Relational Quantum Mechanics

• Relational Quantum Mechanics is an interpretation where quantum states are defined relative to interactions between systems rather than intrinsic properties.
• It employs standard Hilbert space mathematics to reinterpret measurement as correlated interactions, offering a fresh resolution to the traditional measurement problem.
• The framework promotes observer democracy and cross-perspective consistency, addressing paradoxes like Wigner’s friend through a network of relational events.

Relational Quantum Mechanics (RQM) is an interpretation of quantum theory that asserts physical states are not absolute properties of isolated systems, but are always defined relative to other systems with which they have interacted. Originating in the work of Carlo Rovelli (1996), RQM departs from traditional quantum interpretations by replacing the ontology of system-intrinsic wave functions with a network of interaction-dependent, observer-relative state attributions. This relational stance leads to significant revisions in the understanding of measurement, randomness, ontology, and inter-observer consistency, generating both conceptual advances and challenges.

1. Foundational Postulates and Formalism

RQM is grounded on two principal postulates: (1) every physical object, irrespective of its size or complexity, is a quantum system, precluding any fundamental split between quantum and classical domains; (2) the quantum state of a system S is never an absolute attribute, but always a property defined relative to another reference system R with which it has interacted. The state ascribed to S by R is denoted $|\psi_{S|R}\rangle$ in Hilbert space, or more generally as a density operator $\rho_{S|R}$ acting on the relevant subsystem (Gordon, 2022, Rovelli, 2017, Dorato, 2013, Rovelli, 2021, Muciño et al., 2021).

The mathematical formalism of RQM adopts the standard C*-algebraic/Hilbert space structure of quantum mechanics but systematically reinterprets every physical prediction, state assignment, and event as relative to a pair or network of interacting systems. Measurement—traditionally associated with a discontinuous projection—is reformulated as a process by which two quantum systems become correlated, with each updating its record of the other's relevant observables. After such an interaction, the recording system may attribute to the measured system a definite value for the observable—relative to itself—while another system, uninvolved in the interaction, may still assign a superposed or mixed state (Rovelli, 2017, Rovelli, 2021).

The update rule in RQM can be compactly expressed as:

$$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$

where $O$ measures observable $A$ on $S$, $P^A_{\Delta}$ projects onto the eigenspace corresponding to the outcome, and the resulting $\rho'_{S|O}$ is the relational state post-interaction (Rovelli, 2017).

2. Ontology: Events, Facts, and Relational Structure

The primitive ontological elements in RQM are quantum events or "facts," conceived as actualizations of observable values in specific interactions between pairs of systems. These events are sparse—occurring only at interactions—and always relational: e.g., the fact that a variable $V$ of $S$ has value $\rho_{S|R}$0 is meaningful only as actualized during interaction with $\rho_{S|R}$1 and written $\rho_{S|R}$2 (Rovelli, 2021, Rovelli, 2017, Dorato, 2013).

RQM thus denies the existence of an observer- and context-independent list of properties or facts for any system. No universal wave function or global quantum state is postulated—any such construction is ill-defined in RQM’s framework, since there is never an "outside" system to which the universe as a whole is related (Dorato, 2013). The relational event network replaces monistic ontologies: physical reality becomes a web of discrete events labeled by system pairs and the observables actualized in each interaction.

This event ontology is further connected with the Peircean logic of relations and category theory, where individual systems are understood as assemblies of relational arrows rather than bearers of intrinsic substance (Nicolaidis, 2012). This formalization underpins both the epistemic role of the wave function and the fundamental granularity of phase space.

3. Measurement, Randomness, and the Solution to the Measurement Problem

In RQM, measurement is demystified: it is a particular case of quantum interaction, with no ontologically privileged apparatus, consciousness, or classical cut (Gordon, 2022, Rovelli, 2017, Rovelli, 2021). When system $\rho_{S|R}$3 measures system $\rho_{S|R}$4 for some observable $\rho_{S|R}$5, the only event is R's record of the interaction—a definite outcome actualized for R, not for an uninvolved third system. The "collapse" is re-interpreted as a reference change—what changes is the relational state ascribed by $\rho_{S|R}$6, not the absolute state of $\rho_{S|R}$7.

Randomness in RQM is thus relational: when $\rho_{S|R}$8 measures $\rho_{S|R}$9, $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$0 observes outcome $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$1 with probability $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$2, and thereafter ascribes to $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$3 the new relational state $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$4. For another observer $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$5, $$\rho_{S|O} \rightarrow \rho'_{S|O} = \frac{P^A_{\Delta} \rho_{S|O} P^A_{\Delta}}{\operatorname{Tr}(P^A_{\Delta} \rho_{S|O})}$$6 may remain in the initial superposition. There is no global "collapse" or many-worlds branching; definiteness and randomness are the irreducible consequence of pairwise interactions (Gordon, 2022). This re-framing of collapse resolves classic measurement paradoxes (e.g., Wigner’s friend), as different observers' accounts are naturally distinct and require no mysterious physical process or hidden variable (Dorato, 2013).

The double-slit experiment, when read in RQM, exemplifies these principles: the photon maintains a superposition relative to the slits, actualizes a definite position only upon detection by the screen (relative to the detector), and the entire interference pattern emerges from the record of many such relational events (Gordon, 2022).

4. Observer Democracy, Information, and Cross-Perspective Consistency

RQM posits the universality of the observer role—any physical system, regardless of scale or complexity, can act as observer, and all are subject to the same quantum laws (Laudisa, 2017, Dorato, 2013, Muciño et al., 2021). This observer "democracy" avoids any anthropocentric or classical privileging of "measurement," sharply distinguishing RQM from both Copenhagen and agent-centered quantum Bayesian approaches.

A challenge arises regarding consistency across observers: if all facts are relative, what secures agreement when different parties interact or compare records? In response, recent RQM variants introduce the "cross-perspective links" (CPL) postulate, ensuring that when one observer’s physical record is stable and unperturbed, it is accessible—and will yield the same value—when interrogated by another observer in a compatible basis (Adlam et al., 2022, Adlam, 10 Feb 2025). This extends the RQM ontology from purely relational events to an enriched "flash" ontology: pointlike events whose factual content can be made intersubjectively accessible through measurement of a robust record.

The CPL postulate addresses solipsism concerns and, under decoherence, ties together subsystem records within large (macroscopic) observers. However, it also introduces significant interpretational tension and, according to some critiques, potentially undermines RQM's ban on hidden variables or reintroduces contradiction with unitarity when traced through Wigner-friend scenarios (Markiewicz et al., 2023).

5. Critique, Open Problems, and Theoretical Challenges

Despite its conceptual clarity and avoidance of certain dualisms, RQM faces substantive challenges:

• The Measurement Problem (Persistence): RQM’s relational collapse avoids an absolute measurement problem but does not supply a dynamical criterion distinguishing "mere entanglement" from definite event actualization. The choice of pointer basis and the emergence of robust facts in macroscopic contexts remain under-specified (Laudisa, 2017, Muciño et al., 2021).
• The Third-Person and Consistency Dilemma: Standard RQM allows the same system to be described by reduction (collapse) from the first-person viewpoint and by unitarity from a third-person viewpoint, raising ambiguities about when and how accounts should reconcile (Dorato, 2013, Laudisa, 2017). The notion of unrestricted iterative relativization (or its rejection) plays a crucial role in these debates (Riedel, 2024).
• No-Go Theorems and Empirical Constraints: "Qubits are not observers" results highlight that, in the absence of a preferred basis or pointer selection, observer labels may be insufficient to yield a consistent knowledge theory (Brukner, 2021). More stringently, GHZ-type and Wigner-friend-type no-go theorems demonstrate that, under plausible assumptions, the RQM formulation of relative facts cannot always reproduce the empirical Born-rule correlations observed in quantum experiments, thereby challenging the consistency of the framework unless modified or augmented (Lawrence et al., 2023, Lawrence et al., 2022).
• Completeness and Ontology: Some critics argue RQM fails to provide a complete ontology of events or "beables," particularly when attempting to associate facts with localized carriers, resulting in potential ambiguities and incomplete explanations of quantum-to-classical transition (Muciño et al., 2021).
• Temporal Structure and Relativity: The compatibility of event generation with relativity has been debated, with some variants advocating an all-at-once, block-ontology for the event web, while others (rejecting the universality of CPL) demonstrate that temporal generation without global event sets is consistent with both relativity and the relational framework (Toussaint, 2023).

6. Extensions, Mathematical Frameworks, and Ontological Variants

RQM’s relational stance has inspired a spectrum of technical and metaphysical developments:

• Fact-Nets: Proposed as an explicitly relational mathematical framework, fact-nets replace the absolute state with a network where physical facts are edges between system nodes. Amplitude and probability assignments, superposition, entanglement, and Lüders rule updates are all reproduced within the fact-net model (Martin-Dussaud et al., 2022).
• Mereological Bundle Theory: This approach interprets physical systems as fusions (bundles) of properties, with object-hood constituted by the set of relational and intrinsic properties held (observer-)relative to other systems. The framework is compatible with moderate structural realism, unifying causality, relation, and property (Oldofredi, 2020).
• Category-Theoretic and Peircean Foundations: By reconstructing quantum theory from the composition rules of Peircean relation logic—formalized categorically—one recovers key structures such as the Born rule, canonical commutation relations, and the granularity of phase space (Nicolaidis, 2012).

Variants of RQM (e.g., RQM+CPL) differentiate along the axes of "inherent vs. effective" and "fact-based vs. dynamical" relationality. The Adlam–Rovelli approach incorporates inherent, dynamical relationality enforced by CPLs, attempting to bridge gaps in intersubjectivity and events' reality (Adlam, 10 Feb 2025). These amendments, however, have sparked controversy regarding compatibility with core RQM tenets and the introduction of hidden variables (Markiewicz et al., 2023).

7. Philosophical Implications and Interpretational Landscape

RQM introduces a substantial reconfiguration of quantum realism and metaphysics:

• Anti-monistic, Pluralist Reality: The impossibility of defining the state of the universe in toto, and the necessity of local, pairwise relations, lead to a pluralist picture of physical reality as a network or web of local events, in contrast to any global, monist foundationalism (Dorato, 2013).
• Sparse Ontology and Local Becoming: Events/facts are not continuously distributed but arise discretely at spacetime-localized interactions, supporting a process-based rather than block-universal view of quantum reality, compatible with relativity (Dorato, 2013).
• Epistemic Wave Function, Perspectival Facts: The quantum state is not a physical object but a computational device encoding the information relative to specific systems, suggesting an epistemic or perspectival rather than ontological reading of standard quantum states (Oldofredi et al., 2021). Whether or not this stance is compatible with no-go theorems such as PBR depends fundamentally on the rejection of absolute ontic state spaces in RQM (Oldofredi et al., 2021).

Finally, the relational paradigm's viability and internal consistency depend critically on how it balances its powerful explanatory innovations with the need for a precise account of records, intersubjective agreement, and the quantum-to-classical transition. Recent proposals introduce refined notions of observerhood (physical vs. informational), coherence criteria, and elaborate postulates, but tensions persist regarding both consistency and empirical adequacy (Terris, 3 Mar 2026, Adlam et al., 2022, Markiewicz et al., 2023). The interpretation continues to be an active area of foundational inquiry, with ongoing debate regarding its prospects as a complete and empirically sound quantum ontology.