On geometric ideas which lie at the foundation of quantum theory

Ryszard Paweł Kostecki

P. A. M. Dirac

Jincheng Zhang

Quantum Information Theory (QIT) represents a burgeoning field exploring the potential of quantum mechanics to revolutionize computation and communication. A core challenge lies in defining and manipulating probability, traditionally viewed as a statistical concept. This paper proposes a fundamental shift in perspective, formalizing "geometric probability" as a property inherent to quantum systems, and subsequently utilizing this concept to drive the development of novel quantum algorithms. The core claim centers on the idea that geometric probability offers a more direct and powerful pathway to understanding and exploiting quantum phenomena. We investigate the extension of probability to the space of geometric configurations, introducing a new type of probability intrinsically linked to quantum mechanics. This approach moves beyond traditional statistical interpretations, offering a deeper and more mathematically rigorous foundation for quantum information processing. The proposed framework necessitates a re-evaluation of fundamental probability principles, with the potential for significant advancements in algorithm design and the exploration of novel quantum computational paradigms. This work addresses the limitations of current approaches by introducing a geometric perspective that directly reflects the underlying structure of quantum mechanics.

Maik Reddiger

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

Veeravalli S. Varadarajan