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arXiv:2512.07809

Nonlinear Quantum Mechanics and Artificial Intelligence

Jonathan Oppenheim

hep-thquant-phSubmitted Dec 8, 2025• Announced Dec 9, 2025• 0 citations
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Abstract

We examine a criterion for relativistic covariance of nonlinear quantum field theory recently proposed by GPT-5 and published in Physics Letters B. We show that this criterion inadvertently tests a different property -- locality of the Hamiltonian -- and is insensitive to whether the theory is nonlinear. We recall the correct criterion, identified by Gisin and Polchinski thirty-five years ago, and reformulate their result in field-theoretic language.

Comments
2 pages, comments welcome

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We examine a criterion for relativistic covariance of nonlinear quantum field theory recently proposed by GPT-5 and published in Physics Letters B. We show that this criterion inadvertently tests a different property -- locality of the Hamiltonian -- and is insensitive to whether the theory is nonlinear. We recall the correct criterion, identified by Gisin and Polchinski thirty-five years ago, and reformulate their result in field-theoretic language.
We use the Tomonaga-Schwinger (TS) formulation of quantum field theory to determine when state-dependent additions to the local Hamiltonian density (i.e., modifications to linear Schrodinger evolution) violate relativistic covariance. We derive new operator integrability conditions required for foliation independence, including the Frechet derivative terms that arise from state-dependence. Nonlinear modifications of quantum mechanics affect operator relations at spacelike separation, leading to violation of the integrability conditions.
We use the Tomonaga–Schwinger (TS) formulation of quantum field theory to determine when state-dependent additions to the local Hamiltonian density (i.e., modifications to linear Schrödinger evolution) violate relativistic covariance. We derive new operator integrability conditions required for foliation independence, including the Fréchet derivative terms that arise from state-dependence. Nonlinear modifications of quantum mechanics affect operator relations at spacelike separation, leading to violation of the integrability conditions.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a naïve approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detection events and explain how particles can be regarded as emergent phenomena of a relativistic field theory. In particular, I show that effective localization of Hilbert space vectors is equivalent to an approximate locality of observable algebras.