nLab quantum observables as groupoid convolution -- references

Quantum observable algebras as Groupoid algebras

The original derivation of the Heisenberg picture of quantum mechanics, introducing $N \times N$ matrix algebra for transitions between $N$ measurable states of atomic spectra, by

• Werner Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33 (1925) 879–893 [doi:10.1007/BF01328377, Engl. pdf]

was argued by

• Alain Connes, pp 43-44 of: Noncommutative geometry, Academic Press (1994) [ISBN:9780121858605, pdf]

to be fruitfully understood as the groupoid convolution algebra of the pair groupoid of transitions between $N$ elements.

Later but more generally, the “algebra of (selective) measurement” originally envision by

• Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [ISBN:9780738203034, pdf]

• Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [doi:10.1007/978-3-662-04589-3]

is argued to be, in modern language, the groupoid convolution algebras of groupoids whose morphisms reflect transitions between possible quantum measurement-outcomes – by:

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics I: Groupoids, Int. J. Geometric Methods in Modern Physics 16 08 (2019) 1950119 [doi:10.1142/S0219887819501196, arXiv:1905.12274]

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables, Int. J. Geometric Methods in Modern Physics 16 09 (2019) 1950136 [doi:10.1142/S0219887819501366, arXiv:1907.03883]

• Florio M. Ciaglia, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics II: Algebras and Observables, Int. J. Geometric Methods in Modern Physics 16 11 (2019) 1950165 [doi:10.1142/S0219887819501652, arXiv:1909.07265]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schwinger’s Picture of Quantum Mechanics IV: Composition and independence, Int. J. Geometric Methods in Modern Physics 17 04 (2020) 2050058 [doi:10.1142/S0219887820500589, arXiv:2004.02472]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schwinger’s picture of Quantum Mechanics, Int. J. Geometric Methods in Modern Physics 17 04 (2020) 2050054 [doi:10.1142/S0219887820500541, arXiv:2002.09326]

and as such further developed in:

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Schrödinger’s problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics, Entropy 22 11 (2020) [doi:10.3390/e22111292, arXiv:2012.10284]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone: Schwinger’s picture of quantum mechanics: 2-groupoids and symmetries, Journal of Geometric Mechanics 13 3 (2021) [doi:10.3934/jgm.2021008, arXiv:2104.13880]

• Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo: Quantum Tomography and Schwinger’s Picture of Quantum Mechanics, Journal of Physics A: Mathematics and Theoretical 55 27 (2022) [doi:10.1088/1751-8121/ac7591, arXiv:2205.00170]

• Florio M. Ciaglia, Fabio Di Cosmo, Paolo Facchi, Alberto Ibort, Arturo Konderak, Giuseppe Marmo: Groupoid and algebra of the infinite quantum spin chain, Journal of Geometry and Physics 191 (2023) [doi:10.1016/j.geomphys.2023.104901, arXiv:2302.01050],

Proof that non-perturbative quantum observables on Yang-Mill fluxes through a surface form a convolution algebra:

• Hisham Sati, Urs Schreiber, Thm 1 in: Quantum Observables of Quantized Fluxes, Annales Henri Poincaré (2024) [doi:10.1007/s00023-024-01517-z, arXiv:2312.13037]