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Physica A Statistical Mechanics and its Applications

A quantum-like model of political consensus via non self-adjoint Hamiltonians

Fabio Bagarello, Gloria Liarda

Published Jun 20, 2026• 0 citations
Publisher

Abstract

We discuss here how non self-adjoint Hamiltonians, and their related Heisenberg-like dynamics, can be used to model a political system consisting in a coalition C of different parties (forming a government) and by their (original) supporters S . Our aim is to model how the opinion of these supporters changes depending on the efficiency, competence and coherence of the coalition C , as these are perceived by S during their action while governing. After a rather general introduction we propose three specific models, and we describe and comment the dynamical behaviour of the full system, S ∪ C . The role of the so-called balanced Hamiltonians , recently introduced by the authors in connection with integrals of motion, is discussed in details.

DOI
10.1016/j.physa.2026.131777

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We discuss here how non self-adjoint Hamiltonians, and their related Heisenberg-like dynamics, can be used to model a political system consisting in a coalition C of different parties (forming a government) and by their (original) supporters S . Our aim is to model how the opinion of these supporters changes depending on the efficiency, competence and coherence of the coalition C , as these are perceived by S during their action while governing. After a rather general introduction we propose three specific models, and we describe and comment the dynamical behaviour of the full system, S ∪ C . The role of the so-called balanced Hamiltonians , recently introduced by the authors in connection with integrals of motion, is discussed in details.
We discuss here how non self-adjoint Hamiltonians, and their related Heisenberg-like dynamics, can be used to model a political system consisting in a coalition $\C$ of different parties (forming a government) and by their (original) supporters $\Sc$. Our aim is to model how the opinion of these supporters changes depending on the efficiency, competence and coherence of the coalition $\C$, as these are perceived by $\Sc$ during their action while governing. After a rather general introduction we propose three specific models, and we describe and comment the dynamical behaviour of the { full} system, $\Sc\cup\C$. The role of the so-called {\em balanced Hamiltonians}, recently introduced by the authors in connection with integrals of motion, is discussed in details.
In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.
Abstract In a previous paper, we began our analysis on the role of non-self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are brute-force normalized . Our main interest is on conserved quantities and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.
Classical models of opinion dynamics represent individual opinions as scalar or vector values governed by the classical probability theory, either as deterministic quantities or random variables. This framework does not account for empirically observed phenomena such as cognitive ambivalence (where an individual simultaneously holds conflicting views) and order effects (where survey responses depend on the order in which questions are asked). We propose a quantum model of opinion dynamics in which each agent's cognitive state is represented by a density matrix that encodes both the expressed opinion and cognitive ambivalence. Survey questions become non-commuting self-adjoint operators, which provides a principled explanation for order effects. Our model also identifies quantities without classical counterparts, including quantum coherence and pairwise opinion covariances. Under a product state approximation, the quantum model reduces to the classical Friedkin--Johnsen opinion model. We test the framework on synthetic and real-world networks and observe that pairwise correlations follow network-dependent transient dynamics but converge to the same steady state regardless of the network, and that quantum coherence decays exponentially at a rate independent of the network.