A second course in topos quantum theory
One of the main challenges in theoretical physics in the past 50 years has been to define a theory of quantum gravity, i.e., a theory which consistently combines general relativity and quantum theory in order to define a theory of space-time itself seen as a fluctuating field. The reason for such a difficulty is the seemingly incompatible roles of space-time put forward by general relativity and quantum theory. In fact, on the one hand, in general relativity, although the presence of both the 4-dimensional metric and the connection is assumed ab initio, they are both considered to be dynamical quantities and there is no preferred foliation of space-time. On the other hand, quantum theory assumes a fixed (with respect to its differentiable structure and metric) space-time, implied by the mathematical formalism of the theory. For example the Schroedinger's equation describing unitary evolution between measurements is \(i\hbar \frac{\partial {\psi}}{\partial t}=\hat{H}\psi\) where the notion of time is fixed, while the measurement process `reduces' the state vector via the second law of evolution: \({\psi} \rightarrow \frac{\partial {\hat{P} \psi}}{\partial \| {\hat{P} \psi} \|}\) where \({\hat{P}}\) is the projection on the outcome of the measurement. Therefore the state gets projected on a spacelike surface. Clearly, a fixed geometry of spacetime is needed to define both the time \(t\) in the Schroedinger equation and the spacelike surface on which the state vector is reduced. Therefore, although mathematically space-time is treated in an analogous way in both quantum theory and general relativity, its role in these two theories is very different. However, when defining a theory of quantum gravity, the very definition of space-time as a differentiable manifold is put into discussion. In fact, it is believed that at microscopic scales, space-time ceases to be continuous but acquires a discrete nature. Therefore, the continuum structure of space-time suggested by the two main ingredients of quantum gravity seems to be refuted by quantum gravity itself. This might seem an odd predicament, but it might also suggest that the mathematical description of space-time required for quantum gravity should be radically different from the continuum picture put forward by the two ingredient theories. A candidate for an alternative description of space-time is given by the topos approach. In this approach the notion of a space-time point is replaced by the notion of a space-time region. Such regions should be interpreted as defining regions which are occupied by `extended' objects. The interesting feature is that the collection of such `regions' carry a Heyting algebra structure, which is a generalised Boolean algebra where the law of excluded middle does not hold. This mathematical description of space-time in terms of what is technically called a locale fits well with the discrete notion of space-time put forward by quantum gravity. The book contains 14 chapters. In the introduction (Chapter 1) is summarised the main results dealt with in the ``First Course'' [\textit{C. Flori}, A first course in topos quantum theory. Berlin: Springer (2013; Zbl 1280.81001)] so as to refresh the reader with useful concepts which are used throughout this ``Second Course''. In Chapter 2, the author explained some recent results obtained by A. Doering in which it is shown that \(\mathrm{Sub}_{cl} (\underline{\Sigma})\) is not only a complete Heyting algebra, but also a complete co-Heyting algebra, therefore quantum logic is represented by a complete bi-Heyting algebra where two types of implications and negations are present. In Chapter 3, the author explained an alternative way of describing group actions in topos quantum theory and the definition which rests on the idea of flows in the spectral presheaf. The author also defined time evolution in topos quantum theory both in terms of the Heisenberg picture, where propositions evolved in time while states were constant, and the Schroedinger picture, where states evolved in time while propositions remained constant. In Chapter 4, the author introduced the notions of antonymous and observable functions following A. Doering. In Chapter 5, the author showed how it is possible to interpret self-adjoint operators affiliated with a von Neumann algebra \(\mathcal{N}\), as real-valued functions on the projection lattice \(P(\mathcal{N})\) of the algebra. These functions are called \(q\)-observable functions. The author underlined that the novelty of this approach consists in the fact that these real valued functions are related to both the daseinisation map, central to topos quantum theory, and to quantum probabilities. In Chapter 6, the author showed that, if the initial von Neumann algebra \(\mathcal{N}\) is abelian, then it can be completely reconstructed from the poset of its abelian von Neumann subalgebras. However, if the algebra \(\mathcal{N}\) is not abelian, then it can only be reconstructed up to its Jordan structure. This is because both \(\mathcal{N}\) and its opposite \(\mathrm{op}(\mathcal{N})\) have the same collection of subalgebras but they are not necessarily isomorphic to each other. The author also showed how it is possible to retrieve the Jordan information of a von Neumann algebra given the poset of abelian subalgebras of \(\mathcal{N}\). It allows the author to recover the Jordan structure of a von Neumann algebra from its poset of abelian sub algebras. The question that still remains to be answered is if and how the full von Neumann structure can be recovered. This could be done by adding additional information on the poset of abelian subalgebras which would allow the author to retrieve the full von Neumann algebra. In Chapter 7, the author described the topos of sheaves over a category \(\mathcal{C}\) equipped with a Grothendieck topology. In the ``First Course'', the author came across the definition of sheaf over a topological space \(X\). This definition relied solely on the lattice of open sets of the topological space \(X\), i.e. on the topology. In this chapter, the author also extended the notion of topology so as to be able to define sheaves on this more general `topology'. In Chapter 8, the author showed that a locale is just a lattice which mimics the properties of the lattice of open subsets of a topological space, therefore it can be considered as a generalised notion of a topological space. It then follows that it is also possible to define sheaves on a locale. So, the author started with the notion of a presheaf on a locale then extend it to that of a sheaf. In Chapter 9, the author explained how to define categorical notions internally within a topos. This internal description of objects is needed to understand the covariant approach to topos quantum theory explained in Chapter 10 in which the author introduced the notion of a geometric logic. This is the logic of implication between geometric formulas. In Chapter 11, the author described a different way in which topos theory was utilised to describe quantum theory. C. Heunen called this approach as covariant topos quantum theory. The aim of this approach is to combine, on the one hand, algebraic quantum theory by describing a system via a \(\mathcal{C}^{\star}\)-algebra \(\mathcal{A}\) and, on the other hand, Bohr's idea of classical snapshots which enables one to talk about physical quantities, only with respect to a suitable context of compatible physical quantities. There are many similarities between this approach and the topos approach delineated in the ``First Course''. The main difference, however, is the fact that in the covariant approach one defines all physical quantities internally to the topos \([\mathcal{C(A)}, \mathbf{Sets}]\) and any reasoning is done utilising the internal language of \([\mathcal{C(A)}, \mathbf{Sets}]\), which is the Mitchell-Benabou language with the Kripke-Joyal semantics. In Chapter 12, the author proposed an alternative definition of space-time put forward by topos quantum theory. Such a definition consists of modelling space-time in terms of a locale where, now, the fundamental space-time building blocks are regions, not points. This idea reflects the fact that space-time points are not physically meaningful since real objects occupy space-time regions. As explained in the ``First Course'', topos quantum theory is formulated in terms of a topos that depends on the particular physical system under consideration, namely the category of presheaves on the poset of commutative subalgebras of the algebra of observables \(\mathcal{A}\). Instead of working with commutative subalgebras only, in Chapter 13, the author considered all *-homomorphisms \(C(X) \rightarrow \mathcal{A}\) for all commutative C*-algebras \(C(X) \). Doing so means that \(\mathcal{A}\) becomes a functor \(\mathsf{C H a u s} \rightarrow\mathbf{Sets}\). In this way, the author can consider all physical systems as described by objects in the functor category \(\mathbf{Sets}^{\mathsf{C H a u s}}\) or the category of sheaves \(\mathrm{Sh}(\mathsf{C H a u s})\). This will allows the author to consider multiple physical systems at once, shedding light on the problem of composite systems in topos quantum theory. In Chapter 14, the author is analysing how different quantizations can be represented in Topos Quantum Theory. In total, the book is very important and should be in the bookcase of every researcher because it touches enumerable topics of the categorical quantum theory; however the bookcase should be large enough to contain a lot of referenced publications, on the one hand, and some additional structures on presented categories, on the other hand, without which to understand modern quantum theory discussed in the book is impossible.
scientific article; zbMATH DE number 6823466