Consistency of Relative Events in Relational Quantum Mechanics

Relative Quantum Events: AI generated - DALL E 3

Many years ago my "shut up and calculate" attitude to doing quantum physics was disrupted by reading Karl Popper's Quantum Mechanics without "The Observer" [1]. Although I realised (after a time) that Popper's treatment was inadequate, the attitude in that paper remains a guide and talk of conscious observers in quantum physics triggers a critical response. I later hoped that Relational Quantum Mechanics (RQM) [2,3,4] would enable the completion of the programme but it has become, in some recent presentations, more like  "Quantum Mechanics with Only Observers".

As discussed in the previous post the ontology presented by Rovelli for RQM is a world of relative events and therefore relative facts (mixing ontological and epistemological issues). There can be no problem with an event (and therefore a fact) depending on a context but  an event either happens or does not. In RQM events can happen in a quantum system $A$ (consisting of two subsystems) but that same event does not happen in $A$ for system $B$ that does not interact with $A$. If $B$ is a data processing, programmable automaton then if it has stored the initial state of  $A$ can be programmed  to be a unitary quantum predictor. Such a predictor will predict the future state of $A$, using standard unitary quantum theory, but no actual events can be predicted. After a time $B$ interacts with $A$ to measure some property. If this procedure is repeated a sufficient number of times with identically prepared initial states then the statical properties of the measurements will agree with that of the calculated state and application of the Born rule. This means that as $A$ is involved in the $B-A$ interaction $A$'s earlier event (a postulated in RQM) has become a non-event or never took place.  

The paper by Adlam and Rovelli [4] presents an adaption  of the RQM response to criticism. The modified theory is presented in six postulates. Ruth Kastner reminded me that she has published a refutation of RQM [5] that cites the 1996. 2021 versions [2, 4]. The paper [4] does not address this criticism specifically. I will return to this but first some comments on the 6 postulates. The final set of  postulates is [4]: 

1. Relative facts: Events, or facts, can happen relative to any physical system.
2. No hidden variables: Unitary quantum mechanics is complete.
3. Relations are intrinsic: The relation between any two systems $A$ and $B$ is independent of anything that happens outside these systems’ perspectives.
4. Cross-perspective links (CPL): In a scenario where some observer Alice measures a variable $V$ of a system S, then provided that Alice does not undergo any interactions that destroy the information about $V$ stored in Alice’s physical variables, if Bob subsequently measures the physical variable representing Alice’s information about the variable $V$, then Bob’s measurement result will match Alice’s measurement result.
5. Measurement: An interaction between two systems results in a correlation within the interactions between these two systems and a third one; that is, with respect to a third system $W$, the interaction between the two systems $S$ and $F$ is described by a unitary evolution that potentially entangles the quantum states of $S$ and $F$.
6. Internally consistent descriptions: In a scenario where $F$ measures $S$, and $W$ also measures $S$ in the same basis, and $W$ then interacts with $F$ to “check the reading” of a pointer variable (i.e., by measuring $F$ in the appropriate “pointer basis”), the two values found are in agreement.

Adlam and Rovelli do not claim that these postulates are a complete set of axioms for RQM but that they "characterise" the RQM. Whatever  their status they should to be mutually consistent and (as an interpretation) be consistent with the predictions of standard quantum theory (SQT). 

Adlam and Rovelli initially list as Postulate 4:

Relativity of comparisons: It is meaningless to compare the accounts relative to any two systems except by invoking a third system relative to which the comparison is made.

This is taken from earlier work by Rovelli and collaborators [3].

Discussion on the 5 postulates

Postulate 1 It may be pedantic to quibble that evens but not facts happen but if the status the event is clear then there is no issue in stating an "$A$ given $B$" type of fact. That an event occurs relative to a context is also not controversial. It is controversial to say that the event has at the same time  occurred, in a context, and not occurred for physical systems outside (not in interaction with) that context. The events in RQM are a generalisation of the SQT theory projection postulate and the Born rule to  "any physical system".

Postulate 2 This a strong claim if there are objective events. Even if the events are contextual. Unitary quantum mechanics is a complete description of unitary development of quantum states. The von Neumann projection postulate and the Born rule are part of SQT. If this postulate is used, as it is [4], then Postulate 2 can only be true if the projection postulate is not part of the physical theory but just an instrumental  addition that is not a physical mechanism in the theory. 

In the formulation of non-relativistic quantum theory that I prefer there is a theorem:

In Quantum Mechanics, the operators representing the possible values of the properties of a system form the $\sigma$-complex $Q(\mathcal{H})$ of projections of the Hilbert space $\mathcal{H}$ of the system.       

This theorem holds and the mathematical formalism using \(Q(\mathcal{H})\) is equivalent in outcomes to the SQT. The advantage of the formulation is that it makes clear how close SQT is to being a probability theory but equally how it is different. However, the interpretation of this Kochen inspired formalism is that while a quantum object has a complete set of properties they appear in space-time only when the situation invokes one \(\sigma\)-algebra in the complex that allows that property to take a specific set of values. The ad hoc projection postulate and Born rule (PPBR) provide this in SQT. For example, if an event is measured at a point on a screen in the double slit experiment then that context implicitly selects the $\sigma$-algebra for a position event from the complex. But this is an ad hoc story, not a physical theory. The projection is not a unitary mapping. If Postulate 2 is true then a unitary mechanism is needed that mimics PPBR  close enough to be empirically indiscernible. Despite its mathematical sophistication, the Kochin formulation of quantum theory does not solve the measurement problem.

      

Postulate 3 Understanding "relations are intrinsic" depends on what is meant by the systems' perspectives. No explanation of the scope of the term perspective is provided [4], but is provided in an earlier review [6]. A perspective for a system $B$ is the set of interactions and the set of variables that the  system is capable of registering from the set of systems with which it is in interaction. So there seems to be an equivalence between intrinsic relations and the set of interactions between any two systems.

Postulate 4 The revision is an improvement on the earlier appeal to meaninglessness in what is at best an ambiguous statement.  They use the Bob and Alice terminology current in communication theory and cryptology that has been adopted in Quantum Information Theory.  I prefer to think of Alice and Bob as generic physical quantum systems $A$ and $B$ and concentrate on how the theory works without conscious  observers. The Bob/Alice presentation is not used in the other postulates. This is an artifact of the paper [4]. The new Postulate 4 is formulated half way through and they do not list the 6 postulates using consistent terminology. But this is just criticism of the presentation and not the substance of the paper. In a series of papers (all easily found on line) Rovelli has insisted that any quantum system that can register a variable of another system counts as an observer and registering the value of a variable of another system is a measurement for that observer. This claim cannot be maintained with the innovation that is Postulate 4.

Postulate 5 Measurement is not actually explained or even referred to in this postulate. That the interaction between $F$ and $S$ is unitary is given by Postulate 2. In SQT the unitary evolution of  any combination is well define if there is a Hamiltonian for the total system including interaction. This is the case for any number systems in interaction or even for a single system. It is perhaps implied  that there is something special about the two-system interaction. In earlier work the terms event, interaction measurement seems to be used interchangeably or implying that an interaction between two system leads to an event. In the update [4] this no longer seems to be the case. 

Postulate 6 Here there is an actual measurement. My suspicion is that these measurements are just those in SQT using the projection postulate. For any projection operator $\Pi$, $\Pi \cdot \Pi = \Pi$. 

Ruth Kastner's [5] refutation of RQM is based on three objections:

1. Standard Quantum Theory predicts only systems and pointers in improper mixtures.
2. Decoherence in SQT leads only to approximate vanishing of off-diagonals corresponding to interference; but even if perfect diagonalization obtained provide a mechanism for outcomes.
3. In a "Wigner's Friend" scenario RQM cannot provide a consistent history of events

To this I would add a point that is implicit in Ruth Kastner's discussion

• The RQM event ontology depends on the projection postulate as in SQT but, as in SQT there is no physical mechanism to describe how a measurement takes place.

The fundamental point here is that a physical mechanism is needed for an event to take place. As discussed in previous posts RTI provides such a mechanism but SQT and RQM do not. RTI is not a mere interpretation of SQT because it provides a physical mechanism for an event. The paper [4] does not cite Ruth Kastner's paper but it does discuss decoherence in some detail if rather informally. Once again, despite repeated assurance in the RQM literature that the theory does not assume conscious observers the conscious observer model is used throughout to explain the theory. In places this is not simply a presentational choice. 

Ad hoc projection postulates or collapse theories do not provide the relevant physics. Just because the set of possibilities is reduced that does not provide a mechanism for one possibility to happen. RQM has only the observer-based projection postulate and the Born rule. There is no event physics in RQM or in ad hoc collapse theories.   

Although it is not stated in Postulate 4 the creation of Cros-Perspective Links (CPL) depends on decoherence mechanisms. The discussion does not escape Ruth Kastner's criticism.

Cross-Perspective links

The "Cross-Perspective links" postulate is the major innovation introduced by Adlam and Rovelli. As is often the case, there are a lot of words but no definitive statement of what such a link is.  To aid clarification Postulate 4 is repeated here with a change from "Alice and Bob" to "Wigner's Friend" terms used in the other postulates.

4'. Cross-perspective links (CPL): In a scenario where some observer $F$ measures a variable $V$ of a system S, then provided that $F$ does not undergo any interactions that destroy the information about $V$ stored in $F$’s physical variables, if $W$ subsequently measures the physical variable representing $F$’s information about the variable $V$, then $W$’s measurement result will match $F$’s measurement result.

It should now be clear for many discussions of the Wigner's Friend type scenarios that the CPL is not consistent with SQT. While preparing this post I found a preprint by Markiewicz and Żukowski [7] that is structured similarly to this post. There focus however is to show that this modified RQM is inconsistent with SQM and therefore is not an interpretation of it. 

Adlam and Rovelli try to save the day by invoking decoherence but it is well known that decoherence does not provide a physical mechanism to transition from the possibility to the actual. The still have to invoke the ad hoc projection postulate and Born rule. But they realise that having a projection to a definite state wherever an interaction can be defined would be in conflict with SQT and its empirical consequences and they make this concession

.... we do not consider it reasonable to expect that unique definite values will arise in fundamental interactions before any decoherence has taken place.

But now there are no event without first decoherence and then invoking Born's rule (no a physical mechanism) as an instrument for empirical prediction. If a definite value does not "arise" for a system undergoing "fundamental interactions" then there is no definite event or fact for such a system. A physical system undergoing interaction is a physical system. Therefore the consequences of CPL contradict Postulate 1.

Adlam and Rovelli present their innovations as minor modification to RQM, but I agree with  Markiewicz and Żukowski that these are not minor. The modified theory depends so heavily on decoherence theory but adds no new physics. Postulate 4 is initial put forward as a fundament postulate of RQM but then reduces to an empirically not surprising realisation that macroscopic events are objective. However this uncontroversial common-sense claim is not a consequence of unitary quantum theory. Therefore we also have inconsistency between Postulates 2 and 4.

Can RQM be saved? Yes, by postulating a physical theory of how and when interactions give events. If this can be done then at least Postulates 1 and 2 have to go. 

What would such a theory look like? Rather like the Relativistic Transactional Interpretation - I suggest - a theory (not just an interpretation) that replaces but includes the incomplete unitary quantum theory.

References

1. Popper, Karl (1967) Quantum Mechanics without "The Observer" in Quantum theory and reality, ed. Mario Bunge, Springer-Verlag (later adapted as  part of the introduction to  Quantum Theory and the Schism in Physics: From the Postscript to The Logic of Scientific Discovery . Taylor and Francis, 2005).
2. Rovelli, Carlo (1996). Relational quantum mechanics. In: International Journal of Theoretical Physics 35(8), pp. 1637-1678
3. Rovelli, Carlo (2022). The Relational Interpretation of Quantum Physics. In: Oxford Handbook of the History of Interpretation of Quantum Physics. Oxford University Press
4. Adlam, Emily and Rovelli, Carlo, 2023. Information is Physical: Cross-Perspective Links in Relational Quantum Mechanics. Philosophy of Physics,  1(1), p.4
5. Kastner, Ruth (2023), Quantum Theory Needs (And Probably Has) Real Reduction,  eprint={2304.10649}, archivePrefix={arXiv}, primaryClass={quant-ph}
6. Laudisa, Federico, and Carlo Rovelli. 2021. Relational Quantum Mechanics. In The Stanford Encyclopedia of Philosophy, Winter 2021 edition, edited by Edward N. Zalta. Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2021/entries/qm-relational/.
7. Markiewicz, Marcin and Żukowski, (2024), Relational Quantum Mechanics with Cross-Perspective Links Postulate: an Internally Inconsistent Scheme,   arXiv:2312.07056 [quant-ph]