Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics

by Michael Redhead

Aiming to unravel the mystery of quantum mechanics, this book is concerned with questions about action-at-a-distance, holism, and whether quantum mechanics gives a complete account of microphysical reality. With rigorous arguments and clear thinking, the author provides an introduction to the philosophy of physics.

Genres: Physics, Philosophy

200 pages, Paperback

First published July 9, 1987

About the author

Redhead was Centennial Professor in CPNSS (Centre for Philosophy of Natural and Social Science) at the London School of Economics and Political Science.

Redhead was an Emeritus Fellow of Wolfson College, Cambridge, and was Vice-President (1992–1996) and Acting President 1992 and 1993, Wolfson College, and formerly Head, CU Dept of History and Philosophy of Science.

Reviews

[William Bies · Rating 4 out of 5]

One has not to look to the philosophers of physics for them to frame any compelling new scientific ideas, but, with their fondness for technical jargon, they are pretty good at teasing out the fine points of existing theories. Probably no facet of the natural world has attracted their attention more than quantum mechanics, for here we have a theory that not only tells us how many physical processes work with unrivaled empirical success but also which confronts with a radical revision of received concepts. After having reviewed Max Jammer’s staple history of the interpretation of quantum mechanics (see review here), we wish to turn to a notable effort by a philosopher to synthesize the findings of a few decades’ worth of research and experimentation into its foundations, viz., Michael Redhead’s Incompleteness, nonlocality and realism: A prolegomenon to the philosophy of quantum mechanics , first published by the Oxford University Press in 1987.

Redhead means it when he styles his book a prolegomenon. He chooses not to give any more attention to Niels Bohr’s views (complementarity) than two pages [pp. 49-51]: why? He considers Bohr’s restrictions on what is knowable in quantum mechanics dogmatic. Redhead also ignores the measurement problem in order to concentrate on other interpretive aspects of quantum mechanics [p. 56]. There are several nice points about the exposition, which always begins with the simplest possible case and works up to harder ones. Let us mention a few: what Hasse diagrams are all about [pp. 23-26]; the relation between Gleason’s theorem and the Radon-Nikodym derivative [p. 30]; elucidation of the difference between ideal measurement and normal measurement followed by state selection [p. 59]; explanation of the meaning of the uncertainty relations and an elegant derivation of them in finite dimensions [pp. 60-61]; a simple proof that no state can be dispersion-free [p. 62]. Redhead argues that:

The correct way to understand the uncertainty relations is to see that they represent an inherent limitation on the sort of states which can be produced for QM systems, but that they cannot be ‘explained’ or ‘deduced’ by the naive disturbance argument. [p. 69]

As can only be expected, the EPR argument for incompleteness of the quantum-mechanical description of the world looms large. Chapter three enters into an exhaustive treatment, beginning with a delineation of what EPR mean by their criterion of reality [p. 72f]. Note, EPR are not saying every element of reality has a definite value (determinism) but that if something is definitely predictable there must be an element of reality behind it. Redhead uses the spin version of EPR due to David Bohm to reduce the argument to the minimal possible fragment of quantum mechanics [p. 75]. The paradox does not depend on freedom to measure incompatible observables i.e. spin in x direction resp. z direction [p. 78]. In the last section of this chapter, a clean restatement of EPR’s original argument using continuous variables (not Bohm’s spin version) followed by some good discussion [pp. 79-81]. If what physical reality actually is is some kind of spin foam or network of space-time atoms, why should we ever expect also that its particle-like excitations must have simultaneously existing sharply defined position and momentum? This could be true only of a monochromatic wave extending over all of infinite space, what is obviously an idealization. Einstein’s anachronistic attitude seems crazy in this respect.

The other major topic of the book is Bell’s theorem, or class of theorems. Bell’s inequality establishes a dilemma between LOC3 = a form of the locality principle appropriate to a hidden-variable theory and F = the quantum-mechanical formalism. Redhead wants to give a proof of Bell’s inequality that does not appeal to a joint probability distribution for incompatible observables; in fact, the correlation functions actually used in deriving the Bell inequality always refer to compatible (commuting) observables [p. 90]. His presentation of how Bell’s inequality comes about and how quantum mechanics violates it due to angular dependence of correlation coefficients is exceptionally clear [pp. 84-85]. A nice point is his discussion of why there is no Bell inequality in a classical analogue [pp. 86-87]. Then Redhead supplies a proof of Bell’s inequality for hidden variables avoiding counterfactuals [p. 101]. Aspect’s celebrated delayed choice experiment gets around the possibility that Bell locality but not Einstein locality is what has to be rejected (unless one posit some kind of superdeterminism), [p. 110].

Chapters five and six address the so-called Kochen-Specker paradox asserting that it is impossible to assign sharp values to all observables on a Hilbert space of dimension > 2, contradicting one’s straightforward expectation of what ought to hold for a hidden-variables theory, whether local or not. The premise responsible for the paradox is isolated as the functional composition principle [p. 121]. Redhead’s extended heuristic description of why the Kochen-Specker paradox occurs in dimension > 2 is admirably clear [pp. 121-131]. There are three ways of escaping the Kochen-Specker paradox, none ideal [pp. 133-136]. The reasonable-seeming condition of environmental locality = in other words, the value possessed by a local observable cannot be changed by altering the arrangement of a remote piece of apparatus which forms part of the measurement context for the combined system [p. 141], but ‘forms part of the measurement context’ means not remote, after all! What to get out of all of Redhead’s imposing mathematical formalism [pp. 142-150]: why quantum mechanics is said to be contextual, see esp. p. 150 which would repay closer study. Redhead concludes to a kind of ontological holism in which it is impossible to make sense of a realist construal of quantum mechanics which associates properties (observables) independently with each of two separated systems.

Chapter seven seems like an afterthought. To this reviewer, the problems of quantum propositional logic seem to be related to the old question of why quanutm mechanics is formulated in configuration space and not in phase space (the difficulty which renders the geometric quantization program so inelegant). Redhead employs here too much recondite formalism to be clear (compare with Max Jammer, mentioned above, and David Cohen, see our review here). Eventually one will have to master it if one goes into the field but it can be hard to take for a beginner. To this reviewer, Redhead’s treatment does raise a broader question: what does quantum logic mean for our life-world if we allow high order observables? So far, one has applied it only to relatively simple observables such as total spin or total momentum. But as John Bell has in mind in his discussion of decoherence (cf. our review of Roland Omnès on the interpretation of quantum mechanics, here) is that we need not limit ourselves to straightforward and comparatively easy to analyze examples such as these. An observer who knows what to look for could, if skillful enough, construct a measurement apparatus that registers a subtler property of the system by teasing out just the right correlations among its components. Consider for instance the stratagems people have resorted to in order to derive the tertiary conformation of proteins from x-ray crystallographic data: for instance, isotopic substitution, in which the differential between the scattering from the native state to one with another isotope substituted in a metallic center can be exploited to reveal structural information. In principle, one is just comparing two very complicated observables! By now, one has pretty well beaten the EPR experiment to death; if the field is to stay alive, one direction in which to proceed could be to develop techniques to handle substantially more complicated observables. This reviewer has always found it curious that, in the standard textbooks on quantum mechanics, one declares that any self-adjoint operator can be an observable but then, in practice, has recourse to only a handful of stock examples. Nonetheless, after entering into several detailed analyses, Redhead has the presence of mind to pause for a moment and wonder where he is going. He poses a fair question:

But is this sort of talk genuinely helpful in understanding quantum mechanics? It may be argued that, far from helping to resolve the mysteries in quantum mechanics, it merely substitutes one mystery for another, viz., how to make sense of the sort of slogans we have been repeating. Any attempt to produce a proper realist semantics for quantum logic, via A₂-admissible valuations, for example, seems to run foul of the Kochen-Specker paradox. But if we don’t do that, can we really be said to be retaining realism? [p. 167]

What would David Bohm rejoin, for instance? What propels his whole pilot-wave theory is a reluctance to give up on classical realism; in effect he thinks that he can retain classical realism by means of his hidden variables even though they are unmeasurable. The attentive reader of Redhead’s text, however, will have noticed by now that he has made a strong case against hankering after the comfort of a naïve realist stance. Certainly the experimental situation with respect to violations of Bell’s equality, as Redhead reviews, lends support to some kind of contextual non-locality, no matter which horn of the dilemma one elects to follow – what is interesting here, Redhead implicitly seconds Bohr’s main contention that we cannot have direct knowledge of the quantum world, although the former author appears to be temperamentally disinclined to the latter’s Copenhagen interpretation in general. The upshot being that we may heartily concur with Redhead’s conclusion:

So there it is – some sort of action-at-a-distance or (conceptually distinct) nonseparability seems built into any reasonable attempt to understand the quantum view of reality. As Popper has remarked, our theories are ‘nets designed by us to catch the world’. We had better face up to the fact that quantum mechanics has landed some pretty queer fish. [p. 169]

[Steven H · Rating 5 out of 5]

A REVIEW OF SOME DEVELOPMENTS IN QUANTUM MECHANICS

Michael Logan Gonne Redhead (1929-2020) was a British academic and philosopher of physics, who taught at the London School of Economics and Political Science, and Wolfson College, Cambridge.

He wrote in the Preface to this 1987 book, “This book is intended to be useful, to clear metaphysical fog, and to persuade the reader to join in and develop the argument. I am grateful to my students… who asked me to recommend a book on quantum mechanics to them, so that I ended up writing one myself. I hope physicists will read the book as well as philosophers. I have written for both of them. I assume a nodding acquaintance with nonrelativistic quantum mechanics and the elements of linear algebra, but mostly everything is explained as I go along. I have definitely tried never to let technicalities obscure the arguments.”

He says in the Introduction, “The object of this book is to present in as straightforward and uncluttered a manner as possible some of the exciting work that has been done in the philosophy of quantum mechanics during the past few years, in particular since the discovery of the now famous Bell inequality in 1964, and the apparently unrelated but equally important work of Kochen and Speker in 1967… some twenty years later, the dust is beginning to settle, and we can try to take stock of just what has been learned about quantum mechanics, and how it relates to the classic era of debate and discussion between Einstein and Bohr… Bohr was in effect declared the winner in the controversy, and for nearly thirty years Copenhagen orthodoxy reigned supreme… The new developments in the 1960s revived interest in the old debates about ‘realist’ interpretations of quantum mechanics, and showed much more clearly just what were the difficulties in implementing the Einstein programme of a ‘complete’ version of quantum mechanics. Essentially what has ben achieved is not a resolution of the difficulties surrounding the interpretation of quantum mechanics, but a clarification of just what these difficulties are.” (Pg. 1-2)

He explains, “The mathematical scheme for QM consists then in setting up a mathematical structure such that certain elements in that structure are associated with the states of the physical system and certain other elements are associated with the observables. Certain algorithms are then proposed which serve to answer our two basic questions: 1. What values are possible measurement results for any given observable? 2. For any given state and any given observable, what is the probability that one of the possible measurement results will actually turn up when a measurement is performed? We shall refer to the algorithm which answers the first question as the ‘quantization algorithm.’ The algorithm which answers the second question we shall refer to as the ‘statistical algorithm.’” (Pg. 5)

He observes, “first we introduce a NEW sense of interpretation, DIFFERENT from that employed in the minimal instrumentalist interpretation … It is simply some account of the nature of the external world and/or our epistemological relation to is that serves to EXPLAIN how it is that the statistical regularities predicted by the formalism with the minimal instrumentalist interpretation come out the way they do. Of course we could simply accept the regularities as ‘brute facts’ and leave it at that. Interpretations that go beyond the ‘brute facts’ are idle metaphysical baggage that the austere physicist can and should dispense with. That is to go the way of the positivist, the instrumentalist, and anti-realist. Theories in physics ARE just devices for expressing regularities among observations…. It is not part of the purpose of this book to rehearse the arguments for and against such an attitude to theoretical physics…. We shall assume for the purpose of this book that theories that lack an interpretation … simply do not contribute to our UNDERSTANDING of the natural world. Of course this demand for explanation and understanding must not expect too much…” (Pg. 44-45)

He outlines, “We will now discuss … three views in turn. 2.1. View A: HIDDEN VARIABLES: View A says that QM is not really mysterious at all. It is just a glorified statistical mechanics. There exists an objective external world of entities with well-defined properties which are simply discovered by measurement… Nothing is ever unsharp or fuzzy or undefined or meaningless. Of course, we would have to do a lot more work to fill out view A into a full-fledged interpretation… This is the program envisioned in the so-called hidden-variable interpretation of QM.” (Pg. 45-46)

He continues, “2.2. View B: PROPENSITIES AND POTENTIALITIES: … the observable ‘Q’ does not possess a BALUE at all. What the QM system does in reality possess is a propensity or potentiality to produce various possible results on measurement, in respect of the observable Q. View B says that QM is mysterious, in that new concepts over and above those employed in classical physics have to be invoked… An important question must now be considered. Is view B a realist interpretation of QM?... The experimental probings under propensities manifest themselves have nothing to do with human minds or consciousness. Everything could work out in a world without human beings at all. The fact that we design experiments and intend them to furnish us with certain sorts of information about the external world is quite irrelevant in this connection. The conclusions is that view B is perfectly consistent with realism, and certainly gives no arguments at all in favor of idealism.” (Pg. 48-49)

He goes on, “2.3. View C: COMPLEMENTARITY: View C is the view taken by Bohr, and is based on the orthodox Copenhagen complementarity interpretation of QM… Broadly, we can say that QM is mysterious on view C… in the sense of recognizing limitations on the applicability of the familiar concepts of classical physics, which may not be definable in certain contexts… The difficulty with the complementarity interpretation of QM is undoubtedly the fact that Bohr’s own formulation of the general framework of his ideas is vague and ambiguous… the main objection is the finality with which Bohr prohibits even asking certain questions about QM systems…. Setting dogmatic limitations on scientific theorizing, on the basis of obscure philosophical preconceptions, is a dangerous prejudice … It is for this reason that other approaches to the interpretation of QM are the main business of this book.” (Pg. 49-51)

He summarizes, “we presented the Einstein dilemma, that the minimal instrumentalist interpretation F of QM either implied nonlocality or that F was incomplete. Einstein… chose the incompleteness horn of this dilemma and concluded that… observables for which these states were not eigenstates nevertheless possess sharp values…. But then, in a famous paper published in 1964, Bell showed that A, in conjunction with a locality principles appropriate to view A… implied a certain inequality between measurable correlation coefficients … And this inequality, now usually referred to as the Bell inequality, turns out to be in disagreement, over a certain range of conditions, with the predictions of F itself.” (Pg. 82)

He explains, “There is a large body of literature in the philosophy of QM which claims that the real conceptual revolution that should be recognized as engendered by consideration of the ‘paradoxes’ of QM is a revision of logic… The basic idea is this. If ‘L’ denotes the old classical logic, then we seem driven to new ‘paradoxical’ physical conceptions of potentiality, nonlocality, and so on… Clearly, in this program we are thinking of logic as capturing a special sort of proposition---the logical truths---and then we are going to blur the distinction between logical truths and other true propositions about the world. We are not thinking of logic as primarily concerned with the notion of valid consequence, of transmitting truth from the premises to the conclusion in a valid argument.” (Pg. 153)

He summarizes, “In this book we have been mainly concerned with the difficulties encountered by a simple-minded realism of possessed values. The reaction to these difficulties on the part of many philosophers is to say we must not be son simple-minded… If we want to BE simple-minded and keep to a separable realism of possessed values… then action-at-a-distance … must be admitted. This might be thought to raise severe problems for special relativity… But this form of nonlocality, actualizing possibilities at a distance, may be thought sufficiently far removed from the classical stamping ground of special relativity to allow for … ‘peaceful coexistence.’ … So there it is---some sort of action-at-a-distance or … nonseparability seems built into any reasonable attempts to understand the quantum view of reality… We had better face up to the fact that quantum mechanics has landed some pretty queer fish.” (Pg. 168-169)

While hardly a book for ‘beginners,’ this book will be of keen interest to those studying quantum mechanics, and some of the philosophical issues about it.

[TjeerdFokkens · Rating 4 out of 5]

Great book about the philosophy of quantum physics. I like the formal methods used and the connection to logic. The explanation of the Kochen-Specker theorem is excellent and Bell's theorem is explained in a very nuanced way by making the assumptions explicit. The book is a bit technical, though, so sometimes I couldn't follow it and I didn't bother to go in depth to check all the proofs.