Introduction
What quantum padics could be?
Quantum TGD and Hilbert adeles
Earlier attempts to construct quantum arithmetics
Quantum arithmetics
Summary: the three options for quantum padics
Hilbert padics, Hilbert adeles, and TGD
Could the notion of Hilbert mathematics make sense?
Hilbert padics, hierarchy of Planck constants, and finite measurement resolution
Quantum adeles
Generalized Feynman diagrams as quantum arithmetic Feynman diagrams?
Quantum TGD predicts counterparts for ×_{q} and +_{q} vertices
How could quantum numbers of physical states relate to the number theoretic quantum numbers?
Number theoretical quantum numbers and hierarchy of Planck constants
What is the relation to infinite primes?
What selects preferred primes in number theoretical evolution?
Generalized Feynman diagrams and adeles
Quantum Mathematics and Quantum Mechanics
Unitary process and state function reduction in ZEO
ZEO, state function reduction, unitary process, and quantum mathematics
What multiverse branching could mean?
The replacement of a point of Hilbert space with Hilbert space as a second quantization
Speculations related to Hilbert adelization
Hilbert adelization as a manner to realize number theoretical universality
Could number theoretic emergence make sense?
Appendix: Some possibly motivating considerations
Analogies between number theoretic and function field theoretic ramification
Could one assign analog of function field to integers and analogs prime polynomials to primes?
