PT symmetry and a dynamical realization of the SU(1, 1) algebra

Abstract

We show that the elementary modes of the planar harmonic oscillator can be quantized in the framework of quantum mechanics based on pseudo-hermitian Hamiltonians. These quantized modes are demonstrated to act as dynamical structures behind a new Jordan-Schwinger realization of the SU(1, 1) algebra. This analysis complements the conventional Jordan-Schwinger construction of the SU(2) algebra based on hermitian Hamiltonians of a doublet of oscillators.Traditionally quantum mechanics was formulated with hermitian Hamiltonians which ensures real eigenvalues, unitarity, etc. which are the expected attributes of a physically meaningful theory. The observation that non-hermitian operators may also have real eigenvalues was not considered very important in quantum theory. The situation has been drastically altered over the last decade or so. Now we definitely know that consistent quantization is possible with non-hermitian Hamiltonians if the theory is P T symmetric. 1,2 The corresponding Hamiltonians satisfy η −1 H † η = H, where η = P T . a Such operators will henceforth be called η-hermitian. P T symmetric theories are not only of theoretical interest now a days; such theories have been identified in experimental systems in diverse fields e.g. in superconductivity, quantum optics etc. 3 Naturally, study of P T symmetric theories has occupied a However, there exist more general types of complex Hamiltonians which may have real eigenvalues. 1650028-1 Mod. Phys. Lett. A 2016.31. Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 06/29/16. For personal use only.