Free Choice in Quantum Theory: A p-adic View

Area of Science:

• Mathematical Physics
• Quantum Mechanics
• Number Theory

Background:

• Observational data in physics are rational numbers due to measurement errors.
• The interpretation of this data can lead to different conclusions about the fundamental nature of reality.
• Existing models often rely on real number analysis.

Purpose of the Study:

• To rigorously prove that the choice of metric (real or p-adic) dictates whether nature appears discrete or continuous.
• To introduce p-adic 1-Lipschitz maps and Mealy machines as tools for physical modeling.
• To construct wave functions and prove entropic uncertainty relations within this framework.

Main Methods:

• Utilizing p-adic 1-Lipschitz maps, which are continuous with respect to the p-adic metric.
• Employing sequential Mealy machines to define causal functions over discrete time.
• Expanding these maps to continuous real functions for modeling open physical systems.

Main Results:

• Demonstrated that the choice between real and p-adic metrics determines the perceived nature of physical reality.
• Successfully constructed wave functions and proved an entropic uncertainty relation.
• Showcased the applicability of Mealy machines and p-adic analysis in quantum mechanics.

Conclusions:

• The fundamental properties of nature at the smallest scales are not inherent but depend on the mathematical framework applied to experimental data.
• p-adic mathematical physics offers a viable alternative to traditional real-number-based approaches.
• This work provides a novel perspective on quantum mechanics, consistent with superdeterminism concepts.