Contribution
The founding statement of relational quantum mechanics. It turns one textbook measurement, described twice, into the thesis that the state of a system and the values of its variables exist only relative to another system, and it makes that thesis the interpretation: no special observers, no collapse mechanism, no hidden variables, and quantum mechanics taken as complete. It also proposes a programme, to derive the formalism from a few physical postulates as Einstein derived the Lorentz transformations, and carries it part of the way with two information postulates and a third that it does not motivate physically.
Key insight
Velocity is a relation to a reference body, and the reference body “can be a table lamp” (§I, p. 3). Rovelli asks that every quantum value be read the same way: “the electron has spin up” is elliptic for “the electron has spin up with respect to the Stern Gerlac apparatus” (§II.C, p. 7). O’s account (S collapsed to |1⟩, q = 1) and P’s (S–O entangled, q without a value) are “two distinct correct descriptions of the same sequence of events” (§II.A, p. 4). What is relational is the state and the value; what bears them is a system, given beforehand and characterized by the questions that can be asked of it (§III.A, pp. 10–11).
Assumptions
- Hypothesis 1, all systems are equivalent (§II.A, p. 4): “Nothing distinguishes a priori macroscopic systems from quantum systems.” “Of course, I have no proof of hypothesis 1, only plausibility arguments.”
- Hypothesis 2, completeness (§II.C, p. 7): “Quantum mechanics provides a complete and self-consistent scheme of description of the physical world, appropriate to our present level of experimental observations.”
- The world is decomposable into systems (§III.A, p. 10): “Assume that the world can be decomposed (possibly in a variety of ways) in a collection of systems, each of which can be equivalently considered as an observing system or as an observed system.” Not argued.
- Finite-dimensional Hilbert spaces (§III.A, p. 11), “to simplify the mathematical treatment”; “Generalization to infinite systems should not be difficult” (p. 11), not done.
- Information is Shannon’s: “a measure of the number of states in which a system can be –or in which several systems whose states are physically constrained (correlated) can be” (§I, p. 3); no meaning, storage or purpose (§II.E, p. 10).
- The symmetry p(Q_b^(i), Q_c^(j)) = p(Q_c^(j), Q_b^(i)), flagged “a new assumption! … I leave it here as an unjustified assumption at this stage” (§III.C, p. 13).
- Probability is left uninterpreted; the view is said to work on an objective or a subjective reading (footnote ‡, p. 13).
Key results
§I: the programme (pp. 1–3)
The thesis: the notion rejected is “the notion of absolute, or observer-independent, state of a system; equivalently, the notion of observer-independent values of physical quantities” (p. 1). The analogy is Einstein’s reading of the Lorentz transformations: their formalism existed, the interpretation was the contribution, and it came by re-derivation from “two postulates with terse physical content” (p. 2). The programme has three steps: find experimentally true postulates, show they refute a common assumption, derive the formalism from them (p. 2). “In this paper, I do not achieve such a result in a satisfactory manner” (p. 2). Terminology: an “observer” is any physical object; information is Shannon’s, and “a pen on my table has information because it points in this or that direction” (p. 3).
§II.A: the third-person problem (pp. 3–4)
O measures q on S in the state α|1⟩ + β|2⟩ and gets 1. O’s description: α|1⟩ + β|2⟩ → |1⟩ (eq. 1). P, who knows the initial states and does not interact with S–O between t1 and t2, describes it unitarily: (α|1⟩ + β|2⟩) ⊗ |init⟩ → α|1⟩ ⊗ |O1⟩ + β|2⟩ ⊗ |O2⟩ (eq. 2). Quantum mechanics “requires us to distinguish system from observer, but it allows us freedom in drawing the line that distinguishes the two” (pp. 3–4). Main observation: “In quantum mechanics different observers may give different accounts of the same sequence of events” (p. 4), credited also to Zurek 1982 and Kochen 1979.
§II.B: eight objections (pp. 4–6)
Special collapse-producing systems (gravity, minds, macroscopic objects) violate Hypothesis 1 and predict interference discrepancies, small because of decoherence but “different from zero, and thus observable” (p. 4). Taking only outcomes as physical does not help: for P q has no value at t2, for O it does (p. 5). “P is right, O wrong” would mean no measurement has ever been made; “O right, P wrong” predicts unobserved discrepancies (p. 5). Decoherence cannot make q = 1 absolute, because “decoherence depends on which observation P will make” (p. 5). No-collapse without values cannot match observation (Objection 6, p. 5). Everett’s branching needs a preferred apparatus and basis, violating Hypothesis 1, or turns into histories (p. 5). Consistent histories make the probabilities observer-independent but the frameworks available to O and P differ, because “what is data for O is only a possible choice of framework for P” (Objection 8, pp. 6–7).
§II.C: the relational thesis (pp. 6–7)
“A quantum description of the state of a system S exists only if some system O (considered as an observer) is actually ‘describing’ S, or, more precisely, has interacted with S. The quantum state of a system is always a state of that system with respect to a certain other system” (p. 6). “In quantum mechanics all physical variables are relational, as is velocity” (p. 6). With completeness: “Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world” (p. 7). So “a complete description of the world is exhausted by the relevant information that systems have about each other. Namely, there is neither an absolute state of the system, nor absolute properties that the system has at a certain time” (p. 7), and “the notion of a universal description of the state of the world, shared by all observers, is a concept which is physically untenable, on experimental ground” (p. 7).
§II.D: the relation between descriptions (pp. 7–9)
- “The absolute state of affairs of the world is a meaningless notion … In particular, there is no way of deducing the view of one from the view of the other” (p. 8). The relation between views “must be addressed within the view of one of the two observers (or of a third one)” (p. 8), because “the process of comparison is always a physical interaction, and all physical interactions are quantum mechanical in nature” (p. 8).
- That O has measured is a P-relative quantum property. P can know that O knows q, not what O knows (p. 8). The operator M on H_S ⊗ H_O (eq. 3), M(|1⟩⊗|O1⟩) = |1⟩⊗|O1⟩, M(|2⟩⊗|O2⟩) = |2⟩⊗|O2⟩, and 0 on the mismatched products, asks “Is the pointer correctly correlated to q?”; the state (2) is its eigenstate with eigenvalue 1 (p. 8).
- No half-measurements. If S–O is not an eigenstate of M, “there is no half-a-measurement; there is probability one-half that the measurement has been made!” (p. 8), because “whether or not O has measured S is not an absolute property of the S − O state, but a quantum property of the quantum S − O system” (pp. 8–9). The bi-orthogonal decomposition always supplies some such M, which Rovelli dismisses as of “null practical nor theoretical significance” (p. 9).
- Consistency. If P measures q and then O’s pointer, the results are correlated, “as was first noticed by von Neumann, and as is clear from (2)” (p. 9).
§II.E: information (pp. 9–10)
“q has a value relative to O” means “relative to P, there is a certain correlation in the S and O states”, or “O has information about q” (p. 10). Information “can be lost dynamically”; purposeful and accidental correlation are not distinguished; and “any physical system may contain information about another physical system”, so two spin-½ particles with equal spin have information about each other (p. 10). “Observers are not ‘physically special systems’ in any sense” (p. 10). The answer “only shifts the problem by one step” (p. 10). “Physics is the theory of the relative information that systems have about each other. This information exhausts everything we can say about the world” (p. 10).
§III.A–B: questions and the two postulates (pp. 10–12)
- A system S “viewed as an observed system, is characterized by the family of yes/no questions that can be asked to it”, W(S) = {Q_i}, the observables (p. 10); “The general kinematical features of S are representable as relations between the questions Q_i in W(S), that is, structures over W(S)” (p. 11). O’s information about S is a binary string (e1, e2, …) (eq. 4); the relevant information is the part that bears on future answers (p. 11).
- “A notion does not appear here: the state of the system. The absence of this notion is the prime feature of the interpretation considered here. In place of the notion of state, which refers solely to the system, the notion of the information that a system has about another system has been introduced” (p. 11). Footnote †: “The string (5) is essentially the state … The novelty is that this notion of state is relative to the observer that has asked the questions” (p. 11).
- Postulate 1 (limited information): “There is a maximum amount of relevant information that can be extracted from a system” (p. 11). So “each system is characterized by a number N”, the least integer with N ≥ log₂ k, k the Hilbert-space dimension (p. 11). Planck’s constant is then “the transformation coefficient between physical units (position × momentum) and information theoretical units (bits)” (p. 12).
- Postulate 2 (unlimited information): “It is always possible to acquire new information about a system” (p. 12). With Postulate 1, a new undetermined question costs at least one old bit, and the answers “cannot be fully deterministic” (p. 12).
§III.C: the reconstruction and the third postulate (pp. 12–14)
From Postulate 1, with the “(non-trivial) additional assumption” that ∨ and ∧ are defined for every pair, W(S) is an orthomodular lattice (p. 12), containing Boolean algebras generated by the 2^N “complete questions” of each family of N independent questions (eqs. 6–8, pp. 12–13). The transition probabilities p_ij satisfy 0 ≤ p_ij ≤ 1 and row and column sums of 1 (eqs. 10–12; the PDF prints “0 ≥ p_ij ≥ 1”, a typo), which are met by p_ij = |U_ij|² for a unitary U not fixed by p_ij (eq. 13). The phases are fixed by composite questions, p_{i(jk)i} = |U_ij U_ji + U_ik U_ki|² (eq. 17), “(experimentally!)” not the classical sum (p. 13). Postulate 3 (superposition): the unitaries U_cb can be chosen with U_cd = U_cb U_bd and composite questions given by eq. 17 (p. 14). Then “the conventional formalism of quantum mechanics as well as the standard probability rules follow completely from the three postulates” (p. 14); dynamics follows by labelling questions by time, giving U(t) = exp{−i t H} (p. 14). The reconstruction “is not fully successful” (p. 12), and Rovelli suspects that inter-observer consistency “could be taken as the missing input” but has “no definite result” (p. 14).
§III.D: the observer observed (pp. 14–15)
“A statement about the information possessed by O is a statement about the physical state of O”, so it too is relative to a further observer (p. 14). Knowing W(S) includes knowing the Hamiltonian, “because W(S) includes all Heisenberg observables at all times” (p. 15). For P the measurement is a unitary evolution with a non-vanishing interaction Hamiltonian; O’s unitary description of S breaks down “simply because O is not giving a full dynamical description of the interaction”, and “there is no meaning in being correlated with oneself” (p. 15). Two rules, which if kept mean “there is no way in which contradiction may develop” (p. 15): “(i) There is no meaning to the state of a system or the information that a system has, except within the information of a further observer”; “(ii) There is no way a system P may get information about a system O without physically interacting with it”. The no-contradiction claim is left to the reader (“The reader may convince himself”), not proved.
§IV.A: summary (pp. 15–16)
“A physical system is characterized by the structure on the set W(S) of questions that can be asked to the system” (p. 16). “The properties of the systems are to be described by an interrelated net of observations and information collected from observations” (p. 16). A description in toto by a further observer is “deficient in two directions: upward, because an even more general observer is needed to describe the global observer itself, and –more importantly– downward, because the in toto observer knows the content of the information that the single component systems possess about each other only probabilistically” (p. 16). “Any observation requires an observer”, from Maturana and Varela (p. 16). The thesis restated: “Quantum mechanics is the theoretical formalization of the experimental discovery that the descriptions that different observers give of the same events are not universal” (p. 16).
§IV.B: other interpretations (pp. 16–19)
- Dynamics (GRW, Penrose): violate Hypothesis 2. “My effort here is not to modify quantum mechanics to make it consistent with my view of the world, but to modify my view of the world to make it consistent with quantum mechanics” (p. 16).
- Bohr, as one reading of Copenhagen, “assumes a classical world … physically distinct from the microsystems” (p. 17). Fixing a privileged system S_o, “The Observer”, recovers Bohr’s view: “Bohr’s choice is simply the assumption of a set of systems (the classical systems) as privileged observers. This is consistent with the view presented here” (p. 17). “There is nothing incorrect in fixing the preferred observer S_o once and for all”, and quantum mechanics is used “precisely as is it is currently used” (p. 17). Footnote **: why S_o admits a classical description is decoherence’s question (p. 17).
- The projection postulate is explained: unitary evolution breaks down because S interacts with something its equations omit; after Ashtekar, the view prohibits an observer “to be able to give a full description of ‘itself’”, and Breuer’s no-complete-self-measurement theorem is named as related (p. 17).
- Modal interpretations: RQM keeps value assignment but makes it observer-dependent and lets it break unitarity relative to the observer, so “there is collapse in each observer-dependent evolution of expected probabilities”, and “For a fixed observer, the eigenstate-eigenvalue link is maintained” (p. 17). “Actuality is observer dependent” (p. 17).
- Consistent histories: the distinction between a probability-one property in a framework and a datum is what CH leaves unexplained; “there are no (observer independent) data at all” (p. 18).
- Many worlds: Everett’s universal wave function “represents the perspective of a non-existent observer”; “There is no description of the universe in-toto, only a quantum-interrelated net of partial descriptions” (p. 19). The extra values are “‘somehow perspectival’ (but definitely not mental!)” and “‘wholly a matter of physics’” as relativity of simultaneity is (p. 19).
- “Bohr’s interpretation, consistent histories interpretations, as well as the many worlds interpretation, are all correct” from this view; the closest is Heisenberg’s (p. 19). Zurek 1982 is quoted as having reached “identical” conclusions: “correlations between the properties of quantum systems are more basic that the properties themselves” (p. 19).
- Footnote ‡‡ (p. 19): Rovelli declines “to venture into philosophical terrains”, cites Kant (“entirely made up of mere relations”) and notes that observer-independence has become “almost a commonplace” from anthropology to continental philosophy.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | O’s and P’s accounts of one measurement are both correct, so values are observer-relative | moderate | the two-description analysis (§II.A) and the rebuttal of eight objections (§II.B); conditional on Hypotheses 1 and 2, which are assumed, the first with “only plausibility arguments” |
| C2 | Any physical system can be an observer | strong (as a definition) | information as Shannon correlation (§II.E, p. 10); it is the paper’s stipulation, consistent with Hypothesis 1 |
| C3 | That O has measured S is a P-relative quantum property, which dissolves the “precise moment of measurement” problem | moderate | the operator M (eq. 3) and the probabilistic reading of non-eigenstates (§II.D, pp. 8–9) |
| C4 | No contradiction arises among observers’ descriptions if rules (i) and (ii) are kept | weak | shown for the S–O–P correlation case (von Neumann, p. 9); in general left to the reader (§III.D, p. 15) |
| C5 | Two information postulates give W(S) the structure of an orthomodular lattice with Boolean subalgebras | moderate | sketch with “(non-trivial) additional assumption[s]” and quantum-logic results cited, not proved (§III.C, p. 12) |
| C6 | The Hilbert-space formalism and Born rule follow from three postulates | weak | Postulate 3 builds in the unitary composition and eq. 17; a symmetry is assumed unjustified; the author calls the reconstruction “not fully successful” (pp. 12–14) |
| C7 | Planck’s constant converts phase-space units into bits | weak | dimensional argument from Postulate 1 (pp. 11–12) |
| C8 | Bohr, consistent histories and many worlds are all correct, each a special or partial case | weak | §IV.B, by redescription; no derivation |
| C9 | The theory is about systems already given: a system is characterized by W(S) and N, its values are relative | strong (textual) | §III.A, pp. 10–11; §III.D, p. 15; §IV.A, p. 16 |
Concepts
- Observer — any physical system, in the sense of the reference body of Galilean relativity (§I, p. 3); “with more than one state” (p. 10).
- Information — Shannon’s: correlation that reduces the joint alternatives of two systems (§II.E, pp. 9–10). “O has the information that q = 1” is the paper’s “lexical move” for “q = 1 relative to O” (p. 9).
- Relevant information — the part of the record of answers that bears on future answers (§III.A, p. 11).
- W(S) — the family of yes/no questions that can be asked of S, which characterizes it (§III.A, pp. 10–11).
- Information capacity N — the maximal relevant information about S, N ≥ log₂ k (§III.B, p. 11).
- Relative state — the string of O’s answers about S, “essentially the state” but “relative to the observer that has asked the questions” (footnote †, p. 11).
- Main observation — different observers may give different accounts of the same sequence of events (§II.A, p. 4).
Connections
- Rovelli’s later account (LIT-177, read in NOTE-174). The 1996 paper has no “facts” or “events” vocabulary and no sparse ontology: its primitives are systems, questions and information, and its relative state is the record of answers. The later chapter’s “core idea”, that amplitudes give probabilities only between facts relative to the same system, appears here as “For a fixed observer, the eigenstate-eigenvalue link is maintained” and “there is collapse in each observer-dependent evolution” (p. 17), not as a probability rule. The 1996 postulates are stated for finite-dimensional Hilbert spaces (N ≥ log₂ k), not for compact phase space; there are three, not two, and the third is admitted to lack a physical meaning. In 1996 Bohr’s view is “consistent” with RQM, a privileged-observer special case (p. 17); the 2021 chapter’s charge that Copenhagen and QBism build on an approximation is not here.
- Faye and Jaksland (LIT-1672, NOTE-1220) cite this paper for the freedom in drawing the line (pp. 3–4) and for O’s inability to describe its own interaction (p. 15); both are accurate. Their “sparse ontology” and “systems precede the interactions” are quoted from later texts; see Bearing.
- Grangier and Auffèves (NOTE-044) list this paper as a rival reconstruction (their ref. [11]).
- Maturana and Varela (LIT-1656, Unavailable): “Any observation requires an observer” is “freely taken” from them (p. 16). The in-text citation reads “1920”; the reference list gives 1980.
- Zurek 1982 and Kochen 1979 are credited with the same conclusion (pp. 4, 19); neither is held.
Bearing on the record
- THEORY-1071: the attribution holds of the founding paper, as an assumption. Faye and Jaksland’s words, “a sparse ontology of (relational) quantum events” and “the systems precede the interactions”, are not in the 1996 text, which has no ontology of events. But the 1996 paper does keep systems prior to the values: “Assume that the world can be decomposed (possibly in a variety of ways) in a collection of systems” (p. 10), each “characterized by the family of yes/no questions that can be asked to it” (p. 10), with “a number N” (p. 11) and, through W(S), its Hamiltonian (p. 15). A state exists “only if some system O … has interacted with S” (p. 6): the interaction presupposes the two systems. Two qualifications: the priority is assumed, never argued; and the decomposition is not unique (“possibly in a variety of ways”), so the systems are prior to interactions without being uniquely given. Filed as THEORY-1068.
- CLAIM-136: a physical case of relational properties with relata characterized, not individuated, apart from them. All values are relational (“all physical variables are relational, as is velocity”, p. 6), but what characterizes a system, W(S) and N, is not relative to any observer. W(S) fixes a system’s kind (a lattice of subspaces of a k-dimensional Hilbert space); nothing in the paper says what makes two systems of the same kind two. That is the claim’s open question in a physical key, and the 1996 text neither poses nor answers it. RQM in 1996 is non-eliminative, as the claim is, and closer to the weak version only by silence.
- CLAIM-477: W(S) is the set of questions that can be asked. A system is characterized by the interactions it can enter as an observed system, its admissible questions and their lattice relations, which is near the claim’s “partly fixed by the constraints on the configurations of entities and relations it can enter”. Rovelli does not say the questions constitute the system; he says they characterize it, and the decomposition into systems is assumed first. The note’s “Rovelli’s relational quantum mechanics is a second non-eliminative case” stands on the 1996 text.
- QUESTION-021: Rovelli’s relativity analogy refuses the invariant. A18’s first horn borrowed relativity’s structure, “an invariant object underlying coordinate-dependent descriptions”. Rovelli’s analogy is the same one (simultaneity, velocity), and he breaks it at exactly that point: “there is no way of deducing the view of one from the view of the other” (p. 8), the relation between views is itself relative to a third system, and “There is no description of the universe in-toto, only a quantum-interrelated net of partial descriptions” (p. 19). What is frame-independent is the system’s kind, W(S), not any state of it. That is the second horn for states and something like CLAIM-137’s frame-independent kind for systems. It bears on the question by analogy only. Filed as THEORY-1080.
- CLAIM-150: the founding paper concedes the underdetermination. The relational thesis follows from the main observation only “whatever view of quantum theory (consistent with hypothesis 1) one holds” (p. 6) and with completeness assumed (p. 7). Bohm’s theory, which violates Hypothesis 2, is not refuted but set aside (p. 17), and “the considerations in this paper do not suggest any modification to the conventional use of quantum mechanics” (p. 17). So the paper’s own form is an interpretive argument under two hypotheses, not an entailment from the predictions, which is the claim’s point.
- THEORY-1091: what Barad cannot borrow is the relativization, not a disentangling. In 1996 RQM, relative to P the S–O system stays entangled through the measurement (eq. 2), exactly as Barad says it stays (2007, 348, as Faye and Jaksland quote it). What RQM adds is that relative to O, q = 1, and that “whether or not O has measured S is not an absolute property of the S − O state” (pp. 8–9). The difference from agential realism is that Barad’s entanglement is absolute and RQM’s is relative to P. The theory’s wording, “treatment of the wave function as bookkeeping”, is compatible with the 1996 text (“A notion does not appear here: the state of the system”, p. 11), but “bookkeeping” is the later vocabulary.
- THEORY-621: the 1996 form of its rule, and of its gap. The promote_when asks for a result that relative facts cohere. The 1996 paper supplies one case (P’s later measurements of q and of O’s pointer agree, p. 9) and asserts the general case under rules (i) and (ii) without proof (p. 15). It does not derive the rule from independent principles: it assumes it.
- TERM-087 and THEORY-1076: Rovelli reads Bohr’s cut as physical. “Bohr declares explicitly that we must renounce giving a quantum description of the classical world” (p. 4); “In Bohr’s view, this classical world is physically distinct from the microsystems” (p. 17), hedged “at least in one possible reading”. That is the ontological reading of Bohr’s fixed cut that Faye and Jaksland reject for an epistemic one. Rovelli’s own cut is movable (“freedom in drawing the line”, pp. 3–4) and, unlike Barad’s, not ontological: it is the choice of which system’s information is being described.
- No ML-practice content; nothing here bears on the anthology.
Limitations
- Hypothesis 1 has “only plausibility arguments”; Hypothesis 2 is assumed (pp. 4, 7). The relational conclusion is conditional on both.
- The reconstruction is incomplete by the author’s account: Postulate 3 is “the more modest step” (p. 14), and the symmetry of p is unjustified (p. 13). The “derivation” of the formalism is a sketch.
- Inter-observer consistency is shown for one case and asserted in general (p. 15).
- The paper declines philosophy (footnote ‡‡, p. 19). What a system is, and whether the decomposition into systems is itself relative, are not discussed beyond “possibly in a variety of ways”.
- Read in arXiv v2; the journal text is unchecked (see history).
Open questions
- Is the decomposition into systems itself relative to an observer, as the states are? The paper allows several decompositions and says nothing of how they relate.
- Can inter-observer consistency replace Postulate 3, as Rovelli suspects (p. 14)?
- Does the information-capacity characterization survive the infinite-dimensional case, which “should not be difficult” (p. 11)?