(Printed version is on Metascience doi: 10.1007/s11016-022-00807-8; an unedited pre-print is on Academia.com: https://www.academia.edu/86788063/Traces_in_Probability_Space )
Traces in Probability Space
William Demopoulos’ On Theories: Logical Empiricism and the Methodology of Modern Physics, Cambridge: Harvard University Press. xxiv + 247 pp.
William Demopoulos died in May 2017, altogether too young, and with a manuscript nearing completion that needed some tidying editorial attention. This is the volume under review, and Demopoulos’ long-time friend and collaborator, Michael Friedman, has admirably completed the necessary editorial tasks. In addition he has contributed an Editor’s Foreword and Afterword to help give the chapters more continuity — and, more importantly in the case of the Afterword, to trace Demopoulos’ complex path in interpreting quantum theory.
Since the quest to understand quantum mechanics spanned Demopoulos’ career and since nearly half of the present book is given over to it, we focus on this.
In the late 1970s Demopoulos was one of the leading figures in the ‘Western Ontario’ school of quantum logic. It was an exciting period, even if one was unconvinced by the overall plan. (It was during this period that this reviewer, then a graduate student, became familiar with his work; one never had cause to regret reading his essays, which were always sharp and to the point.) By 1982 Demopoulos had become disenchanted with quantum logic, as he had realised that there were classical contradictory propositions that were nevertheless true in quantum logic! Unable to reconcile himself to this, for the next 22 years he turned away from quantum logic to focus on other, unrelated, issues.
He returned to quantum theory only around 2004, around the same time as Itamar Pitowsky published his paper interpreting quantum probabilities through Dutch book arguments (Pitowsky 2003). They were clearly in close communication and the association informs the discussion in the present book.
The basic idea behind the proposal is as follows. Consider Minkowskian space-time and its group of rotations, the Lorentz group. The introduction of this structure takes a phenomenon that looks to be dynamical, namely Lorentz contraction, or time dilation, and reinterprets it as a kinematical phenomenon. It is nothing more than the result of a rotational change of frame. This might be called a structural explanation or, less happily, a principle explanation. In it, something that looks dynamical receives a purely geometrical explanation. We have a similar case, not mentioned by Demopoulos but worth having in front of us, in the Pauli exclusion principle. This originally also looked dynamical — a strange “force” that kept electrons apart from one another — and was then, almost immediately, given a geometrical interpretation: it was due to the reduction of the tensor product space to just the antisymmetric quotient (for fermions), which is now identified as the exterior power, a Grassmann algebra. Or we could consider the way that gravitation became nothing more than a kinematical phenomenon on a curved space-time. Such structural explanations have had considerable success in 20th Century physics!
But how can we understand the baffling probabilities that occur in quantum theory, both for measurement outcomes and the distant correlations? These also have a faux dynamical character and thus look fit for a structural explanation. The crucial theorems for Demopoulos are those of Gleason and Kochen-Specker (KS) — the latter following from the former plus a (logical) compactness claim — and the relevant structure the projection lattice of subspaces of a Hilbert space. Gleason’s theorem tells us that a generalised measure defined on such a lattice of a Hilbert space (with dim ≥ 3) extends uniquely to a linear functional on all bounded operators. In quantum mechanical terms this means that it extends from pure states to apply also to mixed states. This probability measure is the trace of a density matrix. The pure states, the extremal points of the convex set of all states, are in 1 : 1 correspondence with the subspaces of a Hilbert space. This justifies in a striking way the Born Rule of probabilities for pure and mixed states: it ‘drops out’ of the formalism in a natural way. (In the exceptional case of two dimensions it is possible to have a 0 : 1 measure on the basis vectors that can be consistently continued to the whole space, called the ‘Bloch sphere’ — this is sometimes described as ‘colouring the sphere’ with just two colours.)
Whereas Gleason’s theorem shows the global failure of a 0-1 measure, the KS theorem constructs quite small sub-algebras in which the failure is made manifest (Thm. 1 of Kochen and Specker 1967)). Over the years both Gleason’s and the KS theorem have been made progressively more tractable. For KS and a Hilbert space of dim = 4, the number of basis sets needed is only 9, with 20 vectors (q.v. Kernaghan (1994)). Even in dim = 3 it is possible to use only 31 vectors — down from the KS original 117. A consequence of Gleason’s theorem that is stronger than the KS theorem is this: if two pure states a and b are a part of a set Γ of pure states and if they are not orthogonal, then there is no probability measure on Γ that assigns both a and b a probability of zero or one (a truth value) unless they are both assigned 0, i.e. are both false. This is Pitowsky’s Indeterminacy thesis (Pitowsky 1998) as used by Demopoulos.
In the final section of the KS paper they proved that there were classical tautologies that were quantum-false. Even the seemingly harmless associativity law will be satisfiable only under restricted circumstances. proved the dual of this in (1977; 86). He argued that there were classical contradictions that were true. In particular he showed that a finite partial Boolean subalgebra of subspaces can be represented by a Boolean polynomial in 86 variables — less now of course in the light of Kernaghan (1994). For every assignment of either 0 or 1 this Boolean polynomial takes the value 0 – thus it is a classical contradiction. And yet it is satisfiable by some assignment of co-measurable quantum variables and so can be ‘true’. The conclusion Demopoulos reached was to believe that the semantic determinacy of quantum logic cannot be maintained — there have to be disjunctions that are true but where none of the disjuncts are true. And this has led him to the conclusion that the interpretation of the elements of the projection lattice as propositions about some particle must be false, for if it were not, quantum truth would have as a subset some classical contradictions. As he said: ‘Hence, if we think of the laws of logic as “laws of truth” — as a guide to our concept of truth — the quantum logical interpretation’s concept of truth cannot be our classical concept.’ (2010; 380). It would be a step into paraconsistency that Demopoulos is unwilling to take. (This is all, of course, in contrast to intuitionist truth that is a subset of the classical truths.)
This is apparently why Demopoulos abandoned quantum logic for more than 20 years.
But since 2004 he has returned, obviously having thought deeply on these matter in the intervening years. Quantum logic, as originally conceived, cannot work; quantum probability can work, provided that the probabilities are not understood as a measure on a lattice of subspaces conceived as propositions. The problem then is: how should the states be conceived? His answer is that they should be conceived as effects and such effects have probabilities of occurring. An effect is an event, the result of an interaction with another system. Demopoulos says of effects that they ‘are induced by an interaction with an experimental set-up and registered in the experimental apparatus.’ (Demopoulos 2010; 382) If there is no interaction Demopoulos avails himself of the memorable saying of Asher Peres, ‘Unperformed experiments have no results’. The probabilities are interpreted by Demopoulos in a Bayesian fashion using the notion of a Dutch book, but the underlying space is not the standard Boolean event space but the non-Boolean projection lattice and presumably its continuation to all density operators. This idea (Quantum Bayesianism, or “QBism”) was also set out in Pitowsky (2003) but it goes back to the beginning of quantum mechanics and one can find many comments in von Neumann’s work, including his letters, which would support it (albeit inconsistently). The idea is that we have here a structural explanation: the structural change is to the underlying event space.
An obvious question must arise at this point in the mind of any reader: what does this say about the EPR-type distant correlations? How does a Bayesian probability work in this case, where the correlated probabilities seem perfectly objective? This question is taken up in the book under review, based on the ideas we’ve already seen. Drawing on the analogy with the structural role of Minkowski space-time, Demopoulos puts it as follows: ‘… the probabilities exhibited by the EPR correlations are understood as a consequence of the Hilbert space structure of the properties of physical systems, rather than the effect of unknown local causes’ (p. 183). Underneath this there is a reliance on an unpublished formulation of Bell’s theorem and KS due to Pitowsky. On this formulation, if we have a finite set F of Hermitian operators, a state s and a probability distribution Ps on all of the eigenvalues for the Hermitian operators in F, and agreeing with QM probabilities for commuting members in F, then Bell’s theorem says that there is an s (the EPR state) and an F (which in this case are local operators) for which there is no Ps ; and KS says that there is an F such that for any s the Ps does not exist. Thus KS is not dependent on the choice of state and Bell’s theorem follows from KS and so also from Gleason’s theorem. then deploys this against Einstein’s ‘criterion of reality’ and on behalf of Bell’s analysis of the prospects for a completion of QM along the lines Einstein envisaged. All of this one can accept. But I still don’t quite see how the Bayesian framework plus the non-Boolean character of the quantum probability space explains the distant correlations. In the analogous case of Minkowski space-time there is a clear, provable, sense in which it explains a Lorentz contraction and time dilation. In the current case, Minkowski space-time both forbids causality and allows correlations. Does this suggest that another group is needed? Another space for the probabilities?
The cost of making probabilities Bayesian and personal is that one needs a rich theory of the mutual ‘effect’ of a quantum system on a measurement device — the objective causal event we are betting on. The problem here is that there is no such causal theory on offer. Moreover there may be good reasons why nothing is forthcoming. To describe such an interaction you need, at the very least, two Hilbert spaces under a tensor product. But even this simple requirement has posed a great obstacle for quantum lattices and their logics. This was the problem uncovered by Aerts, Foulis and Randall, Pulmannová, in the early 80s and discussed by Boris Ischi more recently. Under the product, one of the lattices is forced to be commutative. The underlying problem is that a tensor product is not a Cartesian product but is vastly larger. And then there is an additional problem that this reviewer has not seen addressed anywhere: we are really interested in the quotient spaces of the tensor product: the symmetric power (for bosons) and the exterior power (for fermions); where is the lattice product corresponding to these? Where the logics and the probabilities? And what about these quantum probabilities? It has been argued (by Gromov and Hrushovski) that probability, measure theory, has a monadic presupposition, and that this naturally creates a ‘space’ that is imposed from above on the algebra of sets. Relations between things are not fundamental, rather they are functions on this space. In a sense our spatial metaphysics makes everything ‘loose and separate’ from the outset, by assumption. The problem has been put this way:
Thus pure probability logic sees, very precisely, the monadic space of the theory; and conditional probabilities of basic relations as functions on this space. Beyond this, or relative to it, the universe is boring; everything is locally statistically independent, no special features can be made out, and no new relations can be described. Hrushovski (2020; 393)
What is missing is a probability theory that puts the right relations in from the get-go. Unrepresented structures, we might say, imitating Peres, have no interpretation.
Unfortunately we do not have either Demopoulos or Pitowsky — who died within 7 years of one another — to address these issues.
But a further interesting aspect of Demopoulos’ discussion in the book has to do with understanding what Einstein believed — and then, by contrast, with what Bohr believed. Commenting in 1949, in the Schilpp volume, Einstein noted that the best explanation of the EPR situation given by the ‘orthodox’ interpretation [the view that the ψ-function is complete] comes from Bohr:
If the partial systems A and B form a total system which is described by its ψ-function ψ/(AB), there is no reason why any mutually independent existence (state of reality) should be ascribed to the partial systems A and B viewed separately, not even if the partial systems are spatially separated from each other at the particular time under consideration. Schilpp (1949)
Demopoulos doubts that this is Bohr’s view. I think he is right about this. But even though Bohr in his published remarks never said anything resembling this (though von Neumann and did) we have good reasons to believe it to be true: the reduced density matrices of A and B are maximally mixed states, so there is a good sense in which no ‘independent existence (state of reality) should be ascribed to the partial systems’, no ‘being-thus’.
I have one further caveat: I think Demopoulos goes too far in trying to enlist Bohr as an anticipator of this kind of structural view he favours. Bohr showed no interest in the algebraic structure of quantum mechanics, preferring instead his own talk of ‘complementarity’, which had no algebraic interface. Bohr’s form of words that was meant to prove that the EPR argument is unsuccessful is, to put it politely, less than transparent: ‘But even at this stage [the stage at which the final measurement is made on a subsystem] there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behaviour of the system.’ Bohr (1935; 700) What is influencing what here, and when? Just as with the above QBism, we are missing a theorem — and Bohr showed no interest in proving such.
This issue aside, Demopoulos has crafted a thoughtful and interesting interpretation of quantum mechanics that completes his earlier work of the mid-70s; it must be hoped that it gets the attention it deserves. (Potential readers are recommended to have Demopoulos (2010) and (2012) on hand.) I turn now to a brief account of the remaining chapters.
The theme of scientific theories and how theoretical terms should be understood runs through the book and its basic tenets are outlined in the introduction. The stress here is on the importance of theory-mediated measurement, something first noted by Newton in the 3rd edition of his Principia, as against the partial interpretation of theories under the logical empiricists, Carnap in particular. The second chapter lays out this argument in detail. It threads through a number of familiar needles: the Putnam model-theoretic argument; the significance of Ramsey sentences; van Fraassen’s constructive empiricist view, as outlined in The Scientific Image; the Newman argument against Russell’s ‘causal theory of perception’ in Analysis of Matter — as resurrected in the 80s by Demopoulos and Friedman. The criticism of constructive empiricism has its roots in a trenchant 1982 review from Demopoulos of The Scientific Image and was continued in a 2003 article. The argument about unobservable entities is continued here with Demopoulos arguing for the inadequacy of a response to him by van Fraassen. This argument with van Fraassen is continued in chapter 2 on Jean Baptiste Perrin. The M.H.A. Newman 12-page argument against Russell is fascinating: Demopoulos has developed it over a number of publications since 1985 in such a way that it has come to seem like one of the central works of 20th century philosophy. It is a new point here that Frank Ramsey developed his idea of Ramsey sentences as a result of being confronted by early chapters of Russell’s Analysis of Matter. It is one of Demopoulos’ major contributions to our understanding.
All in all, this posthumous publication is a wonderful tribute to a very significant philosopher.
References
Bohr, N. 1935. Can Quantum-Mechanical Description of Physical Reality be considered Complete?, Physical Review, 48, 696–702.
Demopoulos, W. (1977). Completeness and realism in quantum mechanics. In: Butts, R., Hintikka, J. (eds.) Foundational Problems in the Special Sciences, 81–88. Reidel, Dordrecht.
Demopoulos, W. (2010). Effects and Propositions. Foundations of Physics, 40(4), 368–389. doi: 10.1007/s10701-009-9321-x
Demopoulos, W. (2012). Generalized Probability Measures and the Framework of Effects. Ch. 13 of Probability in Physics ed. Yemima Ben-Menahem and Meir Hemmo, The Frontiers Collection, Berlin: Springer-Verlag.
Hrushovski, E. (2020). On the Descriptive Power of Probability Logic, in M. Hemmo, O. Shenker (eds.), Quantum, Probability, Logic, Jerusalem Studies in Philosophy and History of Science. doi: 10.1007/978-3-030-34316-3_17
Kernaghan, M. (1994). Bell-Kochen-Specker theorem for 20 vectors. Journal of Physics A: Mathematical and General, 27(21), 829–830. doi: 10.1088/0305-4470/27/21/007
Kochen, S. and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87. doi: 10.1512/iumj.1968.17.17004
Pitowsky, I. (1998). Infinite and finite Gleason’s theorems and the logic of indeterminacy. Journal of Mathematical Physics, 39, 218–228. doi: 10.1063/1.532334
Pitowsky, I. (2003). Betting on the outcomes of measurements: a Bayesian theory of quantum probability. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34(3), 395–414. doi: 10.1016/s1355-2198(03)00035-2
Schilpp, P. A. (ed.) 1949. Albert Einstein, Philosopher-Scientist, Library of Living Philosophers Vol. VII, N.Y.: MJF Books.
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