American Institute of Mathematical Sciences

September  2021, 13(3): 333-354. doi: 10.3934/jgm.2021008

Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

 1 Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany 2 Departamento de Matematicas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain 3 Instituto de Ciencias Matemáticas (CSIC - UAM - UC3M - UCM) ICMAT, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain 4 Dipartimento di Fisica E.Pancini, Universitá degli Studi di Napoli Federico II, Complesso Universitario di Monte S.Angelo, via Cintia, 80126 Napoli, Italy 5 INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, via Cintia, 80126 Napoli, Italy 6 Departamento de Matematicas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain 7 Dipartimento di Matematica e Applicazioni R.Caccioppoli, Universitá degli Studi di Napoli Federico II, Complesso Universitario di Monte S.Angelo, via Cintia, 80126 Napoli, Italy

* Corresponding author

Received  December 2020 Published  September 2021 Early access  May 2021

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced. It is shown that, given a groupoid $G\rightrightarrows \Omega$ associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global". The microscopic point of view leads to the introduction of an additional layer over the grupoid $G$, giving rise to a suitable algebraic structure of 2-groupoid. On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid. The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.

Citation: 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, 2021, 13 (3) : 333-354. doi: 10.3934/jgm.2021008
References:
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References:
 [1] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343-362. [2] M. R. Buneci, Groupoid $C^*$-algebras, Surveys in Mathematics and its Applications, 1 (2006), 71-98. [3] 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 (2018), 1850122. doi: 10.1142/S0217732318501225. [4] 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 (2019), 1950119. doi: 10.1142/S0219887819501196. [5] 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 (2019), 1950136. doi: 10.1142/S0219887819501366. [6] 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 (2019), 1950165. doi: 10.1142/s0219887819501652. [7] 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 (2020) 2050058. doi: 10.1142/S0219887820500589. [8] 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 (2020), 2050054. doi: 10.1142/S0219887820500541. [9] R. P. Feynman and L. M. Brown, Feynman's Thesis: A New Approach to Quantum Theory, World Scientific, Singapore, 2005. [10] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61458-3. [11] P. Hahn, Haar measure for measure groupoids, Trans. Amer. Math.Soc., 242 (1978), 1-33.  doi: 10.1090/S0002-9947-1978-0496796-6. [12] A. Ibort and M. A. Rodriguez, An Introduction to the Theory of Groups, Groupoids and Their Representations, CRC Press, Boca Raton, 2019.  doi: 10.1201/b22019. [13] I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin Heidelberg, 1993. doi: 10.1007/978-3-662-02950-3. [14] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-1680-3. [15] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883. [16] G. W. Mackey, Ergodic theory, group theory and differential geometry, Proc. Nat. Acad. Sci. USA, 50 (1963), 1184-1191.  doi: 10.1073/pnas.50.6.1184. [17] G. W. Mackey, Ergodic theory and virtual groups, Math. Ann., 166 (1966), 187-207.  doi: 10.1007/BF01361167. [18] F. J. Murray and J. Von Neumann, On rings of operators, Ann. Math., 37 (1936), 116-229.  doi: 10.2307/1968693. [19] J. Renault, A Groupoid Approach to $C^{\star }-$Algebras, Springer-Verlag, Berlin, 1980. [20] E. Riehl, Categorical Homotopy Theory, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781107261457. [21] J. Schwinger, Quantum Kinematics and Dynamics, CRC Press, Boca Raton, 2000. [22] J. Schwinger, The theory of quantized fields. I, Physical Reviews, 82 (1951), 914-927.  doi: 10.1103/PhysRev.82.914. [23] R. D. Sorkin, Quantum mechanics as quantum measure theory, Modern Physics Letters A, 9 (1994), 3119-3127.  doi: 10.1142/S021773239400294X. [24] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, Berlin, 2002. [25] A. Weinstein, Groupoids: Unifying internal and external symmetry, Notices of the AMS, 43 (1996), 744–752. [26] E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, London, 1959.
A schematic representation of a bisection: the red line connects the elements of the subset $b_1\subset G$ whilst the dotted arrows represent the map $b_s,b_t,s\mid_b,t\mid_b$. The red arrows denotes the bijective maps $\varphi_{b_1}$ associated with the bisection $b_1$
Schematic representation of the group of bisections $\mathscr{G}$ of the groupoid $C_2(4)$
Multiplication table of the group $\mathscr{G}$ of bisections of the groupoid $C_2(4)\,\rightrightarrows \,\Omega_2$
 0 $b_e$ $b_+$ $b_-$ $b_g$ $b_1$ $b_2$ $b_3$ $b_4$ $b_e$ $b_e$ $b_+$ $b_-$ $b_g$ $b_1$ $b_2$ $b_3$ $b_4$ $b_+$ $b_+$ $b_e$ $b_g$ $b_-$ $b_3$ $b_4$ $b_1$ $b_2$ $b_-$ $b_-$ $b_g$ $b_e$ $b_+$ $b_2$ $b_1$ $b_4$ $b_3$ $b_g$ $b_g$ $b_-$ $b_+$ $b_e$ $b_4$ $b_3$ $b_2$ $b_1$ $b_1$ $b_1$ $b_2$ $b_3$ $b_4$ $b_e$ $b_+$ $b_-$ $b_g$ $b_2$ $b_2$ $b_1$ $b_4$ $b_3$ $b_-$ $b_g$ $b_e$ $b_+$ $b_3$ $b_3$ $b_4$ $b_1$ $b_2$ $b_+$ $b_e$ $b_g$ $b_-$ $b_4$ $b_4$ $b_3$ $b_2$ $b_1$ $b_g$ $b_-$ $b_+$ $b_e$
 0 $b_e$ $b_+$ $b_-$ $b_g$ $b_1$ $b_2$ $b_3$ $b_4$ $b_e$ $b_e$ $b_+$ $b_-$ $b_g$ $b_1$ $b_2$ $b_3$ $b_4$ $b_+$ $b_+$ $b_e$ $b_g$ $b_-$ $b_3$ $b_4$ $b_1$ $b_2$ $b_-$ $b_-$ $b_g$ $b_e$ $b_+$ $b_2$ $b_1$ $b_4$ $b_3$ $b_g$ $b_g$ $b_-$ $b_+$ $b_e$ $b_4$ $b_3$ $b_2$ $b_1$ $b_1$ $b_1$ $b_2$ $b_3$ $b_4$ $b_e$ $b_+$ $b_-$ $b_g$ $b_2$ $b_2$ $b_1$ $b_4$ $b_3$ $b_-$ $b_g$ $b_e$ $b_+$ $b_3$ $b_3$ $b_4$ $b_1$ $b_2$ $b_+$ $b_e$ $b_g$ $b_-$ $b_4$ $b_4$ $b_3$ $b_2$ $b_1$ $b_g$ $b_-$ $b_+$ $b_e$
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