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.
- If one accepts the simplest form of canonical identification I(x): ∑n xnpn → ∑ xnp-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.
- In TGD framework finite measurement resolution realized in terms of pinary cutoff however saves the situation. The canonical identification Ik,l(m/n) = Ik,l(m)/Ikl(n) maps rationals to themselves for m<pk,n<pk. The second pinary cutoff m<pl,n<pl, 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 Ik,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.
- 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.
- 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.
- 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).
- 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.
For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.