We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics. We will not provide rigorous definitions or anything resembling a coherent mathematical theory, but we will take the reader for a journey introducing concepts which are part of category theory in a manner that the physicist will recognize them.
We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics. We will not provide rigorous definitions or anything resembling a coherent mathematical theory, but we will take the reader for a journey introducing concepts which are part of category theory in a manner that the physicist will recognize them.
Abstract Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.
In this paper I shall outline a possibility for incorporating category theory in an essential way into the foundations of quantum theory. The role for category theory which I shall sketch here is strictly foundational in the sense that it offers no new perspectives for a physicist working within the quantum formalism, but rather provides a way of interpreting the algebraic structure by giving an alternative axiomatization of the underlying logic of the theory. The axiomatization is given without recourse to any classical set theory by invoking methods closely connected with the work of F. W. Lawvere. More precisely, then, a possible approach to the foundations of physics which axiomatizes its algebraic structure in a way which is independent of set theory and hence points, as it were, to a structuralist foundation of physical theories, will be proposed.
Introduction Our starting point is a particular “canvas” aimed to “draw” theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this chapter, we consider the conceptual foundations for this canvas and how these can then be converted into mathematical structure. With very little structural effort (i.e., in very abstract terms) and in a very short time span, the categorical quantum mechanics (CQM) research program, initiated by Abramsky and this author, has reproduced a surprisingly large fragment of quantum theory [3, 45, 48, 56, 60–62, 170, 179]. It also provides new insights both in quantum foundations and in quantum information —for example, in [49–51, 58, 59, 64, 79, 80]—and has even resulted in automated reasoning software called quantomatic [71–73], which exploits the deductive power of CQM, which is indeed a categorical quantum logic [77]. In this chapter, we complement the available material by not requiring prior knowledge of category theory and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere—for example, in representation theory [74], quantum algebra [176], knot theory [187], topological quantum field theory (TQFT) [132], and several other areas.
Reviewed by: Quantum Processes: A Whiteheadian Interpretation of Quantum Field Theory Henry J. Folse Quantum Processes: A Whiteheadian Interpretation of Quantum Field Theory. Frank Hättich . Munster: Agenda Verlag, 2004. xiv + 280 pp. Euro 29.90, pbk.3-89688-204-x. Many philosophers and physicists—including the current reviewer—have argued that Whitehead's process philosophy provides a conceptual framework better suited for understanding the quantum theoretical account of the atomic domain than does the substance ontology that has dominated Western metaphysics. However, apart from calling attention to this issue, few (with the notable exceptions of Henry Stapp and Jorge Nobo) have attempted to do anything about it. Frank Hättich has now changed this situation by providing a sustained analysis of quantum field theory (QFT) in terms of Whitehead's ontological scheme. The result is interesting for two reasons: (1) by extending the match between Whitehead's specific metaphysical proposals and the mathematics of algebraic QFT to a level of detail never before attempted, Hättich greatly strengthens the hope that process ontology can help resolve quantum mysteries; and (2) in noting the points of noncompatibility between QFT and Whitehead's metaphysical conclusions, he suggests ways of revising Whitehead's pioneering effort to enhance the match between speculative ontological scheme and contemporary physical theory. Hättich's book is in three unequal parts: the first two are exegetical, and the third, constituting nearly half the book, is synthetic. Part I presents a "simplified version" of those parts of Whitehead's ontology pertinent to the interpretation of QFT to be presented, and Part II presents an algebraic formulation of [End Page 283] QFT susceptible to that interpretation. Although Part I is "self-contained" from the point of view of its intended application, it is unlikely that a reader wholly ignorant of Whitehead's process ontology would gain enough enlightenment to appreciate much of the author's argumentation; but it is equally unlikely that such a person would be reading this book. Part II presumes considerable mathematical sophistication and familiarity with the formalism of at least quantum mechanics and some knowledge of field theory. It is "introductory" only in the sense that it introduces the particular version of QFT, algebraic QFT (AQFT), to which process ontology is to be applied. Of course, both Whitehead's metaphysics and quantum theory have been subjected to many and various rival interpretations, and in selecting among them Hättich has clearly, and reasonably, been led by choices that maximize the connections he wishes to explore. It is misleading to see the goal here as developing yet another argument for why quantum theory must (or must not) be interpreted realistically. Rather, this investigation presupposes a realist interpretation and then gets to work constructing an ontological framework to show how this is possible, thereby suggesting that those who have argued that a realist interpretation of quantum theory is conceptually impossible have done so because they remain committed to ontological presuppositions moored in classical physics. In presenting the ontological framework for his project in Part III, Hättich accounts for the process of each occasion's creative concrescence from pure potentiality to determined actuality in terms of three Whiteheadian concepts: (1) the extensive continuum; (2) eternal objects; and (3) the underlying activity. Each one of these ontological elements is then matched up with an aspect of the mathematical formalism by which AQFT represents the quantum state of a region in spacetime. Hättich then shows that given these identifications, various consequences of the mathematical formalism imply that some of the ontological claims made by Whitehead are either inconsistent or incompatible with the presumption that AQFT gives us a true account of the world. Thus although the final conclusions of this investigation are critical of Whitehead's specific metaphysical conclusions, the extent of the applicability of his ontological categories to quantum mechanics and QFT provides persuasive grounds for hoping that future realist interpretations of quantum theory can learn much from process metaphysics. For this reason, Hättich is to be commended for keeping the project of process ontology alive by proposing specific reasons for revising Whiteheadian conclusions based on our best scientific theories of the microdomain, rather than arguing...