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.

Prediction, indeterminacy, and randomness

 The previous post introduced the ETH (Events, Trees and Histories) approach to the foundations and completion of quantum theory. This post addresses why it is impossible to use a physical theory to predict the future and why quantum mechanics is probabilistic although the Schrödinger is deterministic.

Prediction uses avalable information to get knowledge about what will happen. It is an epistemic concept. The impossibility of prediction does not mean that the world is not deterministic, not governed by probabilistic law or even not governed by any laws at all. However, if we could predict the future reliably then this would be evidence that the world is deterministic. We will examine Fröhlich's [1] arguments.

Impossibility of prediction

It is always possible for someone to make a series of successful guesses about the future but what we are considering here is prediction based on available information and physical theory. To predict an event, we must know of everything that can affect that event. That is what the event is and when and where it takes place. Fröhlich's [1] argument is simple and illustrated by the diagram below.

The diagram uses the standard light-cone representation of space-time. The Future is at the top and the Past at the bottom. The predictor sits at the Present and has in principle access to all information in their Past light-cone. That is, they can access information on all past events but no information on what happens outside their Past light-cone. There is causal structure within the Past light-cone. Events 1 and 2 are space-like separated. They cannot influence each other. Event 3 is in the future of 2, so event 2 can influence 3. Of significance for the argument are the events outside the predictors light-cones (Future and Past) that are in the past light-cone of the predicted event. These events can influence what happens at the space-time point of the predicted event, but the predictor can have no knowledge of them. Therefore, the predictor cannot predict in principle what will happen at a future space time point. In practice it is common knowledge that prediction is seen to work many situations. These situations are controlled to isolate the predicted phenomenon from the influence outside the predictors control.   

Indeterminacy of quantum mechanics

The impossibility of reliable prediction does not imply indeterminacy.

Consider an isolated system. That is, over a period of time its evolution is independent of the rest of the universe. It is only for isolated physical systems that we know how to describe the time evolution of operators representing physical quantities in the Heisenberg picture (in terms of the unitary propagation of the system). 

In the Heisenberg picture states of \(S\) are given by density matrices, \(\rho\), acting on a separable Hilbert space, \(\mathcal{H}\), of pure state vectors of S as in the mathematical formulation presented previously. Let \(\hat{X}\) be a physical property of \(S\), and let \(X(t) = X(t)^∗\) be the self-adjoint linear operator on \(\mathcal{H}\) representing \(\hat{X}\) at time t. Then the operators \(X(t)\) and \(X(t')\) representing \(\hat{X}\) at two different times \(t\) and \(t'\), respectively, are related by a unitary transformation:

$$ X(t) = U(t', t) X(t') U(t,t') $$

where, for each pair of times \(t, t'\), \(U(t, t')\) is the propagator (from \(t'\) to \(t\)) of the system \(S\), which is a unitary operator acting on \(\mathcal{H}\), and \(\{U(t,t')\}_{t,t'} \in \mathbb{R}\) satisfy

$$ U(t, t') · U(t', t'') = U(t, t''), \forall  \mbox{ pairs } t, t', U(t, t) = 1 , \forall  t $$

However, in the Copenhagen interpretation, whenever a measurement is made, at some time t, say the deterministic unitary evolution of the state of \(S\) in the Schrödinger picture is interrupted, and the state collapses into an eigenspace of the selfadjoint operator representing the physical quantity that is measured and over that eigenspace the probabilities are given by Born’s Rule. This is what we have previously called the selection of a \(\sigma\)-algebra fron the \(\sigma\)-complex of \(S\).

As I am working towards a formulation of quantum mechanics that does not give a special status to measurement or observers. In the post Modal categories and quantum chance Born's rule is invoked as follows

The probability measure describes a contingent mode of being for the quantum system with a spectrum of valued that are possible and become actual. What is missing is an understanding of the timing of the actualisation. In all the versions of quantum theory considered so far time behaves the same way as in classical physics.

 While the question of timing is still to be resolved, quantum mechanics should be a theory that incorporates random events that are not derived from the deterministic evolution of the state. However, that evolution does govern the probabilities of becoming actual of property values associated with the spectrum of the self-adjoint operators representing the physical properties of \(S\). 

It should be noted that Bohmian mechanics provides an alternative model in which the uncertainty of the outcome is due to uncertainty in the initial conditions for the dynamical evolution. In the Bohmian theory the uncertainty is epistemic, and the dynamics is deterministic. The Bohmian theory needs to introduce this uncertainty into an otherwise deterministic theory to obtain empirical equivalence to standard quantum mechanics.

The formulation of the quantum mechanics presented in this blog is, so far, consistent with the ETH approach [1]. Some aspects of special relativity have now been introduced even though a relativistic formulation of quantum mechanics has not been presented yet. This will be part of the formulation of a mathematical description of the quantum "event".

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