Een stukje uit het boekwerk van Roger Penrose. Geheten The “road to reality”. Penrose is een meester in het jongleren met imaginaire en complexe getallen. Om dat bij te kunnen houden moet je alle zeilen bijzetten. Maar goed, dit is een terzijde. ’t Gaat ons hier om hoofdstuk 29 “The measurement paradox.” Waarom? Omdat we daar een zinsnede tegenkwamen die misschien als een tegenwicht kan gelden voor de redeneringen van Markus Gabriël en soortgelijke halve garen. Het gaat om de volgende zinsnede: “… This grand superposition is described by a wavefunction ǀφ) for he entire universe. Is is sometimes referred to as the “multiverse”, but I believe that a more appropriate term is the Omnium.
(dus wellicht (of welzeker) meer geëigend dan Sinnfeld?).
29 The measurement paradox
29.1 The conventional ontologies of quantum theory
THERE is no doubt that quantum mechanics has been one of the surpreme achievements of the 20th century. It explains a great many phenomena that had been profoundly puzzling in the 19th, such as the existence of spectral lines, the stability of atoms, the nature of chemical bonds, the strength and colours of materials, ferromagnetisme, solid/liquid gas phase transition, and colours of hot bodies in equilibrium with hot surroundings (black body radiations). Even some puzzling matters of biology, such as the extraordinary reliability of inheritance, are now seen to arise from quantum-mechanical principles. These phenomena - as well as many others which had become known in the 20th century, such as liquid crystals, superconductivity and superfluidity, the behaviour of lasers, Bose-Einstein condensates, the curious non-locality of EPR effects and of quantum teleportation - are now well understood on the basis of of the mathematical formalisme of quantum mechanics. This formalism has, indeed, provided us with a revolution in our picture of the real physical world that is far greater even than that of the curved spacetime of Einstein’s general relativity. Or has it? It is a common view among many of today’s physisists that quantum mechanics provides us with NO picture of “reality” al all! The formalism of quantum mechanics, on this view, is to be taken as just that: a mathematical formalism. This formalism, as many quantum physiscists would argue, tells us essentially nothing about an actual QUANTUM REALITY of the world, but merely allows us to compute probabilities for alternative realities that might occur. Such quantum physicists’ ontology - to the extent that they would be worried by matters of “ontology” at all - would be the view (a): that there is simply no reality expressed in quantum formalism. At the other extreme, there are many quantum physisists who take the (seemingly) diametrically opposite view (b): that the unitarily evolving quantum state completely describes actual reality, with the alarming implication that practically all quantum alternatives must always coexist (in superposition). As already touched upon in §21.8, the basic difficulty that confronts quantum physicists, and that drives many of them to such views, is the conflict between the two quantum processes U and R, where (§22.1) U is the deterministic process of unitary evolution (as can be described by Schrödinger’s equation) and R is the quantum state reduction which takes place when a “measurement” is performed. The U proces, when it was found, was something of the kind familiar to physicists: the clear-cut temporal evolution of a definite mathematical quantity, namely the state vector ǀǀΨǀǀ, controled deterministically by a (partial) differential equation - the temporal evolution of the Schrödinger equation being not unlike that of the classical Maxwell equations (see §21.3 and Exercise [19.2]). On the other hand, the R proces was something quite new to them: a discontinuous random jumping of this same ǀǀΨǀǀ, where only the probabilities of the different outcomes are determined. Had the fysics of the observed world been described simply by a quantity ǀǀΨǀǀ, just acting on its own according to U on its own, then physicists would have no serious trouble with accepting U as providing a “physically real” evolution process for a “physically real” ǀφ). But this not how the observed world behaves. Instead, we seem te perceive a curious combination of U with the interjection of the very different proces R, from time to time! (Recall Fig 22.1.) This made it far harder for physicists to believe that ǀφ) could actually be a description of physical reality after all. The puzzling issue of how R can somehow come about, when the state is supposed to be in accordance with U-evolution, is the measurement problem - or, as I prefer it, measurement paradox – of quantum mechanics (discussed briefly in §23.6, and hinted at in §21.8 and §21.8 and §22.1). The viewpoint (a) is basically the ontology of the Copenhagen interpretation as expressed specifically by Niels Bohr, who regarded ǀφ as not representing a quantum-level reality, but as something to be taken as the merely describing the experimenter’s “knowledge” of a quantum system. The “jumping”, according to R, would then be understood as the experimenter’s simply acquiring more knowledge about the system, so it is the knowledge that jumps, not the physics in the system. According to (a), one should not ask that any “reality” be assigned to quantum-level phenomena, the only acknowledged reality being that of the classical world within which the experimenters apparatus finds his home. As a variant of (a) one might take the view that this “classical world” comes in not at the level of some piece of “macroscopic machinery” that constitutes the observer’s measuring apparatus, but at the level of the observer’s own conscousness. I shall dicuss thes alternatives in more detail shortly. The supporters of alternative (b), on the other hand, do take ǀφ) te represent reality, but they deny that R happens at all. They would argue that when a measurement takes place, all the alternative outcomes actually coexist in reality, in a grand lineair superposition of alternative universes. This grand superposition is described by a wavefunction ǀφ) fort the entire universe. It is sometimes referred to as the “multiverse”, but I believe that a more appropriate term is the omnium. For although this viewpoint is commonly colloqually expressed as a belief in the parallel co-existence of different alternative worlds, this is misleading. The alternative worlds do not really “exist” separately, in this view; only the vast particular superposition expressed by ǀφ) is taken real. Why, according to (b), is the omnium not percieved as actual “reality” by an experimenter? The idea is that the experimenter’s states of mind also coexist in the quantum superposition, these different individual mind states being entangled with different possible results of the measurement being performed. The view is that, accordingly, there is effectively a “different world” for each possible result of the measurement being performed. The view is that, accordingly, there is effectively a “different world” for each different possible result of the measurement, there being a separate “copy” of the experimenter in each of these different Worlds, all these worlds co-existing in quantum superposition. Each copy of the experimenter experencies a different outcome for the experiment, but since these “copies” inhabit different worlds, there is no communication between them, and each thinks that only one result has occured. Proponents of (b) often maintain that it is the requirement that an experimenter have a consistent “awareness state” that forces the impression that there is just “one world” in which R appears to take place. Such a viewpoint was first put forward by Hugh Everitt in 1957 (although I suspect that many others had, not always with conviction, privatly entertained this kind of view earlier - as I had to myself in the mid-1950s – without daring to be open about it!). Despite their diametrical opposing natures , the viewpoints (a) and (b) have some significant points in common, with regard to how φ is taken to relate to our observed “reality” - by which I mean to he seemingly real world that, on a macroscopic scale, we all experience. In this observed world, only one result of an experiment is taken to occur, and we may justly regard it as a job of the physics to explain or to model the thing that we indeed normally refer to as “reality”. Neither according according to (a) nor according to (b) is the scale vector ǀφ) taken to describe that reality. And in each case , we must bring in the perceptions of some human experimenter to make sense of how the formalism relates to the opserved real world. In case (a) it is the state vector ǀφ) itself that is taken to be an artefact of human experimenter’s perception, whereas in case (b), it is “ordinary reality” that is somehow delineated in terms of the perceptions of the experimenter, the state vector ǀφ) now representing some kind of deeper reality (the omnium) that is not directly perceived. In both cases the “jumping” or R is taken to be not physically real, being, in a sense, “all is the mind”! I shall be explaining my own difficulties with both propositions (a) and (b) in due course, bur before doing so, I should mention a further possibility for interpreting conventional quantum mechanics. This, as far I can make out, is the most prevalent of the quantum-mechanical standpoints - that of environmental decoherence (c) - although it is perhaps more a pragmatic than an ontological stance. The idea of (c) is that in any measurement proces, the quantum system under consideration cannot be taken in isolation from its surroundings. Thus, when a measurement is performed, each different outcome does not constitute a quantum state on its own, but must be considered as part of an entangled state (§23.3), where each alternative outcome is entangled with a different state of the environment. Now, the environment will consist of a great many particles, effectively in random motion, and the complete details of their locations and motions must be taken to be totally unobservable in practice. There is a well-defined mathematical procedure for handling this kind of situation where knowledge is fundamentally lacking: one “sums over” the unknown environmental states to obtain a mathematical object to be known as a density matrix, to describe the fysical system under consideration. Density matrices are important fort the general discussion of the measurement problem in quamtum mechanics (and are important also in many other contexts), but their ontological status is hardly ever made clear. I shall explain what a density matrix is very shortly ( in §29.3). However, we shall be seeing later why it is important for the position (c) that the ontology of the density matrix is not completely clear! Holders of viewpoint (c) tend to regard themselves as “positivists” who have no truck with “wishy-washy” issues of ontology in any case, claiming to believe that they have no concern with what is “real” and what is “not real”. As Stephen Hawking has said:
I don’t demand that a theory correspond to reality because I don’t know what it is. Reality is not a quality you can test with litmus paper. All I’m concerned with is that the theory should predict the results of measurements.
My own position, on the other hand, is that the issue of ontology is crucial in quantum mechanics, though it raises some matters that are far from being resolved at present time.