Abstract
The first problems in this chapter deal with basic properties of groups and of group representations. Fundamental results following from Schur lemma are introduced since the beginning in the case of finite groups, with simple applications of character theory, in the study of vibrational levels of symmetric systems. Other problems concern the notion and properties of Lie groups and Lie algebras, mainly oriented to physical examples: rotation groups \(SO_2\), \(SO_3\), \(SU_2\), translations, Euclidean group, Lorentz transformations, dilations, Heisenberg group, \(SU_3\), with their physically relevant representations. The last Section starts with some examples and applications of symmetry properties of differential equations, provides a group-theoretical interpretation of the Zeeman and Stark effects, and finally is devoted to obtaining the symmetry properties of the hydrogen atom (the group \(SO_4\)) and of the 3-dimensional harmonic oscillator (the group \(U_3\)) in quantum mechanics.
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Notes
- 1.
For any fixed \(g\in G\), the coset gH is the set \(\{gh,\,h\in H\}\).
- 2.
It is customary in physics to introduce a factor i in the definition of these operators, in order to have Hermitian operators. E.g., \(A=-i\,d/dx\), which is proportional to the momentum operator \(P=-i\,\hbar d/dx\) in quantum mechanics, as well-known.
- 3.
\(\mathscr{L}\) is the subgroup “connected with the identity” of the full Lorentz group, usually denoted by O(3, 1), which includes also space inversions and time reversal. The same remark holds for the group considered in q.(2) (ii), which is a subgroup of O(2, 1).
- 4.
It can be useful to point out that, differently from all the groups \(SU_n\) with \(n>2\), the “basic” irreducible representations \(\mathscr{R}\) and \(\mathscr{R}^*\) of \(SU_2\), by means of \(2\times 2\) unitary matrices, are equivalent.
- 5.
the equation indeed is left invariant under the transformations of the group, which is called the symmetry group of the equation; symmetric and invariant are synonymous in this context.
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Cicogna, G. (2020). Groups, Lie Algebras, Symmetries in Physics. In: Exercises and Problems in Mathematical Methods of Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-59472-5_4
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DOI: https://doi.org/10.1007/978-3-030-59472-5_4
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