Overviews and specific results for General Probabilistic Theories

General Probabilistic Theories (GPTs) are operationally the broadest framework to describe nature. They have been studied actively in the field of quantum foundations, followed by the intuition that seeing quantum theory from a broader perspective will contribute to elucidating its essence. In this seminar, I explain the mathematical formulation of GPTs to show how they give the most intuitive description of nature. In addition to the fundamental description, I also present results for one of the simplest classes of GPTs called regular polygon theories. Regular polygon theories are GPTs whose state spaces are in the shape of regular polygons and are often regarded naturally as generalizations of the two-level quantum system (the qubit system). I explain whether or how several quantum behaviors such as uncertainty relations and the violation of Bell-CHSH inequality are observed also in regular polygon theories. References: [1] M. Plávala, Phys. Rep. 1033, 1-64 (2023) [2] R. Takakura, arXiv:2202.13834 (2022) [3] R. Takakura, J. Phys. A: Math. Theor. 54, 315302 (2021) [4] R. Takakura, arXiv:2401.04596 (2024)