Waves in Open Systems via Bi-orthogonal Basis

TL;DR

This paper demonstrates that wave dynamics in open dissipative systems can be exactly described using a bi-orthogonal basis framework, extending traditional quantum formalism to non-hermitian Hamiltonians with finite norms.

Contribution

It provides a rigorous foundation for modeling open system wave dynamics with bi-orthogonal bases, including a finite-norm inner product and extensions of quantum formalism.

Findings

Exact representation of open system dynamics using bi-orthogonal basis

Finite norms and inner products without regularization

Extension of perturbation theory and second-quantization to open systems

Abstract

Dissipative quantum systems are sometimes phenomenologically described in terms of a non-hermitian hamiltonian $H$, with different left and right eigenvectors forming a bi-orthogonal basis. It is shown that the dynamics of waves in open systems can be cast exactly into this form, thus providing a well-founded realization of the phenomenological description and at the same time placing these open systems into a well-known framework. The formalism leads to a generalization of norms and inner products for open systems, which in contrast to earlier works is finite without the need for regularization. The inner product allows transcription of much of the formalism for conservative systems, including perturbation theory and second-quantization.