Quantum mechanical settings inspired by RLC circuits

TL;DR

This paper reveals that simple RLC circuits can be analyzed using quantum mechanical concepts, uncovering biorthogonal bases, gain-loss relations, and pseudo-fermionic structures, thus bridging classical circuits and quantum physics.

Contribution

It introduces a novel quantum-inspired analysis of RLC circuits, including biorthogonal bases, gain-loss relations, and the concept of m-equivalence between circuits.

Findings

RLC circuits produce biorthogonal bases linked to their parameters

Loss and gain RLC circuits are naturally related through the Liouville matrix

A pseudo-fermionic framework is applicable to circuit analysis

Abstract

In some recent papers several authors used electronic circuits to construct loss and gain systems. This is particularly interesting in the context of PT-quantum mechanics, where this kind of effects appears quite naturally. The electronic circuits used so far are simple, but not so much. Surprisingly enough, a rather trivial RLC circuit can be analyzed with the same perspective and it produces a variety of unexpected results, both from a mathematical and on a physical side. In this paper we show that this circuit produces two biorthogonal bases associated to the Liouville matrix $\Lc$ used in the treatment of its dynamics, with a biorthogonality which is linked to the value of the parameters of the circuit. We also show that the related loss RLC circuit is naturally associated to a gain RLC circuit, and that the relation between the two is rather naturally encoded in $\Lc$. We propose a pseudo-fermionic analysis of the circuit, and we introduce the notion of $m$-equivalence between electronic circuits.