Two recent arguments for linear dynamics in quantum theory are critically re-examined. Neither argument is found to be satisfactory as it stands, although an improved version of one of the arguments can in fact be given. This improved version turns out to be still not completely unproblematic, but it is argued that it contains only a single actual loophole, which is identical to a loophole that remains in experimental proofs of nonlocality of Bell-type. It is concluded that - within the context of the standard quantum kinematical framework and in agreement with what has been concluded by earlier authors - a nonlinear dynamics of density operators is inconsistent with relativistic causality. However, it is also stressed that this conclusion in itself has little implication for the nature of dynamics at the Hilbert space level - in particular, it does not force dynamics to be linear at this level - nor does it continue to be valid in contexts that go beyond the standard quantum kinematical framework. Despite their seeming triviality, these last two points have not always been appreciated in the literature. Finally, it is also pointed out that the argument for complete positivity, as given in conjunction with one of the two recent arguments for linear dynamics, in fact only establishes a condition that is weaker than complete positivity.
Two recent arguments for linear dynamics in quantum theory are critically re-examined. Neither argument is found to be satisfactory as it stands, although an improved version of one of the arguments can in fact be given. This improved version turns out to be still not completely unproblematic, but it is argued that it contains only a single actual loophole, which is identical to a loophole that remains in experimental proofs of nonlocality of Bell-type. It is concluded that - within the context of the standard quantum kinematical framework and in agreement with what has been concluded by earlier authors - a nonlinear dynamics of density operators is inconsistent with relativistic causality. However, it is also stressed that this conclusion in itself has little implication for the nature of dynamics at the Hilbert space level - in particular, it does not force dynamics to be linear at this level - nor does it continue to be valid in contexts that go beyond the standard quantum kinematical framework. Despite their seeming triviality, these last two points have not always been appreciated in the literature. Finally, it is also pointed out that the argument for complete positivity, as given in conjunction with one of the two recent arguments for linear dynamics, in fact only establishes a condition that is weaker than complete positivity.
Despite their Nobel prize-winning empirical confirmation, the interpretation of the Bell inequality remains controversial. By carefully analyzing Bell's work on nonlocality, we show how the correct appreciation of Bell's rationale calls for reformulating a widespread argument on quantum nonlocality, yielding a more consistent formulation of the problem. We delve into a formal proof of quantum mechanics' violation of local causality strikingly overlooked in favor of less appropriate arguments. For completeness, we also mention a few alternatives that may rationally justify considering quantum mechanics as a local theory.
Quantum dynamics is linear. How do we know? From theory or experiment? The history of this question is reviewed. Nonlinear generalizations of quantum mechanics have been proposed. They predict small but clear nonlinear effects, which very accurate experiments have not seen. Is there a reason in principle why nonlinearity is not found? Is it impossible? Does quantum dynamics have to be linear? Attempts to prove this have not been decisive, because either their assumptions are not compelling or their arguments are not conclusive. The question has been left unsettled. There is a simple answer, based on a simple assumption. It was found in two steps separated by 44 years. They are steps back to simpler and more compelling assumptions. A proof of the assumptions of the Wigner-Bargmann proof has been known since 1962. It assumes that the maps of density matrices in time are linear. For this step, it is also assumed that density matrices are mapped one-to-one onto density matrices. An alternative is to assume that pure states are mapped one-to-one onto pure states and that entropy does not decrease. In a step taken in 2006, it is proved that the maps of density matrices in time are linear. It is assumed, as in the earlier step, that at each time the physical quantities and states are described by the usual linear structures of quantum mechanics, so the question is only about how things change in time. Beyond that, the proof assumes only that the dynamics does not depend on anything outside the system, but must allow the system to be described as part of a larger system.
J.S. Bell's remarkable 1964 theorem showed that any theory sharing the empirical predictions of orthodox quantum mechanics would have to exhibit a surprising – and, from the point of view of relativity theory, very troubling – kind of nonlocality. Unfortunately, even still on this 50th anniversary, many commentators and textbook authors continue to misrepresent Bell's theorem. In particular, one continues to hear the claim that Bell's result leaves open the option of concluding either nonlocality or the failure of some unorthodox “hidden variable” (or “determinism” or “realism”) premise. This mistaken claim is often based on a failure to appreciate the role of the earlier 1935 argument of Einstein, Podolsky, and Rosen in Bell's reasoning. After briefly reviewing this situation, I turn to two alternative versions of quantum theory – the “many worlds” theory of Everett and the quantum Bayesian interpretation of Fuchs, Schack, Caves, and Mermin – that purport to provide actual counterexamples to Bell's claim that nonlocality is required to account for the empirically verified quantum predictions. After analyzing each theory's grounds for claiming to explain the EPR–Bell correlations locally, however, one can see that (despite a number of fundamental differences) the two theories share a common for-all-practical-purposes (FAPP) solipsistic character. This dramatically undermines such theories’ claims to provide a local explanation of the correlations and thus, by concretizing the ridiculous philosophical lengths to which one must go to elude Bell's own conclusion, reinforces the assertion that nonlocality really is required to coherently explain the empirical data. Introduction Fifty years ago, in 1964 [1], John Stewart Bell first proved the theorem that has become widely known as “Bell's Theorem” but that Bell himself instead referred to as the “locality inequality theorem” [2]. In Bell's own view, the theorem showed that the empirical predictions of local theories will be constrained by Bell's inequality (or as Bell himself preferred to call it, the “locality inequality”). Hence, nonlocality is a necessary feature of any theory that shares the empirical predictions of standard quantum mechanics.
This deposit gathers the complete structure-side chapter of the Reid Postulate following its acceptance by the International Journal of Quantum Foundations / Quantum Speculations on 2 May 2026. The framework proposes that the dimensions of physical description are not all the same structural kind. It names the unargued assumption that they are, the Homogeneity Postulate, and lifts it, proposing categorical separation between Type I (spatial), Type II (entropic), and Type III (observational) dimensions. Observation is articulated as a Type III index on a fiber bundle over four-dimensional spacetime, with binary fiber Ω carrying purely indexical distinction between two members of identical Lorentzian content. The Born-compatible measure inherits from Gleason's theorem on the projection lattice. Composability requires jointly describable systems to share index. A companion no-go theorem closes one class of candidate dynamics, articulating a combined kinematic and dynamic no-go boundary. The reader will encounter the framework's articulation in three layers. The Postulate body (Rev 3) and the no-go theorem (2026b) carry the framework's standing content. The synthesis layer (2026c through 2026i) articulates that content at distinct registers, including minimal structure, the structure to fact fence, Gleason compatibility, scope conditionality, algebraic conditions, Hilbert space level conditionality, and the contextuality stack, without extending the framework's claims. The post closure arc (2026j through 2026o) is the chapter's actual completion: the section articulation extension (2026j Rev 4), the synthesis register filing on joint describability failure as structural principle (2026k), the prediction note articulating the framework's structural prediction in Bell type scenarios at the level of object ontology (2026l), and the construction trilogy specifying the algebra to section map (2026m), verifying it on safe regimes (2026n), and exploring whether the framework produces quantitative content of its own (2026o, which finds the framework's reach to be qualitative structural rather than quantitative numerical). The Closing Statement (15 May 2026) marked the chapter procedurally complete. The Position Statement articulates the chapter's substantive final standing after the post closure arc produced what the chapter's articulated content required. The framework stands as a structural account of the indexical dimension of physical description, internally consistent on safe regimes, with load bearing commitments named honestly, falsification conditions stated, and open problems located and not resolved. The framework does not derive standard quantum mechanics but is consistent with it. It does not resolve the measurement problem. It does not produce new empirical predictions beyond the object ontology level. What the framework does not claim is part of what it offers. The work is offered for technical scrutiny. It adds no particles, forces, symmetries, or universes. It operates by removing an assumption rather than adding structure. Submitted for technical scrutiny, not belief.
A basic linearity of quantum dynamics, that density matrices are mapped linearly to density matrices, is proven very simply for a system that does not interact with anything else. It is assumed that at each time the physical quantities and states are described by the usual linear structures of quantum mechanics. Beyond that, the proof assumes only that the dynamics does not depend on anything outside the system but must allow the system to be described as part of a larger system. The basic linearity is linked with previously established results to complete a simple derivation of the linear Schr\"odinger equation. For this it is assumed that density matrices are mapped one-to-one onto density matrices. An alternative is to assume that pure states are mapped one-to-one onto pure states and that entropy does not decrease.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.