Foundations of quantum theory. From classical concepts to operator algebras
The stated aim of this ambitious new book -- which hence ranks it within the time honored field of the foundations of quantum mechanics -- ``is to analyze the foundations of quantum theory from the point of view of classical-quantum duality, using the mathematical formalism of operator algebras on Hilbert space'' (p. viii), the final target being even more aspiring: ``although experiments and detailed theoretical models are lacking so far, this powerful combination of mathematical and philosophical tools leads to a compelling scenario for solving the measurement problem, harboring the hope of finally laying this problem to rest'' (p. 14). This is distinctly commendable indeed because in so doing the book openly challenges the ``unfortunate `shut up and calculate' attitude most physicists seem to have (especially towards the measurement problem)'' (p. 435). According to this program the framework of \(C^*\)-algebras is adopted here in order to exploit the unification they provide of the commutative and the noncommutative worlds into a single mathematical scheme: a choice deemed unavoidable since some physical systems admit a dual description in either classical or quantum-mechanical terms, and in particular the measurement problem of quantum mechanics is supposed to be entailed precisetly by this duality. The development of the quantum theory in the first four decades of the XX century make clear that this unified language smooths the way for the required shifts between classical and quantum-mechanical modes of description which lies at the heart of the foundations of quantum physics: ``this `change of perspective', which roughly speaking amounts to switching (and interpolating) between commutative and noncommutative \(C^*\)-algebras, is added to Dirac's transformation theory (which comes down to switching between generalized bases, or, equivalently, between maximal commutative von Neumann algebras)'' (p. 3) The setting of the book however is admittedly rather philosophical because, while it is mostly written in strict mathematical physics style, ``its real subject is natural philosophy'' (p. viii) and the author is consequently lavish in explicitely stating his ideological inspiring principles. And first of all he articulates his adherence to Bohr's \textit{doctrine of classical concepts}, which in his opinion forms part of the Copenhagen Interpretation of quantum mechanics (here defined strictly as ``a body of ideas shared by Bohr and Heisenberg'' p. 3). In the words of Heisenberg himself this doctrine maintains that ``any experiment in physics, whether it refers to the phenomena of daily life or to atomic events, is to be described in the terms of classical physics.'' Since this doctrine of classical concepts implies in particular that the measuring apparatus is to be described classically, it is essential that one can describe at least some quantum-mechanical devices classically: those for which this is possible include the candidate-apparatuses (i.e. measuring devices). The present book then implements this principle in the language of operator algebras, in the sense that the physically relevant aspects of the noncommutative operator algebras of quantummechanical observables are deemed accessible only through commutative algebras. This program requires then what the author denominates \textit{Bohrification}, namely ``the mathematical interpretation of Bohr's classical concepts by commutative \(C^*\)-algebras, which in turn are studied in their quantum habitat of noncommutative \(C^*\)-algebras'' (p. viii). Several further principles are then evoked from the contemporary literature in the field of the philosophy of science (J. Earman, J. Butterfield\dots) to play a role in the implementation of the said Bohrification program which in fact splits into two parts, differentiated by the precise relationship between a given noncommutative operator algebra \(A\) and the commutative operator algebras that give physical access to \(A\). According to Bohr Complementarity only one classical concept (or one coherent family of classical concepts) applies to the experimental study of some quantum object at a time. Bohr's view on classical concepts comes then back mathematically in the \textit{exact Bohrification} which studies (unital) commutative \(C^*\)-subalgebras \(C\) of a given (unital) noncommutative \(C^*\)-algebra \(A\). Heisenberg's interpretation of the doctrine resurfaces instead in the \textit{asymptotic Bohrification} which involves asymptotic inclusions (more specifically, deformations) of commutative \(C^*\)-algebras into noncommutative ones. Here the classical concepts are recovered either in the classical limit for \(\hbar\to0\), or in the macroscopic limit for \(N\to\infty\) where \(N\) is the number of particles. The more technically innovative part of the book, however, seems to lie in the endeavor of bringing together three (relatively) recent scientific advances in order to gain a glimpse of a breakthrough in the longstanding problem of quantum measurement: 1. The deformation quantization; 2. The mathematical analysis of the BCS-model (Bardeen-Cooper-Schrieffer) of superconductivity; 3. The role of low-lying states and the ensuing instability of ground states under tiny perturbations in the two limits at hand, discovered by \textit{G. Jona-Lasinio} et al. [Commun. Math. Phys. 80, 223--254 (1981; Zbl 0483.60094)] for the classical limit \(\hbar\to0\), and by \textit{T. Koma} and \textit{H. Tasaki} [J. Stat. Phys. 76, No. 3--4, 745--803 (1994; Zbl 0863.46048)] for the macroscopic limit \(N\to\infty\). This array of ideas finally brings new scenarios for the quantum measurement problem: the author in fact rejects the typical way of looking at measurement as a two-step process: first a transition of a pure state to a mixed one, and then the registration of a single outcome. In his opinion, if the proposed scenario is correct, ``the mistaken impression that quantum theory seems to imply the irreducible randomness of nature, then arises because measurement outcomes are merely unpredictable `for all practical purposes', indeed they are unpredictable in a way that dwarfs even the apparent randomness of classical chaotic systems'' (p. 15). This would be indeed a huge conceptual breakthrough because it would revive the deterministic Bohmian mechanics (which apperas to be one of the background inspiring models of this book) and the praticability of \textit{hidden variable} models into it. Landsman is of course aware of the disputability of these statements and candidly adds that his book ``is predicated on the idea that operator algebras provide the right language for quantum theory. If they don't -- for example, if path integrals are really its essence, as researchers especially in quantum gravity seem to believe, and there turns out to be a difference between the two toolkits -- the mathematical underpinning of Bohrification would fall. Since our conceptual program is closely linked to this mathematical language, it would presumably collapse, too'' (p. 19). According to this program, the book is divided in two parts: Part I \textit{``\(C_0(X)\) and \(B(H)\)''} (about 170 pages) gives a mathematical introduction to both classical and quantum mechanics from an operator-algebraic point of view: here these theories are kept separate, whilst mathematical analogies are stressed whenever possible. The longer Part II \textit{``Between \(C_0(X)\) and \(B(H)\)''} (about 300 pages) is devoted instead to the asymptotic Bohrification program. This part centers on the spontaneous symmetry breaking (SSB) and the closely related measurement problem of quantum mechanics which in some sense is the conceptual core of the book and is explicitly addressed in the Chapter 11. It is generally acknowledged that quantum mechanics is unable to account for single outcomes of experiments and has to satisfy physicists with merely probabilistic predictions. Accordingly ``giving up determinism was soon incorporated in the Copenhagen Interpretation of Bohr and Heisenberg\dots and more broadly became part of what might be called `orthodoxy' '' (p.435). Moreover solving the measurement problem within unitary quantum mechanics is generally felt to be impossible because of the so-called \textit{insolubility theorems}: (1) the impossibility of explaining the transition from a pure state to a mixture from the unitary time evolution according to the Schrödinger equation; (2) the impossibility of getting single measurement outcomes in the said unitary evolution. The author on the other hand proposes to outrightly abandon the two-step way of looking at the measurement process, to take instead into account the special nature of measurement devices and finally to attribute some kind of reality to the state of the system (though admittedly this reality cannot be `absolute', as in classical physics). In his opinion a suitable framework for this paradigm is provided by what he calls \textit{asymptotic emergence} that he describes as follows: ``1. A `higher-level theory' \(H\) (which in the context of the measurement problem is either classical mechanics or classical thermodynamics, depending on the measurement setup) is a limiting case of some `lower-level theory' \(L\) (viz. quantum mechanics, including quantum statistical mechanics of a finite system). 2. Theory \(H\) is well defined and understood by itself (typically predating L). 3. Theory \(H\) has `emergent' features that cannot be explained by \(L\), e.g., because \(L\) does not have any property inducing those feature(s) in the limit pertinent to \(H\)'' (p. 452). In this context the proposed collapse mechanism is only effective asymptotically: by taking into account perturbations of the Hamiltonian that are tiny and ineffective in the quantum regime, but become hugely destabilizing in the classical regime, the wave-function of the apparatus will in fact collapse. Crucial to this collapse paradigm appears to be the so called \textit{Flea mechanism} (small perturbations effect): ``The `flea' perturbation might itself be a genuine random process, perhaps ultimately of quantum origin. In that case, the measurement merely amplifies the randomness that was already inherent in the flea by transferring it to the apparatus. Alternatively, the flea might be fundamentally deterministic\dots. In principle, this would open the door to a restoration of determinism: for the flea now transfers its determinism (rather than its randomness) to the apparatus. The mistaken impression that quantum theory implies the irreducible randomness of nature then arises because although measurement outcomes are determined, they are unpredictable `for all practical purposes', even in a way that (because of the exponential sensitivity to the flea in \(1/\hbar\) or \(N\)) dwarfs the unpredictability of classical chaotic systems\dots. The location of the flea plays a similar role to the position variable in Bohmian mechanics, i.e., it is essentially a hidden variable'' (p. 456). Since however Bell's inequalities are violated by quantum mechanics the author argue that any hidden variable theory compatible with quantum mechanics must in some way violate Bell's locality conditions: the possibility of superluminal signaling being however dismissed as inherently tied to an impossible control of the hidden variables involved. The book ends with more than 800 references, and a valuable list of five mathematical appendices which -- albeit, understandably, not completely exhaustive -- are rich enough (they total some 350 pages: a book in itself indeed) to give to the reader a good tutorial on the selected topics focusing on concepts and ideas rather than on technicalities that are, however, not outrightly sidelined. Two are especially detailed: Appendix B: \textit{Basic functional analysis} (130 pages), and Appendix C: \textit{Operator algebras} (132 pages): ``Though primarily meant to support the main body of the book, these appendices may also be of some interest by themselves, especially to philosophers, but even to mathematicians'' (p. 15)
scientific article; zbMATH DE number 6718357