Revisiting Born's Rule through Uhlhorn's and Gleason's Theorems.

Area of Science:

• Quantum mechanics foundations
• Mathematical physics

Background:

• Born's rule is a fundamental principle in quantum mechanics.
• Previous work introduced Contexts, Systems and Modalities (CSM) axioms to derive Born's rule.
• CSM provides a physical justification for Gleason's hypotheses.

Purpose of the Study:

• To extend the derivation of Born's rule within the CSM framework.
• To demonstrate that unitary transforms are not an independent hypothesis.
• To connect the necessity of unitary transforms to Uhlhorn's theorem.

Main Methods:

• Axiomatic approach based on Contexts, Systems and Modalities (CSM).
• Analysis of the interplay between quantized modalities and continuous contexts.
• Application of Uhlhorn's theorem to quantum mechanical transformations.

Main Results:

• The structure of quantum mechanics arises from the interaction of modalities and contexts.
• The requirement of unitary transforms to relate different contexts is shown to be unnecessary as a standalone axiom.
• This necessity is a consequence of Uhlhorn's theorem.

Conclusions:

• The CSM framework offers a deeper physical understanding of quantum mechanics.
• Born's rule can be derived without assuming unitary transforms as a fundamental requirement.
• Uhlhorn's theorem plays a crucial role in understanding transformations between quantum contexts.