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

Potentiality, possibility, and probability

In the previous post I returned to Heisenberg and his discussion of potentia in quantum theory and examined Barbara Vetter's modal metaphysics of potentiality in that context. The intention being to relate this metaphysics to the formulation of quantum mechanics as a \(\sigma\)-complex of potentiae, that captures the possible manifestations of quantum objects. The occurrence of actual events is then governed by the probability of actualisation, which depends on the physical context that the object finds itself.

Heisenberg's book on the philosophy of quantum mechanics is well known and I may have missed some recent work by others that is inspired by or develops his thinking.  A search found a number of papers but one, "Taking Heisenberg’s Potentia Seriously" by R. E. Kastner, Stuart Kauffman and Michael Epperson [1], provided an analysis that is close to the position that I have arrived at 5 years later and goes further in working out some consequences for our understanding of quantum theory. 

To better compare my position with that of Kastner et al, here is a brief list of my main points:

1. Physics is about the physical and not about, information, knowledge, or psychology (of course physics is knowledge and provides information.)
2. Ontology needs to encompass potentiality so that the dispositional nature of quantum processes can be captured in a physical theory
3. The quantum state is represented by a complex of probability models (modified Kochen formulation) and is not merely statistical
4. The manifestation (or actuality) of quantum events depends on the physical context (the mechanism for this is the outstanding puzzle)
5. The process of going from the potential to the actual is a probabilistic transition
6. Measurment is a physical process but there are physical events in the absence of measurement that the theory must be able to describe
7. The mathematical structure of quantum theory can provide important clues to the underpinning ontology.

The main points extracted from Kastner et al are:

1. A realist understanding of quantum mechanics calls for the metaphysical category of res potentia
2. Res potentia and res extensa are interdependent modes of existence
3. Quantum states instantiate in quantifiable form res potentia; ‘Quantum Potentiae’
4. Quantum Potentiae are not spacetime objects, and they do not obey the Law of the Excluded Middle or the Principle of Non-Contradiction.
5. Measurement is a real physical process that transforms Quantum Potentiae into elements of res extensa, in a non-unitary, acausal process
6. Spacetime (the structured set of actual events) emerges from a quantum substratum
7. Spacetime is not all that exists
8. There is a mathematical theory covering the above.

Although the above only provides the briefest summary of both points of view, the similarities should be obvious. 

The modifications I made to Kochen's formulation, although modest, were aimed at eliminating any temptation to think that we are dealing with some non-standard logic. So, point 4 by Kastner et al poses a problem and I don't think that it is well argued in the paper that potentiae do not obey the Law of the Excluded Middle or the Principle of Non-Contradiction (I note that in her book [2] Ruth Kastner makes no mention of this point). I think the appropriate logic for potentiality is far better captured in the proposal of Barbara Vetter. However, within point 4 the proposal that potentiae are not space-time objects is interesting and potentially fruitful. However, it does indicate an ontology that may be as extravagant as the multiverse interpretation. In mitigation the ontology captures a rich substratum of all possibilities rather than infinity of actual universes. 

Where they clearly go beyond my points is with their point 9. Whereas I have been comparing and contrasting standard quantum theory, the propensity interpretation, Kochen's reformulation, various Bohmian proposals, GRW collapse theories, the multiverse interpretation, and Fröhlich's ETH research project; Ruth Kastner has developed a specific theory of quantum potentiality [2]. She starts with an interpretation that I have neglected so far; the Transactional [3]. It is an interpretation that gives physical importance to an aspect of the mathematical formulation that is usually considered a mere calculation device. In the Dirac notation, for standard quantum mechanics, a state \(\Psi\) is denoted by the ket \(|\Psi>\). The observables, represented by Hermitian operators, act on the ket as follows \(\Omega |\Psi>\) and in standard theory it is simply a calculation device to gain the expected value of the observable in the state to use the complex conjugate of the state, the bra, \(<\Psi|\Omega |\Psi>\).

However, in the transactional interpretation \(<\Psi|\) gains a physical significance. It is the confirming echo from the absorber (or detector) to the emitter's potential for observable properties to become actual. Despite this attractive proposal Cramer's original formulation has some issues. There is backward causation. This is what many would consider anti-causation because the effect precedes the effect. There are some other issues that Ruth Kastner proposes a solution for in her book and we will examine later. She takes Cramer's formulation and, building on other work, develops a relativistic formulation. She shows that this is needed to avoid some of Cramer's difficulties. 

I now believe that Kastner's formulation and ontology provides a very promising approach to gaining a deeper understanding of the quantum domain and it will therefore provide the focus of the coming posts. Even if it turns out to have some flaw the analysis should rewarding.

Some of the points I address will be presentation and terminology. For example, Vetter's metaphysics indicates a process going from potential to possibility and (through weighting) probability. So, for me quantum states will represent potentiality with the set of possible events represented by the spectrum of the associated observables. The probabilities are gained by decomposing the potentiality in the basis associated with a particular observable. I will als try to eliminate reference to the wave concept. I will stick by my preference for a Heisenberg picture because the wave concept to too closely tied to the space-time continuum.

1. Ruth  E. Kastner, Stuart Kauffman and Michael Epperson, Taking Heisenberg’s Potentia Seriously, International Journal of Quantum Foundations, March 27, 2018, Volume 4, Issue 2, pages 158-172
2. Ruth E. Kastner, The Transactional Interpretation of Quantum Mechanics - A Relativistic Treatment, Cambridge University Press, 2nd edition 2022
3. J. G. Cramer, (1986). The transactional interpretation of quantum mechanics, Reviews of Modern Physics 58, 647–88.

Metaphysical potentiality and physics

Heisenberg in one of his excursions into philosophy (Physics and Philosophy) discusses epistemology, logic, and ontology. In developing his ideas on how the innovations of quantum mechanics impact on philosophy he introduces (or rather reintroduces with reference to Aristotle) the concepts of potentiality.

He comes to potentiality through considering the logic required to refer to things that quantum mechanics describes. That is, it ...

... concerns the ontology that underlies the modified logical patterns. If the pair of complex numbers represents a "statement" in the sense just described, there should exist a "state" or a "situation" in nature in which the statement is correct. We will use the word "state" in this connection. The "states" corresponding to complementary statements are then called "coexistent states" by Weizsäcker. This term "coexistent" describes the situation correctly; it would in fact be difficult to call them "different states," since every state contains to some extent also the other "coexistent states." This concept of "state" would then form a first definition concerning the ontology of quantum theory. One sees at once that this use of the word "state," especially the term "coexistent state," is so different from the usual materialistic ontology that one may doubt whether one is using a convenient terminology. On the other hand, if one considers the word "state" as describing some potentiality rather than a reality —one may even simply replace the term "state" by the term "potentiality" —then the concept of "coexistent potentialities" is quite plausible, since one potentiality `may involve or overlap other potentialities.

In the version of quantum mechanics that I have been developing these "coexistent potentialities" should be understood in relation to the \(\sigma\)-complex. Despite his hints, Heisenberg does not succeed in including potentiality in the required logic. What is needed is a modal logic that includes a POTENTIALITY operator. Barabara Vetter constructs such a modal logic and defends it with modal metaphysical arguments in her book Potentiality: From Dispositions to Modality. There is a need for a theory going beyond the naive position that there are objects that simply have the potential to have certain properties. In quantum mechanics in particular it is not sufficient to say than an electron has the potential to being in some place. If this is elaborated to saying that it has some position with a certain probability, we get a deceptive oversimplification. The theory of potentiality developed by Vetter give potentialities the status of properties of objects. Contrary to Heisenberg 's view that "one considers the word "state" as describing some potentiality rather than a reality". Vetter's theory allows us to enlarge the domain of what is real to potentialities. 

In standard quantum mechanics observables can take possible values. The possibility is to be described by the theory and the values are represented by the spectrum of the self-adjoint operator that in turn represents the observable.  Vetter proposes, in general, to define possibility as follows: 

POSSIBILITY: It is possible that p \(=_{\mbox{df}} \) Something has an iterated potentiality for it to be the case that p.

We have now moved on to potentiality and more generally its iteration.  I have discussed dispositions in previous posts and now explore Vetter's proposal that we call those properties which form the metaphysical background for disposition ascriptions potentialities. In the definition the something also need explanation. Something can be an object or any collection of objects. 

Potentialities are individuated, not by a pair of stimulus and manifestation and a corresponding conditional, but by their manifestation alone. The manifestation of a potentiality is the property that the potentiality’s possessor would have if the potentiality were to be manifested. A manifestation can be a property of and object (or collection of objects) or a relation that the object (collection) has with something else. 

Joint potentiality

The relation between a potentiality and its manifestation, the relation ‘...is a potentiality to ...’. Manifestations individuate potentialities and therefore the classification of manifestations as relations or properties provides a classification of the joint potentialities:

Type 1 joint potentialities are joint potentialities whose manifestation is a relation between, or a plural property of, all its possessors. 

Type 2 joint potentialities are joint potentialities whose manifestation consists in a property or relation of only some, but not all, of its possessors.

In type 1 joint potentialities, the manifestation is a relation (or plural property) holding non-trivially between all its possessors, such as for an example of the door key's joint potentiality with the door for the former to unlock the later. In type 2 joint potentialities, however, the manifestation of a joint potentiality concerns only some of its possessors; such are the cases of 

• a uranium pile (with a potential to go unstable) plus the rods and fail-safe mechanism
• a glass (with a potential to brake) but protected by styrofoam,
• a city (with a potential to be destroyed) and its defence system. 

In all three only one of the objects manifests the disposition or potentiality but they have jointly the potentiality, to a certain degree.

Extrinsic potentiality

An object has an extrinsic potentiality if the object has a joint potentiality whose co-possessors are the dependees of the extrinsic potentiality. To return to the key and door example, in the case of the key’s extrinsic potentiality to open the door, the dependee and co-possessor (the door) was part of the potentiality’s manifestation (unlocking the door). In general, where a potentiality’s manifestation consists in a relation to a particular other object—such as opening this particular door, as opposed to opening some door with a lock of a specific shape—the potentiality will be extrinsic, and the object involved in the manifestation will function as its dependee and as the co-possessor of the relevant joint potentiality.

Iterated potentiality

As well as joint or extrinsic the potentialities can be iterated. That is, objects can have the potential to have potentials.

Objects and collections of objects have potentialities to possess properties. Potentialities themselves are properties. So, things should have potentialities to have potentialities and the latter potentialities might themselves be potentialities to have potentialities. So, there is nothing to prevent things from having potentialities to have potentialities to have potentialities, or potentialities to have potentialities to have potentialities to have potentialities ... and so forth. These are iterated potentialities.

Formalising potentiality 

In Vetter's theory potentialities include dispositions and abilities as properties of individual objects or collections of objects. However, in this theory POTENTIALITY cannot be defined because it is the metaphysical underpinning of dispositions and consequently possibilities. Vetter uses examples such as those I have discussed in explaining dispositions in earlier posts. Ordinary language semantics is also used as when the potential to do or be something is discussed.  It is on this bases that potentiality is then posited to be metaphysically fundamental. 

Although POTENTIALITY cannot be formally defined it can be formalised to show how it operates. Just as there are modal operators for necessity and possibility that act on propositions, the modal potentiality operator is called POT. 

POT must [therefore] be a predicate operator which takes a predicate—specifying the potentiality’s manifestation—to form another predicate, which can then be used to ascribe the specified potentiality to an object.

Formally, where upper case Greek letters are predicates,

$$ \mbox{POT} [\Phi](t)$$

ascribes to t the potentiality to \(\Phi\), where \(\Phi\) is an \(n\)-place singular predicate and \(t\) is a singular term, or \(\Phi\) is an \(n\)-place plural predicate and \(t\) is a plural term. Potentialities can be described by more general sesntences rather than just predicated. For the POT operator to work with sentences rather than predicates, let the sentence \(\phi\) be for something to be such that \(\phi\) is true is equivalent to it have the potentiality to \(\Phi\).  To express logically complex predicates and to turn closed sentences into ‘such that’ predicates, we need to introduce a standard predicate-forming operator, \(\lambda\).

$$ \mbox{POT} [\Phi](t) \rightarrow  \mbox{POT} [\lambda x.\phi](t)$$

 Where \(\phi\) is a sentence, open or closed, and \(\phi[t/x]\) is the result of substituting a term \(t\) for any free occurrence of \(x\) in \(\phi\), the sentence \(\lambda x.\phi (t)\) is true just in case \(\phi[t/x]\) is true. Intuitively, \(\lambda x.\phi\) turns the sentence \(\phi\) into a predicate meaning ‘is such that \(\phi\)’. 

Although there are further subtleties it is now possible to formulate a simple form of iterated potentiality.

$$ \mbox{POT}[\lambda x. \mbox{POT}[F](x)](a) $$

which ascribes to \(a\) an iterated potentiality to be \(F\). This formalisation can be iterated further.

The electron again

Returning to an electron described by quantum mechanics. An electron has the potential to be somewhere, to have a momentum, to have charge and a mass. But the electron description in quantum mechanics has even more structure. It appears somewhere with a certain probability in certain circumstances. Quantum mechanics does not just provide bare probability distributions for each potential property but has those probability models combined in a \(\sigma\)-complex; corresponding, I propose, to the "coexistent states" of Weizsäcker. The electron has a joint potentiality with other objects to manifest one of the probability distributions from the complex, in certain circumstances. Then that probability distribution weights the appearance of a specific set of values of the property with a numerical probability. Therefore, the probability that the electron will appear in a specific region of space in certain circumstance is an iterated potentiality. The electron has the potential to manifest one of a complex of probability distributions and a probability distribution captures in mathematical form the potential for the electron to appear in a region of space.

This example indicates how the concept of potentiality, as formalised, can be used to describe quantum processes. It remains to be seen how useful this will be when we introduce considerations of spin, entanglement, interference etc. This will be explored in future posts.

 

Events in quantum mechanics: a simple proposal

In standard quantum mechanics events are observations. The occurrence of an event needs an experimental set up if not an actual observer, but an actual observer participates in thought experiments or conundrums such as Wigner’s friend or Schrödinger’s Cat. I do not deny that observing experimenters exist but claim that they are not essential to the workings of the physical world. I have examined well known experiments:

• Double slit interference
• Stern-Gerlach

In these previous posts it has been shown that a local interaction of a quantum particle with a pointer or a Stern-Gerlach setup can select a $\sigma$-algebra from the $\sigma$-complex that describes the possible properties of the quantum particle. This selection happens prior to detection. The interaction causes the complex of potential $\sigma$-algebras to reduce to one.

Once there is a selected $\sigma$-algebra we have a classical probability description of the situation. In this case we can propose that an event is simply drawn from the probability distribution. This is like sampling in classical statistics. The actualisation form potential values is caused by the interaction and is the immediate next step once the complex has been reduced to one algebra.
After this actualisation, the particle may proceed to a detector and its properties recorded in accord with the purpose of the experiment.

Therefore, this outline theory of quantum events agrees with the empirical content of standard quantum mechanics.

Locality and quantum mechanics

Until now, I have concentrated on trying to free quantum mechanics, as far as possible, from reference to measurement but quantum mechanics also has a problem with locality. However, firstly it is worth remembering that classical mechanics also had a locality problem. This is exemplified by the Newton's theory of gravity followed by Coulomb's law of electric charge attraction and repulsion. In both cases any local change in mass or charge, whether magnitude or position, had an instantaneous effect everywhere. There was no mechanism in the physics for propagation of the effect. The solution to this was found first for electricity in combination with magnetism. Faraday proposed the existence of a field. The mathematical formulation of this concept by Maxwell led to the classical electromagnetic theory and provided a propagation mechanism. 

The success of electromagnetic theory brought to the fore two problems with classical dynamics. The space and time translation invariance in classical Newtonian dynamics did not follow the same transformation rules as in the electromagnetic theory and there was still no mechanism for the propagation of gravitational effect. As is well known, Einstein solved both anomalies with first his special and then his general theory of relativity. 

By the time the general theory of relativity was formulated it was evident that classical theory had a further deep problem; it could not explain atomic and other micro phenomena. To tackle this problem solutions were found for specific situations. Max Plank introduce his constant \(\hbar\) to resolve the problem of the ultraviolet singularity in the black body radiation spectrum through energy quantisation. This same constant came to be fundamental in explaining atomic energy levels, the photoelectric effect role and more generally the quantisation of action.

Quantum theory took shape is the 1920's with the rival formations that agreed with experiment, by Heisenberg and Schrödinger (with much help from others), shown to be formally equivalent.  The space time translation symmetry of special relativity was also built into an equation for the electron proposed by Dirac that in turn implies the existence of anti-matter. But a fully relativistic quantum mechanics remains a research topic. 

To combine particle theory with electromagnetism quantum electrodynamics was developed. This theory was remarkably successful in its empirical confirmation but relied on some dubious mathematical manipulation. To deal with this the mathematical foundations of quantum field theory were examined. It is at this point that the first type of locality that we are going to consider appears in quantum theory in mathematically precise form.

Causal locality

 A basic characteristic of physics in the context of special relativity and general relativity is that causal influences on a Lorentzian manifold spacetime propagate in timelike or light-like directions but not space-like. Space-like separated points in space-time lie outside each other's light cone, which means that no influence can pass from one to the other. 

A further way of considering causality is that influences only propagate into the future in time-like and light-like directions, but this is not simple to dealt with in either classical special relativity or standard quantum mechanics because of their time reversal symmetry.  One approach would be to treat irreversible processes through coarse-grained entropy in statistical physics. But this seems more like a mathematical trick or treats irreversibility because of a lack of access to the detailed microscopic reversable dynamics. That is, as an illusion. A more fundamental approach is to develop a new physics as is being attempted by Fröhlich [1] and hopefully in this blog.

To return to Einstein causality, any two space-like-separated regions of spacetime should behave like independent subsystems. This causal locality is, with a slightly stronger technical definition, Einstein causality. This concept of locality when adopted in relativistic quantum theory (algebraic theory) implies that space-like separated local self-adjoint operators commute. This is sometimes known as microcausality. Microcausality is causal locality at the atomic level and below.

In quantum theory, where operators represent physical quantities, the microcausality condition requires that any operators commute that pertain to two points of space-time if these points cannot be linked by light propagation. This commutation means, as in standard quantum mechanics, that the physical quantities to which these operators correspond can be precisely determined locally, independently, and simultaneously. However, the operators in standard quantum theories and the non-relativistic alternatives discussed so far in this blog don't have a natural definition of an operator that is local in space-time.   For example, the position operator is not at any point in space. The points in space are held as potential values in the quantum state that is represented mathematically by the density matrix.  How these potential values become actual is dealt with in standard quantum mechanics by the Born criterion, which is, however, tied to measurement situations. To remove this dependence on measurement situations is a major aim of this blog and we will see that measurement only need be invoked when discussing how various form of locality and non-locality are known about.

As the introduction of classical fields cured Newtonian dynamics of action at a distance and eventually modified then replaced it with General Relativity, the development of quantum field theory could cure standard quantum mechanics of its causal locality problem. Local quantum theory as set out in the book by Haag [2] tackles this challenge. The technical details involved are too advanced to deal with here.

Although dealing with these questions coherently within non-relativistic quantum theory is not strictly valid it is possible to explore specific examples. Following Fröhlich [1], it is natural to consider the spin of the particle to be local to that particle. Therefore, the spin operators, whether represented by Pauli matrices or by projection operators that project states associated with some subsets of the spin spectrum, can be assigned unambiguously to one particle or another. 

In a situation where two particles are prepared so that they propagated in opposite directions their local interactions with other entities will eventually be space-like separated. The spin operators of one commute with the spin operators of the other. The local interaction of one cannot then be influence by the local interaction of the other. This is a specific example of microcausality. 

But what if the preparation of the two particles entangles their quantum states? This entanglement may persist over any subsequent separation, if the particle does not first undergo any interaction with other particles or fields. 

We note that entanglement is a state property whereas microcausality is an operator property and proceed to a discussion of entanglement and its consequences in a developed version of the two-particle example.

Entanglement and non-locality

The two-particle example we have been discussing only needs the introduction of a local spin measurement mechanism for each particle for it to become the version of the Einstein Podolsky, Rosen thought experiment formulated by David Bohm [3]. This post will follow Bohm's mathematical treatment closely but will avoid as far as possible invoking the results of measurements. Bohm's discussion follows the Copenhagen interpretation but also uses the concept of potentiality as developed by Heisenberg [4].

The system in this example consists of experimental setups (described below) for two atoms (\(1\) and \(2\)) with spin \(1/2\) (up/down or \(\pm\hbar\)). The \(z\) direction spin aspect of the state of the total system consists of four basic wavefunctions

$$ | a> = |+,z,1>| +,z,2) > $$

$$ | b> = |-,z, 1>| -,z, 2 > $$

$$ | c> = |+, z,1>| -,z, 2 > $$

$$ | d> = |-,z, 1>| +,z, 2 >, $$

it will be shown below that although the choice of the \(z\) direction is convenient the results of the analysis do not depend on it.

  

If the total system is prepared in a zero-spin state, then it is represented by the linear combination

$$ \tag{1} | 0> = |c> - |d>.$$    

This correlation of the spin states of the particles is an example of quantum entanglement.                                 

Each particle also has associated with its spin state a wavefunction that describes its motion and position. Theses space wavefunctions will not be shown explicitly here but are important conceptually because the particles aways move away from each other. The description of the thought experiment is completed by each particle undergoing a Stern-Gerlach experimental interaction at space-like separated regions of space-time, as shown below.

Two space-like separated Stern-Gerlach interaction situations.

The detecting screen is a part of an experimental setup that is need for confirming the predictions of the theory but not the physics of the effects. Here we are primarily concerned with the interaction of the particles with the magnetic field \(\mathfrak{H}\). The component of the system Hamiltonian for the interaction of the particle spin with the magnetic field is, from Bohm [3],

$$ \mathcal{H}= \mu (\mathfrak{H}_0 + z_1 \mathfrak{H}'_0 )\sigma_{1,z} +\mu (\mathfrak{H}_0 + z_2 \mathfrak{H}'_0 )\sigma_{2,z} $$

where \(\mu = \frac{e \hbar}{2mc} \), \(\mathfrak{H}_0 = (\mathfrak{H}_z)_{z=0}\) and \(\mathfrak{H}'_0 =(\frac{\partial \mathfrak{H}_z}{\partial z})_{z=0}\).  \(m\) and \(e\) are the electron mass and charge. \(c\) is the speed of light in vacuum. We also assume the magnetic fields have the same strength and spatial form in both regions but this not essential. It is also assumed that each particle interacts with its own local magnetic field at the same time. This is not a limiting assumption, but it is essential to assume that the time of the interaction is short enough for the local space-time regions to remain space-lie separated.

The Schrödinger equation can now be solved for a wavefunction of the form

$$ |\psi> = f_c |c> + f_d |d>$$

with initial conditions given by equation (1). The result is, once the particles have passed through the region with non-zero magnetic field strength

$$f_c = \frac{1}{\sqrt{2}}e^{-i \frac{\mu\mathfrak{H}'_0}{\hbar}(z_1-z_2) \Delta t} $$

and

$$f_d = - \frac{1}{\sqrt{2}}e^{i \frac{\mu\mathfrak{H}'_0}{\hbar}(z_1-z_2) \Delta t}. $$

Where \(\Delta t\) is the time it takes for the particles to pass through the magnetic field.

Inserting the above results into the equation for \(|\psi>\) gives the post interaction wavefunction

$$ |\psi>=\frac{1}{\sqrt{2}}e^{-i \frac{\mu\mathfrak{H}'_0}{\hbar}(z_1-z_2) \Delta t} |c> - \frac{1}{\sqrt{2}}e^{i \frac{\mu\mathfrak{H}'_0}{\hbar}(z_1-z_2) \Delta t}|d>.$$

Therefore, for a system prepared with total spin zero undergoing local interactions in space-like separated regions, as shown in the figure, there is equal probability for each particle to be deflected either up or down. However, because of the correlation when one is deflected up the other is deflected down. 

This may seem unsurprising because the total spin is prepared to be zero. No more surprising than taking a green card and a blue card, putting them in identical envelopes, shuffling them and then giving one to a friend to take far away. Opening the envelope you kept and seeing a green card means that the distant envelope contains a blues card. This, clearly, is not a non-local influence.

However, this is not the end of the story. As mentioned above, there is nothing special about the \(z\) direction of spin. The same analysis can be carried with \(x\) direction states, as follows 

$$ | a'> = |+, x,1>| +, x,2) > $$

$$ | b'> = |-, x, 1>| -, x, 2 > $$

$$ | c'> = |+, x,1>| -, x, 2 > $$

$$ | d'> = |-, x, 1>| +,x, 2 > $$ 

and again, the zero total spin state is

$$\tag{2} | 0'> = |c'> - |d'>.$$

Using the standard spin state relations (valid for both particles one and two, by introducing the appropriate tags (1 or 2), see Bohm [3])

\( |+,x> = \frac{1}{\sqrt{2}}(|+,z> + |-,z>)\) and \( |-,x> = \frac{1}{\sqrt{2}}(|+,z> - |-,z>)\)

Inserting into equation (2), with some algebra, it can be shown that 

$$ |0'> = |0>. $$

Therefore, if the Stern-Gerlach setup is rotated to measure the \(x\) component of spin, exactly the same analysis can be carried out as for the \(z\) component giving the same anti-correlation effect. It must be stressed that we are discussing physical effects and not the results of experiments or the experimenter's knowledge of events at this point.

In general, there is no reason for the two space-like separated setups to be chosen in the same direction.  If the choice is effectively random then when the direction of interaction does not coincide there will be no correlation between the outcomes but if they happen to be in the same direction, then there will be the \(\pm\) anti-correlation. Locally the spin operators for the \(x, y\) and \(z\) do not commute. Their values are potential rather than actual and remain non-actual after the interactions. The situation is not like the classical coloured cards in envelopes example. There is no direction of spin fixed by the initial state preparation. Indeed, that would be inconsistent with a total spin zero state preparation. What the interaction does is chose a \(\sigma\)-algebra from the local \(\sigma\)-complex but the spin state of the system remains entangled.

As far as local effects are concerned, each particle behaves as expected for a spin \(1/2\) particle. This is causal locality. It is only if someone gets access to a sequence of measurements from both regions (here is the only place where detection enters this description of the physics of this situation) that the anti-correlation effect can be confirmed. 

The effect depends on the preparation of the initial total system state. There is persistent correlation across any distance just as in the green and blue card example, but it is mysterious because the initial state does not hold an actual value of each spin component for each particle, unlike the actuality green and blue card example. There is no way for the one particle to be influenced by the choice of direction of measurement at the region where the other particle is, but a correlation of potentiality persists that depends on the details of the total quantum state.

It is perhaps too early to simply accept that there are non-causal, non-classical correlations of potentialities between two space-like separated regions.  That would be a quantum generalisation of the blue and green card example. What the theory does predict is that the effect due to entanglement is not just epistemic but physical once potentiality is accepted as an aspect of the ontology.

References

\(\mbox{[1] }\)	Fröhlich, J. (2021). Relativistic Quantum Theory. In: Allori, V., Bassi, A., Dürr, D., Zanghi, N.(eds) Do Wave Functions Jump? . Fundamental Theories of Physics, vol 198. Springer, Cham.    https://doi.org/10.1007/978-3-030-46777-7_19
\(\mbox{[2] }\)	Haag, R. (1996). Local Quantum Physics: Fields, Particles, Algebras, 2nd revised edition, Springer Verlag
\(\mbox{[3] }\)	Bohm, D. (1951). Quantum Theory, Prentice Hall
\(\mbox{[4] }\)	Heisenberg, W (1958). Physics and Philosophy: The Revolution in Modern Science. New York: Harper.