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Foundations of Physics

Non-relativistic Twistor Theory: Newtonian Limits and Gravitationally Induced Collapse

Eleanor March, James Alexander Mabyn Read

Published Jul 13, 2026• 0 citations
Publisher

Abstract

Abstract Recently, Dunajski and Gundry (Commun. Math. Phys., 342 (3), 2016) have developed a version of twistor theory for the non-relativistic domain. Unlike relativistic twistor theory, their approach is able to reproduce the entire space of models of Newton–Cartan theory. We critically assess the significance of non-relativistic twistors, in particular with respect to proposals by Dunajski and Penrose (Ann. Phys., 451 , 2023) that using non-relativistic twistors to describe gravitationally induced collapse could play a part in solving the quantum measurement problem.

DOI
10.1007/s10701-026-00940-y

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Abstract Recently, Dunajski and Gundry (Commun. Math. Phys., 342 (3), 2016) have developed a version of twistor theory for the non-relativistic domain. Unlike relativistic twistor theory, their approach is able to reproduce the entire space of models of Newton–Cartan theory. We critically assess the significance of non-relativistic twistors, in particular with respect to proposals by Dunajski and Penrose (Ann. Phys., 451 , 2023) that using non-relativistic twistors to describe gravitationally induced collapse could play a part in solving the quantum measurement problem.
Quantum-gravity-induced entanglement of massive systems (QGEM) is commonly approximated in the nonrelativistic static limit by a Newtonian interaction between spatially separated masses. In this work, we reformulate the gravitationally mediated interaction phase in a conformally invariant twistor framework in which no notion of spacetime distance is assumed. We show that the bilocal phase responsible for entanglement generation remains well-defined and non-factorizable even in the absence of spacetime geometry. The familiar Newtonian 1/r1/r1/r phase, relevant for QGEM protocols, arises only after the conformal invariance is broken by introducing the infinity twistor, which selects a particular spacetime representation of the underlying bilocal quantum interaction. Our results isolate the genuinely quantum content of QGEM protocols and clarify the contingent role played by spacetime geometry in mediating entanglement.
Non-relativistic objective collapse theories have been remarkably successful in addressing the conceptual problems of standard quantum mechanics. Despite substantial efforts, the project of extending them to the relativistic domain remains burdened by significant conceptual objections and technical challenges, often taken to cast doubt on the viability of the program as a whole. On the conceptual side, relativistic collapse theories have been claimed to face challenges involving tension between instantaneous collapse and relativity, frame-dependence of property values and probabilities, the possibility of superluminal signaling and the failure of narratability. On the technical side, persistent infinities, difficulties in constructing fully covariant frameworks and the apparent need for non-standard fields have hindered the development of workable models. In this paper, we offer a systematic rebuttal of the conceptual objections and provide a structured account of the remaining technical challenges. We conclude that relativistic collapse theories do provide a promising route toward a fully relativistic quantum framework that overcomes the conceptual limitations of standard quantum theory.
This is the first in a two-part series in which we extend non-relativistic stochastic mechanics, in the ZSM formulation [1, 2], to semiclassical Newtonian gravity (ZSM-Newton) and semiclassical Newtonian electrodynamics (ZSM-Coulomb), under the assumption that the gravitational and electromagnetic fields are fundamentally classical (i.e., not independently quantized fields). Our key findings are: (1) a derivation of the usual NNN-particle Schr{ö}dinger equation for many particles interacting through operator-valued gravitational or Coulomb potentials, and (2) recovery of the `single-body' Schr{ö}dinger-Newton and Schr{ö}dinger-Coulomb equations as mean-field equations valid for systems of gravitationally and electrostatically interacting identical particles, respectively, in the weak-coupling large N limit. We also compare ZSM-Newton/Coulomb to semiclassical Newtonian gravity/electrodynamics approaches based on standard quantum theory, dynamical collapse theories, and the de Broglie-Bohm theory.
A relativistic version of the Schr{ö}dinger-Newton equation is analyzed within the recently proposed Grave de Peralta approach [L. Grave de Peralta, {\em Results Phys.} {\bf 18} (2020) 103318], which include relativistic effects by a parametrization of the non-relativistic hamiltonian, so as to impose that the average kinetic energy of the system coincide with its relativistic kinetic energy. The reliability of this method is tested for the particle in a box. By applying this method to the Schr{ö}dinger-Newton equation we shows that the characteristic length of the model [L. Di{ó}si, {\em Phys. Lett}. {\bf 105A} (1984) 199] goes to zero for a mass of the order of the Planck mass, suggesting a collapse of the wavefuncton, induced by gravity.
An elementary prediction of the quantization of the gravitational field is that the Newtonian interaction can entangle pairs of massive objects. Conversely, in models of gravity in which the field is not quantized, the gravitational interaction necessarily comes with some level of noise, i.e., non-reversibility. Here, we give a systematic classification of all possible such models consistent with the basic requirements that the non-relativistic limit is Galilean invariant and reproduces the Newtonian interaction on average. We demonstrate that for any such model to be non-entangling, a quantifiable, minimal amount of noise must be injected into any experimental system. Thus, measuring gravitating systems at noise levels below this threshold would be equivalent to demonstrating that Newtonian gravity is entangling. As concrete examples, we analyze our general predictions in a number of experimental setups, and test it on the classical-quantum gravity models of Oppenheim et al., as well as on a recent model of Newtonian gravity as an entropic force.