Gp,n in a number theoretically universal manner? Could one think of identifying discrete subgroups of Gp,n allowing also an interpretation as real groups?
- The easy way to solve the problem is to interpret the hierarchy of continuous p-adic Lie groups Gp,n as analogs of gauge groups. But if the wave functions are invariant under Gp,n, what is the situation with respect to Gp,m for m<n? Infinitesimally one obtains that the commutator algebras [Gp,k,Gp,l] ⊂ Gp,k+l must annihilate the functions for k+l ≥ n. Does also Gp,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 Gp,n would have no physical implications. This would be disappointing.
- 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 Gp,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 pm, 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.
- Exponentiation implies that in matrix representation the elements of Gp,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.
- 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.
- The inclusion hierarchy Gp,n ⊃ Gp,n+1 would in the case of SL(2,C) have interpretation in terms of finite measurement resolution for four-momentum. If Gp,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 Gp,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 Gp,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 Ik,l suggests a natural solution to the problem. The following is a rough sketch leaving a lot of details open.
- Discrete p-adic subgroup G0 corresponds as such to its real counterpart represented by matrices in algebraic extension of rationals. Gp,n can be coordinatized separately by Lie algebra parameters for each element of G0 and canonical identification maps each Gp,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.
- One should map the p-adic exponentials of Lie-group elements of Gp,n to their real counterparts by some form of canonical identification.
- Consider first the basic form I=I0, ∞ 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 Gp,n associated with elements gm and gn of G0 would be defined by the condition gm I(exp(itaTa)= gn I(exp(ivaTa). Here ta and va are Lie-algebra coordinates for the groups at gm and gn. 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 gn. This condition allows to solve va in terms of ta and va= va(tb) 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.
- The use of Ik,n instead of I gives hopes about analytic chart-to chart maps on both sides. One must however restrict Ik,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 pk. 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 Gp,k at different points of G0 using Ik,l(exp(ivaTa)= gn-1gm × Ik,l(exp(itaTa). When real charts intersect, this correspondence should allow solutions va,tb 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 Gp,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.
- p-Adic Lie-groups Gp,k for an inclusion hierarchy with size characterized by p-k. For large values of k the canonical image of Gp,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 pk 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 see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.