In this talk, I will expound a point of view on the theoretical investigation of the foundations and mathematical formalism of quantum mechanics which is based on Schwinger’s “Symbolism of atomic measurement” [8] on the physical side, and on the notion of groupoid on the mathematical side. I will start by reviewing the “development” of quantum mechanics and its formalism starting from Schrödinger’s wave mechanics, passing through the Hilbert space quantum mechanics, and arriving at the $C^∗$-algebraic formulation of quantum mechanics in order to give an intuitive idea of what is the “place” of the groupoid-based approach to quantum theories presented here. Then, after (what I hope will be) a highly digestible introduction to the notion of groupoid, I will review two historic experimental instances in which the shadow of the structure of groupoid may be glimpsed, namely, the Ritz-Rydberg combination principle, and the Stern-Gerlach experiment. The last part of the talk will be devoted to building a bridge between the groupoid-based approach to quantum mechanics and the more familiar $C^∗$-algebraic one by analysing how to obtain a (possibly) non-commutative algebra out of a given groupoid. Two relevant examples will be discussed, and some comment on future directions (e.g., the composition of systems) will close the talk. The material presented is part of an ongoing project developed together with Dr. F. Di Cosmo, Prof. A. Ibort, and Prof. G. Marmo. In particular, the discrete-countable theory has already appeared in [1, 2, 3, 4, 5, 6, 7].

References

[1] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics. Entropy, 11(22):1292 – 18, 2020.

[2] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics. International Journal of Geometric Methods in Modern Physics, 17(04):2050054 (14), 2020.

[3] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics IV: Composition and independence. International Journal of Geometric Methods in Modern Physics, 17(04):2050058 (34), 2020.

[4] F. M. Ciaglia, A. Ibort, and G. Marmo. A gentle introduction to Schwinger’s formulation of quantum mechanics: the groupoid picture. Modern Physics Letters A, 33(20):1850122–8, 2018.

[5] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics I: Groupoids. International Journal of Geometric Methods in Modern Physics, 16(08):1950119 (31), 2019.

[6] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables. International Journal of Geometric Methods in Modern Physics, 16(09):1950136 (32), 2019.

[7] F. M. Ciaglia, A. Ibort, and G. Marmo. Schwinger’s Picture of Quantum Mechanics III: The Statistical Interpretation. International Journal of Geometric Methods in Modern Physics, 16(11):1950165 (37), 2019.

[8] J. Schwinger. Quantum Mechanics, Symbolism of Atomic Measurements. Springer-Verlag, Berlin, 2001.