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A description of pseudo-bosons in terms of nilpotent Lie algebras

Francesco G. RussoFabio BagarelloFabio Bagarello

subject

Pure mathematicsSwanson modelDynamical systems theoryLie algebraStructure (category theory)FOS: Physical sciencesGeneral Physics and AstronomyContext (language use)01 natural sciencesPhysics and Astronomy (all)Pseudo-bosonic operator0103 physical sciencesLie algebraMathematical Physic0101 mathematicsAbelian group010306 general physicsQuantumSettore MAT/07 - Fisica MatematicaMathematical PhysicsMathematicsQuantum PhysicsSchur multiplier010102 general mathematicsHilbert spaceMathematical Physics (math-ph)NilpotentLadder operatorGeometry and TopologyQuantum Physics (quant-ph)

description

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

yearjournalcountryeditionlanguage
2017-12-13
10.1016/j.geomphys.2017.12.002http://arxiv.org/abs/1712.04631