Putting the Fire in the Equations;    Generating multilevel dynamical processes in Physics and Psychology

 

10.3 Waves, Particles and Complementarity One feature of the present account of substances is that they are not necessarily located in small volumes of space, as, for example, the corpuscles or `particles' of classical physics would be. The propensity fields that have been defined do not even have any special `centre' distinguishable from all the other places in the field. They have no centre which could be regarded as the `true substance', so that the surrounding field could be regarded as just the `sphere of influence' of the central substance.   This was Boscovich's conception, and it slowly percolated into physics, resulting in the `dynamic matter' of the mid-nineteenth century. This view is best summarised by the aphorism ``No matter without force, no force without matter''. Our propensity fields, though, have no special continuing centre: the only `point source' which could perhaps be identified is the source event, which must have a definite location in space and time. The field is therefore only localised very briefly, if at all, at times just after this source event. The `continuants' we define are thus occasionally, but never necessarily, strongly localised. For most of the time they have significant spatial extensions. Various reconsiderations to ideas of particles have been suggested by quantum physics have over the last sixty years, but for the most part come fitfully and in scattered parts of which few physicists or philosophers were fully aware in a critical sense. 1.           Margenau [1950, 1954] advocates a `latency interpretation' of quantum mechanics. In this interpretation, what is real is the state function $\psi$ function, whereas the classical state observables, such as position, momentum or energy may be said to be latent in $\psi$ , in the sense that their values emerge only in response to measurement. The distinction between primary and secondary qualities is superseded by that between `possessed' and `latent' qualities or observables, the former exemplified by mass or charge, the latter by the observables with which the quantum theory of measurement is concerned. These later ``are not `always there', [but] take on values when an act of measurement, a perception, forces them out of indiscriminacy or latency.'' In the theory of propensities, Margenau's `possessed' qualities are parameters which describe strengths of propensities, and his `latent' qualities are parameters which describe the outcomes of operations of propensities. 2.     Heisenberg further notes10.4 that what a typical physicist of today tends to think is rather close to Aristotelean `potentia', even if unwittingly. The meanings of words such as `particle' have moreover gradually changed in these sixty years. 3.         Kaempffer [1965], for instance, after pointing out the `erosion of naive pictures of particles', goes on to suggest that the word particle stand for a quantum mechanical state [a wave field], characterised by a set of quantum numbers, which is associated, in principle, with an identifiable event such as the momentum transfer in a ``collision''. This conception of a particle (as a wave field associated with a definite event) has come a long way from the corpusclar theory, and is remarkably similar to the present account of a continuant as a propensity field which extends over the various places possible for actualising events. 4.   Müller [1974] is trying to formulate an objective interpretation that is materialistic as much as is possible. He interprets probabilities in terms of `potentially possible events', though he `doesn't wish to establish any ontological order'. He wants to describe objects while they are not being measured: `the pre-measurement condition of the object can be characterised only by the probability distribution of the potentially possible states' 10.5 .     Wave Behaviour? The point to note is that the continuant-field does not have a fixed spatial size. Sometimes it behaves more like a spread-out wave, and when at other times it interacts, it behaves like a localised particle. In fact, propensity fields can have practically any extensive shape over the places that are possible for it, as they are subject only to some field equation. We can allow that propensity fields are described by Schrödinger's time dependent equation, including the interaction potentials. This then allows them to propagate in interesting manners around obstacles or potentials which would stop any classical atoms. They can even tunnel through barriers, as the probability for a definite interaction may be reduced but non-zero. It becomes reasonable to expect the diffraction, interference and tunnelling effects we know in quantum physics from the solutions of Schrödinger's equation.       Particle Behaviour? On the basis of our account of propensity fields as continuants: There are no such things as small particles like corpuscles with definite properties. Nor are there such things as small particles with uncertain or indeterminate properties. Measurements are not the process of assigning values to properties of particles 10.6 , even if we allow that they are `peculiar particles' in not having definite properties at all past times. Nor are measurements the momentary production of particles with definite properties for that moment. To believe any of these is to believe that somewhere, as it were hidden away behind the propensities, there really exist particles waiting to appear. This is not the case. Questions like `Where is the electron and what is its speed?' have no answer, because there never exists such a thing as a small corpuscular electron. The only things that exist are propensity fields and the actualities they produce. Propensity fields are not like vague, indeterminate or smeared-out particles, but are perfectly definite entities in their own right. It may not be determinate in advance which actualities a propensity field will produce, but that does not mean that the propensity field is any the less real or definite when considered as a thing in itself. Its field structure can be described using perfectly definite mathematics. Its existence is as real and substantial as any existing object. In fact, as we saw in chapter 9, propensity fields are the very substances out of which all things are made! Nothing can be more substantial than them. It is only given all this, that we can follow Kaempffer above when he redefines the meaning of the word `particle' to refer to (something like) propensity fields! We may appear to be using particle ideas when we talk of `observables' such as position, energy and momentum. The meaning of these concepts seems to be irrevocably connected with the classical ideas of particles as atoms. In fact, position, energy and momentum can be given self-sufficient definitions as `variable' properties of propensity fields. They are variable, in the sense that they vary across the extent of the field.   `Position' obviously varies across the field, as fields were initially defined in terms of the range of places or positions within their range. If the field is a $\psi$ -function following Schrödinger's equation, then the probability distribution of `position' $\vec{x}$ at any given time is proportional to $\mid \psi (\vec{x} , t) \mid ^{2} .$  `Momentum' is a `variable' which also changes according to where you are in the field. If the field follows Schrödinger's equation, then the `local' component of momentum in a given direction is simply the logarithmic derivative of $\psi ( \vec{x} ,t)$ in that direction: $$p_j = \frac{\hbar}{i} ~ \frac{\partial}{\partial x_j} \ln \ps... ...ac{\hbar}{i} ~ {1}{\psi} ~ \frac{\partial \psi }{\partial x_j}$$   Similarly, the local value of the `energy' variable is the logarithmic derivative of the $\psi$ -function in the time direction of the extensive continuum: $$E = i \hbar ~ \frac{\partial}{\partial t} \ln \psi ( \vec{x} ,t) = \frac{i \hbar}{\psi} ~ \frac{\partial \psi }{\partial t}$$ The so-called `classical variables' of position, momentum and energy are thus functions or derivatives of the propensity distribution at each point. If it were to actualise at that point, then those derivatives can be carried over to the new field that is formed after the actualisation. It is also possible to define mean or expectation values for the variables of position, momentum and energy, by averaging with appropriate weights over the whole field. Thus a mean position $\overline{\vec{x}} ,$ mean momenta $\bar{p_j}$ and a mean energy $\bar{E}$ can be defined according to $$\overline{\vec{x}} ~=~ \int \vec{x} \mid \psi (\vec{x} , t) \mid ^{2} d \vec{x}$$ (10.1) $$\bar{p_j} ~=~ \int p_j \mid \psi (\vec{x} , t) \mid ^{2} d \vec{x} ~=~ \int \p... ...st \frac{\hbar}{i} ~ \frac{\partial}{\partial x_j} \psi ( \vec{x} ,t) d \vec{x}$$ (10.2) $$\bar{E} ~=~ \int E \mid \psi (\vec{x} , t) \mid ^{2} d \vec{x} ~=~ \int \psi... ...} ,t) ^ \ast i \hbar ~ \frac{\partial}{\partial t} \psi ( \vec{x} ,t) d \vec{x}$$ (10.3) The values of $\overline{\vec{x}} , \bar{p_j} \& \bar{E}$ for the whole quantum substance are only defined in this mean or secondary sense, by averaging local logarithmic derivatives over the extent of the propensity fields.         Complementarity? The conjunction of an extensive field with some actualising event also corresponds, I believe, to what Niels Bohr has called10.7 the basic `quantum phenomenon', being an `undivided' and `closed' occurrence. It is `undivided' because between the source and realising events is a single extensive propensity field, and not any intervening actual events which could constitute some kind of unknown connection. It is `closed' because once a place in a propensity field has become realised, the field no longer exists: its history is closed. Bohr's `complementarity' of the wave and particle aspects of the quantum phenomenom arises because although a propensity field can be regarded as propagating through space and time like an oscillating wave and as obeying a wave equation, it is in fact a single field which can produce only one actual event. This event must be at one definite place, just as a strongly-localised particle would produce. If we were not aware of the notion of a `distribution of propensity for a definite event', we would be confused because sometimes the continuant behaves like a wave, and sometimes like a particle. This quantum substances are capable of acting like a wave, and capable of acting like a particle, as Bohr's complementarity asserts. Once it is determined, however, what events actually occur, then, at each time, a quantum substance is quite definitely either a wave or a particle. At any given time, it is either a field (with wave-like patterns), or undergoing a localisation event (with a sort of particle-like localisation). It in fact never behaves like a traditional `corpuscle', in that it never has a localised extension which is constant for any finite (non-zero) duration. This is because the localising events (as the production of purely actual past-events) have no finite and divisible duration.     Two-Slit Experiment Consider the two-slit intereference experiment with beams of electrons. The emission of electrons produces a field of propensity that extends through space (in the direction of the beam) and endures through time. The exact form of this distribution will be governed by something like Schrödinger's time-dependent equation. It will have wave-like characteristics according to the variation of momentum (see above) in different parts of the field.   As in Maxwell [1988, p. 16], when the electron field overlaps with the field of the two-slitted screen, either the electron is absorbed by the screen and there is an actual event (an instantaneous, probabilistic collapse of the wave packet), or the electron wave packet passes through both slits towards the photographic film. In that case, the electron field extends across in the intervening space, and the parts of the field that came through different slits will interfere with each other constructively or destructively in a wave-like manner. Eventually the recombined electron-field will overlap with the photographic plate, more specifically with all the photographic grains in the plate.   An actual event can necessarily occur on interaction with only one of these grains: this amounts to a reduction of the wave packet to a highly localised form trapped in the grain. Thus, one particular grain is definitely exposed, and none of the other grains will be exposed. If the whole process is repeated with many electrons over a period of time, then a statistical distribution of exposed grains will be built up according to the probabilities resulting from the propensity distribution of the recombined electron-field. The exact sufficient condition for an actualising event will be discussed in chapter 12; Maxwell proposes that is the difference in energy between the exposed and unexposed grains which triggers the actual localisation of the propensity fields.       Superpositions and Mixtures Until an actualising event occurs, the propensity fields may well extend over a wide range of possibilities. In the language of quantum mechanics, we would say that the quantum system is in a superposition of these different possibilities. After an actualising event occurs, then only one possibility can be realised for each propensity field (each quantum substance), and there will be different probabilities for the different outcomes. What this corresponds to in the language of quantum mechanics depends on whether we know what the actual outcome is. If we do know, then a measurement is said to occur.   If we do not know the particular result, but only that some definite outcome did obtain, then the system is said to be in a statistical mixture of the different possibilities. The statistics arise here because of our ignorance as to what actually happened.   Summary It would appear in summary, then, that the present conception of substance is able at least qualitatively to account for several of the features of nature that have been captured by quantum physics, and which are mysterious or impossible in classical physics. We can see how there might arise a `wave-particle complementarity', indeterminacies, objective probabilities, diffraction, interference and tunnelling effects.

    

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Author: I.J. Thompson, Dec 2007.

Email: Ian@generativescience.org