Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics

Abstract

By adding an imaginary interacting term proportional to ip_1p_2 to the Hamiltonian of a free anisotropic planar oscillator, we construct a new model which is described by the PT-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our PT-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positive and negative energy levels in our model. Moreover, we find that the permutation symmetry of two dimensions in our Hamiltonian corresponds to the identity (not in magnitude but in attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, and reveal that the unequal-frequency condition imposed as a prerequisite upon the Pais-Uhlenbeck oscillator can reasonably be explained as the spontaneous breaking of this identity.

16 pages, no figures; v2: 21 pages, clarifications and references added; v3: minor modification, to appear in Phys. Rev. A

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Comments on the Dynamics of the Pais-Uhlenbeck Oscillator
• Interactions of Hermitian and non-Hermitian Hamiltonians
• Imaginary-Scaling versus Indefinite-Metric Quantization of the Pais-Uhlenbeck Oscillator
• A possible method for non-Hermitian and non-$PT$-symmetric Hamiltonian systems

Cited by in corpus (4)

• Investigation of non-Hermitian Hamiltonians in the Heisenberg picture
• Investigation of PT-symmetric Hamiltonian systems from an alternative point of view
• A possible method for non-Hermitian and non-$PT$-symmetric Hamiltonian systems
• Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions