Quantum arithmetics is a notion which emerged as a possible resolution of long-lived challenge of finding mathematical justification for the canonical identification mapping p-adics to reals playing key role in p-adic mass calculations. The model for Shnoll effect was the bridge leading to the discovery of quantum arithmetics.
I have been gradually developing the notion of quantum p-adics and during the weekend made quite a step of progress in understanding the concept and dare say that the notion now rests on a sound basis.
- What quantum arithmetics suggests is a modification of p-adic numbers by replacing p-adic pinary expansions with their quantum counterparts allowing the coefficients of prime powers to be integers not divisible by p. A further important constraint is that the factors of coefficients are primes smaller than p. If the coefficients are smaller than p, one obtains something reducing effectively to ordinary p-adic number field.
- A further constraint is that quantum integers respect the decomposition of integer to powers of prime. Quantum p-adic integers are to p-adic integers what the integers in the extension of number field are for the number field and one can indeed identify Galois group Gp for each prime p and form adelic counterpart of this group as Cartesian product of all Gp:s.
- After various trials it turned out (this is what motivated this posting!) that quantum p-adics are indeed quantal in the sense that one can assign to given quantum p-adic integer n a wave function at the orbit of corresponding Galois group decomposing to Galois groups of its prime factors of n.
- The basic conditions are that ×q and +q satisfy the basic associativity and distributivity laws. These conditions are extremely powerful and can be formulated in terms of number theoretic Feynman diagrams assignable to sequences of arithmetical operations and their co-algebra counterparts. This brings in physical insight.
- One can interpret ×q and +q and their co-algebra operations as 3-vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD, namely stringy vertices in which 3-surface splits and vertices analogous to those of Feynman diagrams in which lines join along their 3-D ends. Only the latter vertices correspond to particle decays and fusions whereas stringy vertices correspond to decay of particle to path and simultaneous propagation along both paths: this is by the way one of the fundamental differences between quantum TGD and string models. This plus the assumption that Galois groups
associated with primes define symmetries of the vertices allows to deduce very precise information about the symmetries of the two kinds of vertices needed to satisfy the associativity and distributivity and actually fix them highly uniquely, and therefore determine corresponding zero energy states having collections of integers as counterparts of incoming positive energy (or negative energy) particles.
- Zero energy ontology leads naturally zero energy states for which time reversal symmetry is broken in the sense that either positive or negative energy part corresponds to a single collection of integers as incoming lines. What is fascinating is the the prime decomposition of integer corresponds to a decomposition of braid to strands. C and P have interpretation as formations of multiplicative and additive inverses of quantum integers and CP=T changes the positive and negative energy parts of the number theoretic zero energy states.
- This gives strong support for the old conjecture that generalized Feynman diagrams have number theoretic interpretation and allow moves transforming them to tree diagrams - also this generalization of old-fashioned string duality is old romantic idea of quantum TGD, which I however gave up as too "romantic". I noticed the analogy of Feynman diagrams with the algebraic expressions but failed to realize how extremely concrete the connection could be. What was left from the idea were some brief comments in Appendix A: Quantum Groups and Related Structures to one of the chapters of "Towards M-matrix".
The moves for generalized Feynman diagrams would code for associativity and distributivity of quantum arithmetics and we have actually learned them in elementary school as a process simplifying algebraic expressions! Also braidings with strands labeled by the primes dividing the integer emerge naturally so that the connection with quantum TGD proper becomes very strong and consistent with the earlier conjecture inspired by the construction of infinite primes stating that transition amplitudes have purely number theoretic meaning in ZEO.
- Canonical identification finds a fundamental role in the definition of the norm for both quantum p-adics and quantum adeles. The construction is also consistent with the notion of number theoretic entropy which can have also negative values (this is what makes living systems living!).
- There are arguments suggesting that quantum p-adics form a field - one might say "quantum field" - so that also differential calculus and even integral calculus would make sense since quantum p-adics inherit almost well-ordering from reals via canonical identification.
- One can also generalize the construction to algebraic extensions of rationals. In this case the coefficients of quantum adeles are replaced by rationals in the extension and only those p-adic number fields for which the p-adic prime does not split into a product of primes of algebraic extension are kept in the quantum adele associated with rationals. This construction gives first argument in favor of the crazy conjecture that the Absolute Galois group (AGG) is isomorphic with the Galois group of quantum adeles.
To sum up, the vision abut "Physics as generalized number theory" can be also transformed to "Number theory as quantum physics"!
For detais see the new chapter Quantum Adeles.