Hermiticity of the Hamiltonian is a foundational assumption of standard quantum mechanics, guaranteeing real eigenvalues, unitary evolution, and a consistent probability interpretation. We argue that this assumption, while epistemically indispensable in the low-energy limit, may not be ontologically fundamental. Working within the Quantum Reality Framework (QRF), in which physical structure is derived from a causal set (E, ≺, A), we propose to identify the relational amplitude kernel A(x,y) with the Benincasa–Dowker operator and suggest a route toward recovering aspects of the kinematical structure of quantum theory — Hilbert space, dispersion relation, and momentum operator — in the continuum limit. A fundamental tension then arises: the BD kernel is intrinsically time-directed and non-symmetric, while quantum mechanics requires a Hermitian kernel. We propose that Hermiticity is an emergent approximate symmetry valid in the limit E ≪ E_P, and that the appropriate mathematical framework is pseudo-Hermitian quantum mechanics (Bender & Boettcher 1998; Mostafazadeh 2002). This leads to potentially falsifiable consequences, partly model-dependent in their precise form: a qubit/qutrit decoherence ratio τ(qubit)/τ(qutrit) = 3/5 in a minimal implementation, quadratic Lorentz invariance violation δv/v ∼ −(E/E_P)²/6, mass-dependent deviations in matter-wave interference, and possible CPT violation at order (m/m_P)². Keywords: causal sets · quantum foundations · Hermiticity · PT-symmetry · time asymmetry · emergent quantum mechanics
Hermiticity of the Hamiltonian is a foundational assumption of standard quantum mechanics, guaranteeing real eigenvalues, unitary evolution, and a consistent probability interpretation. We argue that this assumption, while epistemically indispensable in the low-energy limit, may not be ontologically fundamental. Working within the Quantum Reality Framework (QRF), in which physical structure is derived from a causal set (E, ≺, A), we propose to identify the relational amplitude kernel A(x,y) with the Benincasa–Dowker operator and suggest a route toward recovering aspects of the kinematical structure of quantum theory — Hilbert space, dispersion relation, and momentum operator — in the continuum limit. A fundamental tension then arises: the BD kernel is intrinsically time-directed and non-symmetric, while quantum mechanics requires a Hermitian kernel. We propose that Hermiticity is an emergent approximate symmetry valid in the limit E ≪ E_P, and that the appropriate mathematical framework is pseudo-Hermitian quantum mechanics (Bender & Boettcher 1998; Mostafazadeh 2002). This leads to potentially falsifiable consequences, partly model-dependent in their precise form: a qubit/qutrit decoherence ratio τ(qubit)/τ(qutrit) = 3/5 in a minimal implementation, quadratic Lorentz invariance violation δv/v ∼ −(E/E_P)²/6, mass-dependent deviations in matter-wave interference, and possible CPT violation at order (m/m_P)². Keywords: causal sets · quantum foundations · Hermiticity · PT-symmetry · time asymmetry · emergent quantum mechanics
Non-unitary dynamics in quantum mechanics arise in open quantum systems, where the interaction between the system and its environment is not negligible. One way this coupling can be realised is through measurement, where the backaction from the measurement apparatus induces non-unitary dynamics in the system. An effective non-Hermitian description arises within the no-jump approximation of a measured system, in which the time-evolved states are explicitly normalised at all times. Non-Hermiticity in quantum Hamiltonians gives rise to non-unitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this thesis, we explore these two interconnected concepts related to non-unitary dynamics: non-Hermiticity and measurements. However, instead of effective non-Hermiticity, we consider the cases where non-Hermiticity is realised exactly. This setup necessitates a treatment under the self-consistent frameworks of non-Hermitian quantum mechanics in order to restore its probabilistic interpretation. Concretely, the self-consistent metric formalism restores unitarity of the evolution through a modified Hilbert space endowed with a dynamical metric. Through comparative studies using exactly solvable models, we find that the physics of the two approaches of non-Hermitian quantum mechanics is fundamentally different. This has a profound physical impact in terms of defect production in quenches across an exceptional point (EP). We demonstrate this through a non-Hermitian system hosting both PT-symmetric and PT-broken modes subject to a linear quench. Our results highlight the role played by the EP and the spontaneous breaking of the PT-symmetry in contributing to a non-zero defect density, even in the limit of an infinitely slow drive. Further, we uncover generalised transformations underpinning the results, which are incorporated in the dynamical metric. Going beyond purely theoretical considerations, we implement two novel dilation schemes to emulate the metric dynamics in a digital quantum simulator. Our protocols enable controlled access to all PT-symmetry regimes and to a direct tomography of the dynamical metric, opening the possibility of studying a new kind of non-Hermitian physics in engineered matter. To explore the second theme of the thesis, we investigate the dynamics of entanglement entropy in the continuously monitored Bose-Hubbard model. Our numerical data reveals a counterintuitive enhancement of superfluid correlations in a parameter regime, in which the entanglement entropy increases with the onsite repulsion strength in the presence of measurement. This phenomenon arising from the interplay between hopping, interaction, and measurement, is also present in the effective central charge extracted from a fit of the logarithmic scaling form of the entropy. In summary, this thesis reveals the richness of non-unitary quantum dynamics. Through an in-depth study of the different aspects of non-Hermitian systems, we illustrate the non-trivial physics uncovered only with the consideration of the dynamical metric. By bridging theoretical insights and physical realisation through novel dilation protocols realisable in engineered matter, our work lays the groundwork for further exploration of dynamical aspects in non-Hermitian and monitored quantum systems.
Abstract PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
This article is the first part of a three-part series developing Ontological Field Theory (OFT), a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a Hamiltonian, one writes H = H †, where the symbol † denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate. In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space – time reflection symmetry (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian 𝒫𝒯-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian 𝒫𝒯-symmetric quantum theories.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.
This work presents a rigorous, first-principles audit of unitarity in quantum mechanics, treating unitarity not as a foundational axiom but as a hypothesis whose validity depends on explicit physical and mathematical assumptions. The analysis is conducted entirely within the operational framework of open quantum systems, using completely positive trace-preserving (CPTP) maps to describe the dynamics of reduced subsystems. By systematically isolating and auditing every prerequisite required for unitary evolution—including perfect isolation, Hilbert-space factorization, linearity, the Born rule, and time parameterization—the manuscript demonstrates that unitarity is neither fundamental nor universal, but instead an emergent, scale-dependent consistency condition. The paper distinguishes clearly between mathematical bookkeeping identities and physically meaningful conservation laws, showing that probability conservation does not require unitarity, and that reversibility and energy conservation are not guaranteed by unitary evolution alone. Explicit, mathematically consistent non-unitary evolution models are constructed, including Lindblad dynamics, non-Hermitian Hamiltonians, stochastic Schrödinger equations, non-Markovian evolution, and quantum Darwinism. The work further identifies the precise regimes in which unitary dynamics emerges as an effective approximation—such as weak coupling, short timescales, or limited observational resolution—and delineates the conditions under which it fails. Empirical evidence from atomic spectroscopy, quantum information processing, decoherence experiments, and measurement theory is used to anchor the analysis without assuming fundamental unitarity. The central conclusion is unambiguous: no physically accessible open or effective subsystem undergoes strictly unitary evolution. Unitarity is shown to be an emergent, conditional, and scale-dependent property rather than a fundamental law of nature. This manuscript is intended as a foundational clarification of the status of unitarity in quantum theory and is suitable for researchers in quantum foundations, open quantum systems, quantum information, and quantum gravity. FUSION OS INNOVATIONS