The quantum Rabi model and Lie algebra representations of sl2

Masato Wakayama, Taishi Yamasaki

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of sl2(R). We define a second order element of the universal enveloping algebra u( sl2) of sl2(R), which, through the image of a principal series representation of sl2(R), provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Judd's eigenstates) of the Rabi Hamiltonian due to Kus̈ in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of sl2(R).

Original languageEnglish
Article number335203
JournalJournal of Physics A: Mathematical and Theoretical
Issue number33
Publication statusPublished - 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • General Physics and Astronomy


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