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11.1 Point Events

Up to now we have been constructing a philosophical framework for the earliest kind of quantum mechanics, and have not paid any special attention to interaction events. The actualising events considered so far have been simpler, and more like the spontaneous acting of a single homogeneous propensity field.     This field could reasonably be imagined to follow Schrödinger's equation, but we have not decided between whether the Schrödinger wave function is for `free field' substances (between interactions), or whether it is a many-body wave function including interaction potentials. The free-field equation for the p'th substance could be
$\displaystyle - \frac{\hbar^2}{2m_p} \nabla ^ 2 \psi ( \vec{x}_p , t)
= i \hbar \frac{\partial \psi (\vec{x}_p , t)}{\partial t} .$     (11.1)

  Interaction potentials of the form $ V_{pq} (\vec{x}_p ,
\vec{x}_q) $ between substances p and q mean that the many-body wave function $\Psi ( \vec{x}_1 , \vec{x}_2 , \ldots ,t)$could satisfy the Schrödinger's equation
$\displaystyle \sum_p [ - \frac{\hbar^2}{2m_p} \nabla ^ 2_p \Psi
+ \sum_q V_{pq} (\vec{x}_p , \vec{x}_q) \Psi ]
= i \hbar \frac{\partial \Psi}{\partial t} .$     (11.2)

The potentials Vpq may describe electromagnetic, gravitational or one of the stronger nuclear interactions.   The wave function needs to be defined over the configuration space $( \vec{x}_1 , \vec{x}_2 , \ldots ,t) $ of all possible places for interactions, as mentioned in chapter 4,     since all the interacting substances will in general have propensities for actualisation that are correlated with each other (in the EPR sense), because of previous interactions. Equations (11.1) and (11.2) are the Schrödinger form of quantum mechanics, but the same propensity distributions between times t1 and t2 could equally well be described in the Heisenberg representation,     or in terms of a Lagrangian density ${\cal L}(\vec{x})$ subject to a least action principle
$\displaystyle \delta \int _ {t_1} ^ {t_2} dt
\int {\cal L}(\vec{x}) d \vec{x} ~ = ~ 0 .$     (11.3)

For a particle of mass m in a potential $V(\vec{x})$, the Lagrangian density (to be subject to variations in $\psi$ and $\dot{\psi} $) is

\begin{displaymath}{\cal L}(\vec{x}) = i \psi ^\ast \dot{\psi}- \frac{1}{2m} \nabla \psi ^\ast
\nabla \psi - V \psi ^\ast \psi

Whether propensities contained interactions or not, point actualisations were assumed to occur, whereby they became momentarily localised at a point in spacetime. After that actual localisation, the field dispersed again according to one of the Schrödinger's equations above, and continued spreading out (for some finite time interval) until the next localisation event.     These localisations fulfilled the role of `measurement events' in ordinary QM, in being definite choices and having definite characteristics once they occurred.   As in N. Maxwell [1976], measurement processes in quantum physics were held to essentially involve these localisation events, and all kinds of measurements of different physical variables were held to be ultimately reducible to different combinations of `position measurements' in our actual localisation processes. Measurements are not assumed to essentially involve any conscious observer, but could be done by computer for example, as the localisation events were a spontaneous natural process that occurred because of the very nature of quantum substances as fields of propensity for actual events of that kind.


Can all interactions be actual events?

If we consider the first option, wherein the propensity fields describe substances while they are not interacting, then interactions must arise because of the actualising events. The actual events are thus the interacting, or the acting of one propensity field on another.   As Leclerc [1972, ch. 23] points out, `actings on' always involve a reciprocal capacity to `receive' in that which is acted upon, so that aspects of capability or propensity are required in both the agent and in the patient. That most events are interactions is also supported by quantum physics, where events which are the spontaneous action of a single particle are comparatively rare, because events such as spontaneous decays are not the most common type.
Figure: Successive Localisations of Propensity Fields via Overlap Interactions

  If an interaction between two propensity fields is to result in an actual event at a particular place, it will be necessary for the two propensity fields to overlap in at least that region.   This would be a generalisation for propensity fields of the principle of `interaction by contact', and follows in the philosophy of nature using the argument of section 8.2 that the mutual interaction of distinct substances presupposes that they have an extensive relation with respect to one another. It may be concluded from these two arguments that two propensity fields may only interact if they overlap in spacetime, and that, if they do interact to produce an actual event, then, as shown in 11.1, both fields are reduced in spatial extent at the time of the event. The likelihood for specific interaction events occurring will again depend on the form or distribution of the measures of the two propensity fields, and it is plausible that the probability of interaction will depend on something like the product of the two fields. Exactly how it works is of course for physics to determine.

The difficulty with this option is that it does not seem to be true of the quantum world.   There are many instances of interactions between quantum mechanical substances which do not appear to be definite outcomes, in the sense of definitely occurring, or definitely not occurring.   One such kind of interaction is between a low energy neutron and the atoms in a crystal from which it is being diffracted. The neutron is reflected from the crystal as though from a mirror or a diffraction grating, but, although this reflection arises from its interactions with all the atoms, no single interaction can be said to definitely occur or not occur. Rather, all the interactions appear to be operating in some conjoint fashion, as the subsequent propensity field of the neutron does not radiate out from any single atom, but remains spread out, as if it interacted with all the atoms together. It did not interact with any single atom actually and irrevocably.

  Another such kind of interaction occurs in an interferometer between the photons (or neutrons) and the various mirror surfaces. These reflection interactions cannot be actual, with only one alternative actually occurring, otherwise interference effects would never be observed.   A third kind of non-actualising interaction occurs when the diffraction pattern of an electron is deformed as it passes through subsequent electric or magnetic fields. If we treat electromagnetic fields as interactions with photons, then these photons are able to interact with the electron without a specific localisation, by means of events which are `virtual' rather than actual.

      Quantum field theory, the more advanced form of quantum mechanics, has all potentials arising from the exchange of `virtual particles'. These would be photons (for electromagnetic potentials), gluons (for nuclear interactions), W and Z bosons (for weak nuclear interactions), and presumably gravitons too for gravitational forces. The potential fields are not now continuously variable, but are composed of discrete `field quanta' (i.e. virtual particles), and these interact with ordinary substances via virtual rather than actual events.

    It thus appears that propensity fields can interact with each other, but without producing a fully actual result. These more gentle interactions are called `virtual events' in quantum physics, and are as it were the `hidden ingredients' of the many-body wave function $\Psi$, so that it includes implicitly the necessary kinds of interaction potentials. If propensity fields can interact with each other by means of vitual events, as distinct from actual events, then the propensity that makes up quantum substances cannot be just for pure actual entities, but must have some more complicated characterisation.

From the examples above, it does not seem empirically tenable to hold that all interactions must be actual events, although it may be logically possible. However much it may be a simple application of our `logic of process', we are obliged to consider other alternatives.  

Point localisations of the $\Psi$ function?

According to the second option, `virtual events' are already included in the quantum wave function $\Psi ( \vec{x}_1 , \vec{x}_2 , \ldots ,t)$, and actual events are the point localisations of this field.   That is, we take over Schrödinger's equation unchanged, and supplement by an actualising process which intermittently localises the dispersed wave field to successive points in space and time. We now want to examine the consequences of having point localisations. This assumption was made in chapter 8, where it was explicitly assumed that actualities were at point places in our four-dimensional space and time.   This assumption was a `contingent identification' of a term in the process logic, and is hence not absolutely necessary. In fact, it is almost certainly wrong, as we shall now see.

    The principal difficulty is that if point localisation did occur, then, though the propensity field would then (momentarily) have a completely determinate spatiotemporal form, its momentum (i.e. velocity) and energy distributions would become completely indeterminate. This is a simple consequence of the fact that a point distribution is that of a delta-function, and hence has a Fourier spectrum with all frequencies without abatement.

This indeterminacy in momentum would perhaps be acceptable if it existed only momentarily, like the spatiotemporal localisation. But since energy and momentum are conserved, the momentum and energy distributions will be unchanged with time. The infinite spread of momentum will thus persist, and the field will, in `hardly any time at all', spread out over the entire universe!

This infinite range of momentum and energy is a problem independent of another fact in ordinary quantum mechanics. There, it is impossible to localise wave packets to points, as that would require the measuring apparatus to have particles with infinitely small wavelengths, and hence infinite energies. This is an independent problem, because the point localisations being proposed for propensity fields are the characteristic products of these fields, and do not depend on the point localisation of any other fields.   If we had to use potentials in Schrödinger's equation, then point localisation would require an infinite amount of energy prior to the event, but the spontaneous point actualisation of propensity fields (as being considered) does not require the input of any energy at all. The trouble is rather that point actualisation seems to produce an infinite amount of energy in the fields after the event.

    In general, we would wish to see, for any proposed scheme for actualising, that the effects of the actualisation are a selection of the field extent not only at the time of the actualisation, but also at all subsequent times. We could call this a Principle of Selection, as an expression of the facts that actualities are a limited form of potentiality, and that propensity fields are the potentialities for future possibilities over a wide range of times. The principle is clearly violated by the point localisations being considered here, as immediately following, the propensity fields rapidly spread out independently of what they had been previously.

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Prof Ian Thompson


Author: I.J. Thompson (except as stated)