Modal categories and quantum chance

 The motivation for examining modal categories is to provide a more refined account of the ontology of dispositions and chance in quantum mechanics. Therefore, the emphasis will be on the real sphere although the modes apply also in the ideal sphere but in a way appropriate to that sphere. The modes of being are [1]

Necessity Not being able to be otherwise

Actuality Being this way and not otherwise

Possibility Being able to be one way or not

Contingency Not being necessary (also being able to be different)

Non actuality Not being so

Impossibility Not able to be so.

Intuitively necessity is more than actuality, actuality is more than possibility. This provides sense of direction because the lower mode is contained in the higher: what is actual must at least be possible, and what is necessary must at least be actual.  

With the negative modes impossibility is a minimum of being, extreme non-being.  Contingency is but the non-existence of necessity. The definiteness of the way of being is less even in non-actuality than in contingency, and even less in impossibility than in non-actuality?

 Of the three negative modes it is contingency that will be the most significant for our needs in quantum mechanics because it is a border case with a trace of positivity and provides a minimum opening to being-so. From the discussion of spin in the post on Quantum Objects it is contingent (not necessary) for a spin component to take only one of two values \(\pm \hbar/2 \). 

The language of uncertainty is inappropriate in the case of a world described by standard quantum mechanics. The values are not merely hidden but have no actual being until they come into existence.  Standard quantum mechanics does not provide a mechanism to describe the transition from contingency to actuality. It goes no further than to describe possibilities and the probability of them becoming actual.

In a world described by Bohmian Mechanics, the spin wavefunction guides the electron to take one of the two values after following a determined trajectory. For every exact starting point of the electron the end spin state is determined. It is only due to the practical impossibility of determining the starting point that makes the spin value seem contingent. It can be said to be epistemically contingent but not in reality. In the exactly specified situation the value that become actual become so necessarily.

If the world is as describe by Ghirardi, Rimini and Weber theory with the wavefunction still providing probabilities of outcome, the ontological situation is like standard quantum mechanics. This may seem puzzling because this theory is intended to dispel the mystery of quantum measurement (or, more generally, actuality), but even after the spontaneous wavepacket reduction, given by the GWR mechanism, some contingency remains or if the spin component value is actualised due to the spontaneous reduction nothing is explained by it. It is merely posited.  There is distinct version of this theory in which the wavefunction provides a mass density of the particle rather than a probability density. In typical situations the mass density will spontaneously reduce to a situation consistent with either \(+\hbar/2\) or \(-\hbar/2\). 

In the "may-worlds" description, the world is infinitely larger than can be observed. Only one of an infinite set of sub-worlds is observable. The spin component takes both values on two sub-worlds that are empirically separate although through corelation its known that if in one the spin takes value \(+\hbar/2\) then in the other it is \(-\hbar/2\). In this description of the world all values are actualised by necessity.

There are at least five descriptions of the world, including standard quantum mechanics, which are modally distinct in the real sphere.

Quantum chance

In an earlier post the mathematical formulation of quantum theory was presented in a way that brought probabilities rather than wavefunctions to the fore. In discussions of the meaning of probability the term chance is usually used when the probability does not describe a state of uncertainty or partial belief but objective random events. Before the event occurs the complete description of the state of the system is described by a probability measure.

Using the ontological terms adopted in this blog the probability measure over the \(\sigma\)-complex of the quantum system, an object in the inorganic level of real being, describes the objective chance that its properties, in the same inorganic level and represented by self-adjoint operators in the theory, will take one of a spectrum of possibilities that will become actual in that same inorganic level of reality.

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.

The timing of the actualisation of the possible values of the observable and the physical cause (at the inorganic level of reality) of this actualisation is the major open problem. 

The analysis of events in quantum tunnelling may provide some insight into this problem.

[1]   Hartmann, N. (2010) Möglichkeit und Wirklichkeit. 3rd edn. De Gruyter. Available at: https://www.perlego.com/book/1159964/mglichkeit-und-wirklichkeit-pdf (Accessed: 16 August 2022).