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