A multiplet analysis of spectra in the presence of broken symmetriesA multiplet analysis of spectra in the presence of broken symmetries
Faculty of Sciences. Physics

Mathematical Physics

article

2012Bristol, 2012

Physics

Journal of physics : conference series. - Bristol

343(2012), p. 1-14

1742-6588

1742-6596

012084

E-only publicatie

English (eng)

University of Antwerp

We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R. In Theorem 1 these generalised symmetries are characterised in terms of repeated commutators of H with M. Breaking supersymmetry by adding a term linear in the supercharges is discussed as a motivating example. The complex parameter γ which appears in the definition of a generalised symmetry is necessarily real when the spectrum of M is discrete. Theorem 2 shows that γ must also be real when the spectrum of H is fully discrete and R and R are bounded operators. Any generalised symmetry induces a partitioning of the spectrum of H in what we call M-multiplets. The hydrogen atom in the presence of a symmetry breaking external field is discussed as an example. The notion of stability of eigenvectors of H relative to the generalised symmetry M is discussed. A characterisation of stable eigenvectors is given in Theorem 3.

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