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]

Helgoland or the case for the Relational Interpretation

Helgoland (Heligoland) is important in the history of quantum mechanics. The young Heisenberg went to this barren rock in the North Sea because of his severe hay fever and in this barren place developed matrix mechanics theory of quantum physics. At the time this was a radical departure from the practice of theoretical physics. Schrödinger returned to more familiar mathematics with his wave theory. A return to Heisenberg has been a theme of this blog and been a major influence on Carlo Rovelli and his relational interpretation of quantum theory [1].

A few years ago Carlo Rovelli published "Helgoland" and he goes back to the Heisenberg story to set the back ground for a historical and philosophical tour that finishes modestly with his reinterpretation of quantum physics. Having read Rovelli's and other papers on this interpretation several years ago, I felt little need to read his "popular" presentation. A few weeks ago it came to my attention again and I thought it might provide some insights into the interpretation that I have been drawn to but in the end have found unsatisfying. Reading it has confirmed my view that his interpretation does not solve the problems associated with quantum physics. In this and in following post I want to set out why I think this.

Searching back through these blog post I was surprised that I have not mentioned the Relational  Interpretation (RI) of quantum mechanics formulated and defended by Carlo Rovelli. The initial motivation for the discussion of quantum physics presented in this blog was the attempt by Popper to provide a statistical dispositional interpretation of quantum mechanics. It became clear that Popper's attempt failed. Then for a time I had hopes for the relational quantum mechanics developed by Rovelli [1] . However, in trying to develop an ontology for quantum theory in which particles exist and have objective properties it became clear that Rovelli's relational approach would not do. Rovelli's presentation is dependent on an informational analysis and in Rovelli [2], the central importance of information is confirmed. Information is an epistemological concept rather than ontological. Rovelli takes inspiration from Wheeler, 1991, and his concept of "It from bit" that attempts to bridge from the epistemic to the ontological domain. Without falling into idealism, it is difficult to see how a bit of information can exist without a physical substrate. Let us now concentrate on the physics. 

The relationships in the RI are between physical subsystems. These subsystems are entities that are described by Hamiltonian operators in standard quantum mechanics. If the type of physical interaction between such subsystems is known then a Hamiltonian operator for the combined system can be obtained. In standard quantum mechanics the Hamiltonian governs dynamics before measurement of the system (this is not the place to go into the technicalities of operator algebras and state spaces - that has been covered in earlier posts). There is a further process in standard quantum mechanics in which a measurement is made, and the result is generated by sampling a probability distribution obtained from the system state space. These are two distinct processes in standard quantum mechanics. The Relativistic Transactional Interpretation (RTI) of Quantum Mechanics (see previous posts) brings both processes together in one coherent theory but I do not think the RI manages this. In the RI not only are interactions relative to the participating subsystems but events are. This is made clear by Rovelli's discussion of Schrödinger's cat. In Helgoland [4] in a subsection called "Facts are Relative" he argues this point:

Suppose that you are the cat in Schrödinger’s thought experiment. You are shut in a box and a quantum mechanism has a one in two probability of releasing the sleeping drug. You perceive whether the drug has been released, or not released. In the first case you sleep; in the second you remain awake. For you, the drug was delivered, or it was not delivered. There are no doubts. As far as you are concerned, you are asleep, or you are awake. You are certainly not both at once.

I, on the other hand, am outside the box and do not interact either with the bottle of sleeping draught or with you. Later on I can observe interference phenomena between you-awake and you-asleep: phenomena that would not have been produced if I had seen you asleep, or if I had seen you awake. In this sense, for me you are neither asleep nor awake. This is what it means to say that you are ‘in a superposition of sleeping and waking’.

For you the soporific is released or not, and you are asleep or awake. For me you are neither awake nor asleep. For me, ‘there is a quantum superposition’. For you, there is the reality of being awake, or of not being so. The relational perspective allows both things to be true: each relates to interactions with respect to distinct observers: you and me.

Is it possible that a fact might be real with respect to you and not real with respect to me?

Quantum theory, I believe, is the discovery that the answer is yes. Facts that are real with respect to an object are not necessarily so with respect to another. A property may be real with respect to a stone, and not real with respect to another stone.

In this well-known thought experiment (sanitised for sensitive souls by having person (You) sleep rather than a cat be put to sleep) we have three macroscopic subsystems (Bottle (B), You (Y) and Rovelli (R) and one triggering subsystem (T), usually taken to be a radioactive source. Using the unitary process in standard QM we can in principle describe the evolution of the total system. If there is no interaction between R and any of the other subsystems then the system T+B+Y will evolve independently of R. It is even possible to include the Hamiltonian describing Rovelli but because there is no interaction between R and the others, R's evolution will take place in his orthogonal state space. The system T+B+Y will evolve unitarily with no reduction to Y being in a sleeping or a wake state. Contrary to what Rovelli claims there is no definite outcome for you under quantum evolution. To get a definite outcome the "measurement process" needs to be invoked as in standard QM as Rovelli knows. In a footnote Rovelli writes:

The problem of quantum mechanics is the apparent contradiction between two laws of the theory: one describes what happens in a ‘measurement’, and the other in the ‘unitary’ evolution, namely when there is no measurement. The relational interpretation is the idea that both are correct: the first regards the events relative to the systems in interaction, the second regards the events relative to other systems.

So the unitary evolution of  T+B+Y is its evolution relative to any subsystem that is not in "interaction" with it. I put interaction in quotation marks because it is now clear that Rovelli is using the term for a type of quantum measurement process and not for the type of interaction that is including in the Hamiltonian for the total interacting system unitary dynamics.

What is it that determines whether a combined system produces a state reduction event rather than an interacting unitary evolution? RTI provides a specific transactional quantum mechanism, but RI does not more than the state reduction of standard quantum mechanics but generalised. Rovelli's position is that any quantum subsystem that can register a change can act as a measurement system. The registered change in the subsystem is an event that happens for that subsystem and all subsystems in interaction with it. For the rest of the universe nothing has happened. 

In the example above let us assume R knows the initial prepared state of T+B+Y then has no further interaction but has the well tested quantum theory. R can model how T+B+Y changes in time. R will know from that model that after a certain time has passed that the bottle will have been broken and Y will be asleep. This knowledge is model based not measurement based. Rovelli claims that unless he physically interacts with T+B+Y the event will not happen for R. Not just that R will not know about it. So we have a situation where there is a model base prediction, that only requires reliable information on radioactive decay rate and the glass breaking mechanism, saying that it is highly probable that an event will have happened and a relativist ontological theory that says it has not happened for R but has probably happened for Y. If the discussion is restricted to what Y and R know then nothing would be puzzling here. It is the relativistic ontology that is strange. But strange does not mean wrong.

Rovelli's QM is relative rather than relational. RTI is relational and describes which relations give rise to events. The events in RTI are not relative in the sense of them having occurred for one system but not for that same system from the perspective of another system.

The “being” of things is indifferent to whatever things might be “for someone”.

 Above is this blog's moto. The ontology proposed by Rovelli contradicts this statement. Is Rovelli's ontology free from contradiction and if so true? This is a challenge, but it turns out that Rovelli has found it necessary to modify his interpretation of QM [5]. On this, more to follow.

References

1.   Rovelli, Carlo (1996). Relational quantum mechanics. In: International Journal of Theoretical Physics 35(8), pp. 1637-1678.

2.   Rovelli, Carlo (2022). The Relational Interpretation of Quantum Physics. In: Oxford Handbook of the History of Interpretation of Quantum Physics. Oxford University Press.

3.   Wheeler, John A (1991). It from bit. In: Sakharov memorial lectures in physics. Vol. 2, p. 751.

4.   Rovelli, Carlo (2021) Helgoland: The Strange and Beautiful Story of Quantum Physics. Penguin Books  

5.   Adlam, Emily and Rovelli, Carlo, 2023. Information is Physical: Cross-Perspective Links in Relational Quantum Mechanics. Philosophy of Physics,  1(1), p.4.DOI: https://doi.org/10.31389/pop.8

 

Events, quantum field theory and the quantum substrate

The concepts of "event" and "measurement" in quantum physics have been discussed previously. In working physics, for practical purposes, the Born rule is used and is adequate for laboratory predictions. It is nonetheless mysterious what the measuring apparatus is doing physically. How does it sample a probability distribution? Outside the laboratory what triggers spontaneous events?  RTI [1] claims to answer these questions through an extension of transactional quantum mechanics to quantum field theory.

Quantum field theory (QFT) is usually considered a more fundamental development of non-relativistic quantum physics. And it is the case that it treats subjects such as relativistic invariance and particle creation and annihilation that standard quantum theory cannot cover. This has led some to tackle the formal ontological problems of quantum physics via field theory [2]. In addition, Kuhlmann argues for taking a formal representation of quantum field theory (Algebraic QFT) as fundamental for an ontological understanding. Whether or not this is the best approach is open to debate, but it is a worthwhile exploration with Kuhlmann taking analytical ontology seriously. One of the problems with Algebraic QFT, however, is that space-time regions are primary. That is, the physical information in quantum field theories is not contained in individual algebras but in the mapping \(\mathcal{O} \rightarrow \mathcal{A(O)}\) from spacetime regions \(\mathcal{O}\) to algebras \(\mathcal{A(O)}\) of local observables where the \(\mathcal{O}\)s are open and bounded regions in Minkowski spacetime [2]. Considering the previous posts covering relativistic transactional quantum theory (RTQT - which has been referred to as RTI, but we are dealing with more than just a reinterpretation) the role of the quantum substrate and the emergence of space-time was discussed but in what way is QFT ontologically more fundamental if it needs spacetime to get started. 

A field theory works with a function from a domain over which the field lies to whatever form the field values take. Classical electromagnetic theory provides a clear example. The domain is four-dimensional spacetime and the field values are three dimensional vectors.  This view of fields is usually carried over to the quantum domain. If space-time is needed to define the quantum fields, then the quantum substrate would be ontologically more fundamental than QFT.

I think Ruth Kastner avoids this. Particle position is also an important concept for the quantum substrate but here in the mathematical representation the position operators and the state of the system together constrain the set of possible spatial positions. This status of position as potentiality can be carried over to QFT. I think this agrees with Ruth Kastner's stance [1]. Interestingly, although he takes a very different approach, Kuhlmann also ends up proposing a dispositional ontology with observable take potential values that have a probabilistic propensity to occur [2].

So, how does RTQT use QFT to clarify the concept of quantum event? To understand this, I have found it useful to supplement the book length presentation of RTQT [1] with some of Ruth Kastner's papers. For the topics in this post, I found "On Real and Virtual Photons in the Davies Theory of Time-Symmetric Quantum Electrodynamics" [3] particularly enlightening. The discussion in the paper deals with photons but I believe the physics can be generalised to other particles.

Here are the main points extracted from [3] and wider reading on physical transaction in the relativistic quantum domain.

• A real photon is one that transfers empirically detectable energy 
• An absorber is a subsystem that can receive a transfer of energy while respecting conservation laws
• There may be many potential absorbers with the above characteristics and the probability of transaction will, in general, vary across them.
• The response of the absorber is what gives rise to the ‘free field’ that in the quantum domain is considered a ‘real photon’ 
• The virtual photon is just the potential for possible transaction between two currents—but one that was not realised
• A transaction is only attainable for virtual photons that satisfy the energy and momentum conservation constraints for the initial and final states of the system.
• The realisation of a photon happens due to an absorber response and random selection governed by the transaction probabilities
• In a realised transaction a virtual photon is elevated to a real photon
  
• In a situation that is not a prepared experiment it will not be possible to identify the subsystems that qualify as absorbers for a potential transaction.

In the sense understood here a transaction is an event. I believe this model can be generalised and the discussion of matter fields is important for this [4]. The creation and annihilation of photons is implicit in the points made above, but the role of creation and annihilation events needs to be made explicit in the generalisation to all particle types.

References

1. Ruth E. Kastner, The Transactional Interpretation of Quantum Mechanics - A Relativistic Treatment, Cambridge University Press, second edition 2022
2.  Meinard Kuhlmann, The Ultimate Constituents of the Material World: In Search of an Ontology for Fundamental Physics, Ontos Verlag, 2010
3. Ruth E. Kastner, On Real and Virtual Photons in the Davies Theory of Time-Symmetric Quantum Electrodynamics, [v2] 2016 
4. Ruth E. Kastner, Antimatter in the Direct-Action Theory of Fields, 2015

There's lots of room left in Hilbert Space (*): events in the quantum substrate and empirical space

From the perspective of this blog, the most important advance provided by the Relativistic Transactional Interpretation (RTI, [1]) of quantum physics is the replacement of the ad hoc Born Rule with a physical process and freeing from its ties to measurement. This process involves an interplay between the quantum substrate (QS) and the space-time layer of empirical events, as discussed previously. Not only are there empirical events in space-time (constructed from past events) but also non-empirical QS events outside space-time. As shown in the previous post, the quantum state changes from that obtained through unitary propagation to a state associated with the projection on to the space-time event due to conditioning on that projection. So, there is an observable stochastic process in space-time and a hidden (not observable) QS stochastic process governed by the same transition probabilities. This interplay between the QS and space-time events needs further discussion.

Ruth Kastner explains in detail [1] how RTI quantum theory developed from the classical Wheeler-Feynman Absorber Theory of Radiation [2, 3]. In chapter 5 she reaches the technical core of the book that builds on the quantum adaptation of absorber theory by Davies [4,5]. The historical progression will not be our concern here but the formal structure of the argument.  In the book two further references are provided and they help in presenting the structure. They are a joint paper with John Cramer [6] and a paper explicitly on the measurement problem [7]. 

It is a theme of this blog that events in the micro-domain cannot be restricted to measurements so the argument will focus on absorption events with measurements as a special case. A helpful list of RTI terminology has been provided [7] and it will be adapted here to begin the structured argument. In a further adaptation I will seek to avoid mention of waves because pictures of wave propagation in empirical space have been a major source of confusion.

1. micro-emitter: an excited quantum object with the propensity to release of quantum of energy (\(\hbar \omega\)) 
2. micro-absorber: a quantum object with a propensity to be excited from a lower energy state to a higher one on receiving offered the quantum energy
3. macro-emitter: a collection of \(N_E : N_E \gg1 \) micro-emitters 
4. macro-absorber: a collection of \(N_A : N_A \gg 1\) micro-absorbers
5. emission: a micro-emitter generates an offer \(|\omega>\) of energy \(\hbar \omega\)
6. absorber response: a micro-absorber generates a confirmation (\(<\omega , k|\))
1. corresponding to the component of the offer received by it
2. instantiating a non-unitary transition 
7.  absorption: an actualised transaction in which conserved quantities (e.g. energy) are transferred from the emitter to a particular micro-absorber, resulting in excitation of the latter. This is irreversible (non-unitary) and random.

Setting aside the macro items for the moment, 1-2 indicate the elements in place to carry out a transaction. 5-7 is the transaction process resulting in absorption and a transfer of energy. The main part of the physics missing is the creation and annihilation of the boson that carries the quantum from the emitter to an absorber. In point 6 the presence of \(k\) was not explained. It has to do with the role of the photon. 

The photon

Although we are now working with a physical ontology consisting of a quantum substrate and a supervening space-time of events, there is a strong intuitional pull towards thinking of photons as quantised elements of electromagnetic fields in space-time. In standard classical electrodynamics it is possible to construct free waves solutions of the homogeneous (source free) Maxwell equations. A commonly held physical intuition would picture these free waves as having some distant source but this source only has a role in the physics narrative and does not affect the future propagation and eventual interaction of the wave. There are also solutions to the inhomogeneous Maxwell equations that show how the propagating fields are created and there is a theory of wave interaction with matter (charges). In contrast the Wheeler-Feynman classical theory of electromagnetism places emission and absorption in centre stage and not only diminishes but eliminates the free field [2, 3]. 

Both forms of classical electrodynamics are now replaced with some form of Quantum Electrodynamics (QED). Standard quantum electrodynamics [e.g., 8] carries over some intuitions from standard classical electrodynamics. Quantisation produces two kinds of photon commonly known as real and virtual. The real photon corresponds to quantising the freely propagating classical wave. The virtual photon mediates interaction between moving charges. The virtual photon is created and destroyed with a finite lifetime as it mediates the interaction. This virtual photon is sometimes referred to as internal because of how it is portrayed in Feynman diagrammes, as shown below for the electron-electron interaction [8, Fig. 7-3]

Davies recognises this virtual and real terminology is confusing [5] in direct action quantum theory. I can only agree. Transactional quantum mechanics has taken strong guidance from the mechanisms of classical Wheeler-Feynmann electrodynamics. In clarifying the physics of the transaction RTI builds on the direct-action QED developed by Davies [4, 5]. It is a theory of photon exchange but with ontological ambiguity about the status of the photon. As indicated above, in the RTI the physical stratum of the ontology includes two interplaying layers: the quantum substrate (QS) and empirical space-time with its entities. To decide on which layer entities belong requires some analysis. In all mathematical physical theory, there are constructs that correspond to entities in the ontology and some that do not - they are useful conceptual scaffolding. Just as in standard quantum mechanics, there may be versions of RTI that gives ontological preference to a wave picture or one that gives preference to a statistical transition picture. Ruth Kastner's presentation makes use of a wave picture, with offer and confirmation waves. I am interested in seeing how far a probabilistic transition picture can be developed. To help in this, rather than carrying over the "real" and "virtual" terminology from standard QED, it may be more productive to embrace fully the language of potentiality. 

A quantum subsystem (QS entity) may have the potential to emit a quantum of energy. For example, it may be a QS atom in an excited state. It may also have potential to do other things such as bond with another atom to form a molecule. How that potential may or may not be realised depends not only on the atom but its environment. In the QS we must not think about this environment in four-dimensional space-time terms. Here the generalised probability formulation of quantum mechanics may help.  In this formulation the quantum sub-system of interest has potentialities that are conditioned on other entities in the QS. It is proposed that this conditioning is a QS physical effect and not just a mathematical manipulation in the theory. It is transactional. So, a quantum sub-system has a set of potentialities, and these potentialities form a complex that can separate into subsets of possibilities that have well defined probability spaces, in their mathematical description. Physically they are events that are conditioned to occur. Using a concept from Barbara Vetter's theory of Potentiality, we have a two-step iterated potentiality:

1. The potential possibilities that the sub-system could manifest are encapsulated within a complex (represented mathematically by the \(\sigma\)-complex)
2.  In the first step, the QS environment interacts with the sub-systems to realise sub-sets of statistical potentiality. This can be thought of as a step to propensity with statistical weightings realised through conditioning
3. In the second step the possibilities of the statistical potentiality are sampled by the other entities in the QS, and this can, but need not, lead to a space-time event. 

I have used the term "sampled" as it is familiar from statistics. It should be emphasised that this is not the observational sampling that statistical offices undertake or that a moderator may do by extracting a numbered ball from an urn in a lottery game. Here we have a physical active transactional sampling of one part of the QS by another. This can also be considered to be the physical interpretation of the role of projection operators associated with the sub-system's manifestation possibilities. This is an alternative to Ruth Kastner's symmetry breaking explanation as to why only one of a set of possibilities is realised. More generally [6] a representation of the physics is formulated in terms of creation and annihilation operators for photons acting on a Fock space [8]. This in turn has a natural representation in terms of projection operators, as used in the adapted Kochen re-formulation of standard quantum mechanics. 

The rich mathematical structure describing the QS cannot be accommodated in four-dimensional space-time. This was recognised previously (*) but then usually neglected to concentrate on empirical and technological success. Or work with an otology restricted to "be-ables" in four-dimensions. The four-dimensional space-time does, however, seem to represent the empirical space where events take place and experiments are carried out.  If empirical space supervenes on the quantum substrate, then to what extend does the QS determine the space-time structure? 

The structural laws of space-time may emerge in the empirical space and not be determined in anyway by the structure and laws of the QS but this should not be assumed prior to investigation. For both standard QED and RTI quantum theory the space-time structures developed in relativity theory strongly influenced the mathematical form of the theories. So, the relativistic mathematical structure of space-time has been inserted, or pre-supposed, in constructing the theories. As we only have direct access to events at the empirical level it is inevitable that the space-time structures describing physics at that level inform the mathematical representation of the QS. However, it has transpired that while locally observed quantum events obey relativistic locality the statistical corelations between space separated events do not always do so.  So, despite having used space-time relativistic structure to guide the construction of the mathematical representation of the QS, theory leads to predicted violation of relativistic principles at the empirical level. That is, robust statistical correlation is observed between space-like separated events. It is only through the construction of experiments based on the EPR "paradox", motivated by quantum theory, which has led to the confirmation of this effect. 

What has been developed in RTI quantum theory is a theory of emission and absorption of photons that provides a mechanism for no-unitary system dynamics.  However, more generally, other particles can be emitted and absorbed (or so it seems from second quantisation of standard quantum mechanics). For example, the electron dynamics also can be described using creation and annihilation operators acting on a Fock space. What we cannot do here is be guided by classical wave theory of emission and absorption of point particles - there is no such thing. Tackling these issues will need a deeper dive into the mathematical structure of the theory.  

 

(*) Gentleman: there's lots of room left in Hilbert Space. Sanders Mac Lane

References

1. Ruth E. Kastner, The Transactional Interpretation of Quantum Mechanics - A Relativistic Treatment, Cambridge University Press, second edition 2022
2. John A. Wheeler, Richard P. Feynman, (1945). Interaction with the Absorber as the Mechanism of Radiation. Reviews of Modern Physics. 17 (2–3): 157–181.
3. John A. Wheeler, Richard P. Feynman, (1949). "Classical Electrodynamics in Terms of Direct Interparticle Action". Reviews of Modern Physics. 21 (3): 425–433. 
4. Paul C. W. Davies, (1971) Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain I. Scattering Processes, J. Phys. A: Gen. Phys. 4, 836.
5. Paul C. W. Davies, (1972) Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain II. Emission Processes, J. Phys. A: Gen. Phys. 5, 1025-1036.
6. Ruth E. Kastner, and John G. Cramer, (2018). Quantifying absorption in the Transactional Interpretation. International Journal of Quantum Foundations 4:3, 210–22.
7. Ruth E. Kastner, (2018) On the status of the measurement problem: recalling the relativistic transactional interpretation. International Journal of Quantum Foundations 4:1, 128–41.
8. James D. Bjorken and Sidney D. Drell, (1964) Relativistic Quantum Mechanics, McGraw-Hill Book Company

Conditional states, potentiality and experiment

In this post some of my earlier posts will be reviewed using insights from Ruth Kastner's Relativistic Transactional Interpretation (RTI) [1] of quantum physics, which was discussed in the previous post. In fact, the term interpretation under sells this reformulation and its physical insights. Relativistic Transactional Formulation or even Relativistic Transactional Theory could be more appropriate.  However, it is referred to, I will be using it to re-examine the concept of quantum conditional probability and Bohm's version of the Einstein, Podolsky, Rosen (EPR) thought experiment, that use space-like separated Stern-Gerlach detectors. Of course, the experiment has now been carried out using various physical implementations and the results are generally considered to be robust.

In an earlier post, on quantum chance, I used the Kochen formulation of quantum mechanics [2] to discuss a generalisation of conditional probability,

$$\begin{eqnarray}\label{eq2:reduction}
p(X\mid Y) =  \text{tr}(Y \rho YX)/ \text{tr}(Y \rho Y).  \kern 4pc 
\end{eqnarray}$$

Where $p$ is a state of the system of interest on the \(\sigma\)-complex $Q(\mathcal{H})$ and $Y$ a projection operator in $ Q(\mathcal{H}) $ such that $p(Y) \ne  0$. For the mathematical background see the post A Mathematical Foundation of Quantum Mechanics or Kochen's paper [2]. The equation is shown to have the form of classical conditional probability when \(X\) and \(Y\) commute. Otherwise there is an additional interference term. Kochen uses this quantum conditional probability to analyse the Measurement Problem. Conditioning the outcome on the set of possible detections is a mathematically elegant way of seeming to derive the Born Rule. Kochen recognises that unitary evolution of the system state does not give results in agreement with what is observed but then he goes on to say

The present interpretation stands the orthodox interpretation on its head. We do not begin with the unitary development of an isolated system, but rather with the results of a measurement, or, more generally, of a decoherent interaction.

p(Y)≠0

That is, the observable \(X\) of the system of interest is conditioned by the projection on to the set of detection projection operators \({Y_i}_i\). This is interpreted by Kochen as state reduction. However, just as in classical probability the quantum generalisation of conditional probability describes an association rather than causation. Kochen provides no physical mechanism for measurement to cause the reduction or why one of the set of possible outcomes occurs. The mathematical formalism shows that if an event is observed then the description of the system is reduced to the corresponding state. He does then say

... symmetry-breaking processes do take place in isolated compound systems with internal decoherent interactions during reduction of state.

 However, decoherence is at best a research topic rather than an established mechanism. Kochen's is firstly a mathematical description but can be interpreted as the detector taking an active role, just as in the Transactional Interpretation, but he does not provide the physics for this.

The Relativistic Transactional Interpretation (RTI) [1] does provide an explanation for the physical mechanism for identifying what set of physical objects play the active detector role for an object of interest. It does this by going outside non-relativistic quantum theory. Relativistic quantum theory provides a description of particle creation and annihilation, and boson mediated interaction (or transaction). However, it is still possible to do useful calculation in the non-relativistic formulation, but it is an approximate theory in a new sense now. It has long been recognised that it is a low energy theory but in addition it must now be recognised that the Born rule (von Neumann-Luders Projection Rule) is explained outside of formulation although it can be used mathematically within the non-relativistic formulation through conditioning the quantum state on an event. Or more precisely conditioning on the projection on to the event.

As an aside, the term that Kochen uses to refer to the properties in the \(\sigma\) -complex is extrinsic and for properties of the type familiar from classical physics intrinsic. They look like they correspond to the quantum substrate (QS) of potential properties and the actual space-time events, respectively, RTI theory. Kochen provides a mathematical description of consequences of an actual event but not the physics.

Now that we have a theory (RTI) that provides a physics of actual events it will be instructive to revisit a specific example.

In Locality and Quantum Mechanics, the experimental configuration of two space-like separated Stern-Gerlach measurement systems was discussed, as illustrated below.

The mathematical treatment will not be repeated here but commented on using the language of quantum substrates, potentiality, possibility, actuality, and events. However, the first use that I will make of the language is to augment the ontology presented in Strata of Real Being. The inorganic stratum on which all other levels supervene now gets split into two levels. They are the more fundamental quantum substrate and supervening space-time. It is possible that eventually there may be the need to introduce a yet more fundamental level but ontology strata have considerable autonomy and so we can put that consideration to one side for now. For example, understanding and explanation of most of social life, through social entities such institutions, individuals, and groups, need not take account of quantum physics. 

The discussion of a specific experiment should help to develop a better understanding of the relationship between these two levels in the inorganic layer.

The experiment

A system of two particles is prepared in a state of total spin zero. This state exists in the quantum substrate. Its preparation would, however, involve apparatus and presumably scientists existing and acting at higher levels of the ontology. The experiment as a whole involves the preparation of two Stern-Gerlach setups so that they are space-like separated when the particles arrive. This arrangement is situated in the space-time domain of actual events.  In the analysis, the magnetic field is treated classically, this is an approximation to the field description in quantum electrodynamics and can be traced to its origin in the QS although that plays no role in the analysis where the magnetic field is is an actual field. This allows the Schrödinger equation to be solved using a semi-classical coupling of the magnetic field to the spin operators of the particles.  The \(x\) coordinate (see the figure) plays no explicit role in the mathematical analysis. The fact that the two Stern-Gerlach arrangements are space-like separated is merely part of the thought experiment narrative. The spin operators and the associated spinor states have no \(x\) component. The position coordinate \(z\) comes into play only through the coupling to the classical magnetic field. The coupling parameter for the magnetic field in the Hamiltonian \(\mu = \frac{e \hbar}{2 m c}\) is made up of what are usually termed fundamental constants. RTI puts \(e\) and \(m\) (the electron charge and rest mass) in the QS. \(c\) is the speed of light and, as RTI puts actualised photons in the space-time level, it is an actual space-time constant. \(\hbar\) is associated with the dynamics in the QS.  

At least in the non-relativistic approximation, the QS may not be a domain of pure potentiality because \(e\) and \(m\) always have actual values (there is no uncertainty). However, Barbara Vetter's theory of potentiality [3] allows for a potential property that has only one possible actualisation. This could be interpreted just as a form tidiness in the associated modal logic, but it should be recognised that such degenerate potentialities are permitted and can exist in the QS.

In RTI what seemed like non-causal non-local effects in the EPR scenario, using standard quantum theory, are robust correlations in the quantum substrate. In the QS what is going on is not much more mysterious than blue and green card example at a less fundamental level of reality. In the card example, as discussed before, the two cards are put in enveloped each. An associate takes one of the envelopes at random and leaves with it. I open my envelope and know immediately the colour of the card my associate has.  At higher levels of the ontological hierarchy properties are actual (cards have a specific colour) in the QS potential property values can be robustly correlated prior to taking actual values. The RTI theory predicts that this correlation makes an appearance in the statistical correlation of the space-time events that are the result of an EPR experiment. Other formulations give this result, but they do not provide an explanation in physics for the appearance of the events.

References

1. Ruth E. Kastner, The Transactional Interpretation of Quantum Mechanics - A Relativistic Treatment, Cambridge University Press, second edition 2022
2. Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).
3. Barbara Vetter, Potentiality: From Dispositions to Modality, OUP, Oxford 2015