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