Introduction

1. What quantum p-adics could be?

2. Quantum TGD and Hilbert adeles

Earlier attempts to construct quantum arithmetics

1. Quantum arithmetics

2. Summary: the three options for quantum p-adics

Hilbert p-adics, Hilbert adeles, and TGD

1. Could the notion of Hilbert mathematics make sense?

2. Hilbert p-adics, hierarchy of Planck constants, and finite measurement resolution

3. Quantum adeles

Generalized Feynman diagrams as quantum arithmetic Feynman diagrams?

1. Quantum TGD predicts counterparts for ×_q and +_q vertices

2. How could quantum numbers of physical states relate to the number theoretic quantum numbers?

3. Number theoretical quantum numbers and hierarchy of Planck constants

4. What is the relation to infinite primes?

5. What selects preferred primes in number theoretical evolution?

6. Generalized Feynman diagrams and adeles

Quantum Mathematics and Quantum Mechanics

1. Unitary process and state function reduction in ZEO

2. ZEO, state function reduction, unitary process, and quantum mathematics

3. What multiverse branching could mean?

4. The replacement of a point of Hilbert space with Hilbert space as a second quantization

Speculations related to Hilbert adelization

1. Hilbert adelization as a manner to realize number theoretical universality

2. Could number theoretic emergence make sense?

Appendix: Some possibly motivating considerations

1. Analogies between number theoretic and function field theoretic ramification

2. Could one assign analog of function field to integers and analogs prime polynomials to primes?