Quantum Adeles

Quantum arithmetics provides a possible resolution of a long-lasting challenge of finding a mathematical justification for the canonical identification mapping p-adics to reals playing a key role in TGD in particular in p-adic mass calculations. p-Adic numbers have p-adic pinary expansions ∑ anp n satisfying an < p. of powers p n to be products of primes p1 < p satisfying an < p for ordinary p-adic numbers. One could map this expansion to its quantum counterpart by replacing an with their counterpart and by canonical identification map p→ 1/p the expansion to real number. This definition might be criticized as being essentially equivalent with ordinary p-adic numbers since one can argue that the map of coefficients an to their quantum counterparts takes place only in the canonical identification map to reals. One could however modify this recipe. Represent integer n as a product of primes l and allow for l all expansions for which the coefficients an consist of primes p1 < p but give up the condition an < p. This would give 1-to-many correspondence between ordinary p-adic numbers and their quantum counterparts. It took time to realize that l < p condition might be necessary in which case the quantization in this sense if present at all could be associated with the canonical identification map to reals. It would correspond only to the process taking into account finite measurement resolution rather than replacement of p-adic number field with something new, hopefully a field. At this step one might perhaps allow l > p so that one would obtain several real images under canonical