Abstract
In this short chapter we discuss the implementation of symmetries in a quantum-mechanical context.
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- 1.
Recall that a metric space is complete if any Cauchy sequence converges.
- 2.
We will not take into account issues related to the domains of operators.
- 3.
Throughout this thesis representations of groups are denoted by the letters \({\mathcal R}\), \({\mathcal S}\), \({\mathcal T}\), etc. The letter G denotes a group whose elements are written f, g, h, etc. The identity in G is denoted e.
- 4.
Beware: a manifold being multiply connected means that it has a non-trivial fundamental group, and not that it has several connected components.
- 5.
Throughout this thesis the elements of a Lie algebra \(\mathfrak {g}\) will be denoted as X, Y, etc. Representations of Lie algebras will be denoted by script capital letters such as \(\mathscr {R}\), \(\mathscr {S}\), \(\mathscr {T}\).
- 6.
Cochains on Lie algebras will be denoted by lowercase sans serif letters such as \(\mathsf {c}\), \(\mathsf {s}\), etc.
- 7.
It is understood that the relevant representation of \(\mathfrak {g}\) in this case is the adjoint, \(\mathscr {T}[X]\cdot Y\equiv [X,Y]\).
- 8.
Cochains on a group will be denoted by capital sans serif symbols such as \(\mathsf {C}\), \(\mathsf {S}\), etc.
References
S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995)
V. Ovsienko, S. Tabachnikov, Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups (Cambridge University Press, Cambridge Tracts in Mathematics, 2004)
B. Khesin, R. Wendt, The Geometry of Infinite-Dimensional Groups (A series of modern surveys in mathematics. Springer, Berlin Heidelberg, 2008)
L. Guieu, C. Roger, L’algèbre et le groupe de Virasoro (Publications du CRM, Université de Montréal, 2007)
G. Tuynman, W. Wiegerinck, Central extensions and physics. J. Geom. Phys. 4(2), 207–258 (1987)
A. Weinstein, Groupoids: unifying internal and external symmetry - a tour through some examples. Not. AMS 43, 744–752 (1996)
M. Crainic, R.L. Fernandes, “Lectures on integrability of Lie brackets,” Lectures on Poisson Geometry. Geom. Topol. Monogr. 17, 1–107 (2011)
G. Barnich, A Note on gauge systems from the point of view of Lie algebroids, AIP Conf. Proc. 1307 7–18 (2010), arXiv: 1010.0899
G. Barnich, C. Troessaert,. BMS charge algebra. JHEP 12, p. 105 (2011), arXiv: 1106.0213
E. P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. (Springer, 1931)
M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley Publishing Company, Advanced book classics, 1995)
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Oblak, B. (2017). Quantum Mechanics and Central Extensions. In: BMS Particles in Three Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61878-4_2
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DOI: https://doi.org/10.1007/978-3-319-61878-4_2
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