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:
[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. 

show all references

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. 
Figure 1.  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 $
Figure 2.  Schematic representation of the group of bisections $ \mathscr{G} $ of the groupoid $ C_2(4) $
Table 1.  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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