Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the (...) theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix. (shrink)

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller, have thought that the answer to this question is No - that the status of individual continuous quantities is very different in quantum mechanics than in classical (...) mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. (shrink)

The centerpiece of Jeffrey Bub's book Interpreting the Quantum World is a theorem (Bub and Clifton 1996) which correlates each member of a large class of no-collapse interpretations with some 'privileged observable'. In particular, the Bub-Clifton theorem determines the unique maximal sublattice L(R,e) of propositions such that (a) elements of L(R,e) can be simultaneously determinate in state e, (b) L(R,e) contains the spectral projections of the privileged observable R, and (c) L(R,e) is picked out by R and e alone. (...) In this paper, we explore the issue of maximal determinate sets of observables using the tools provided by the algebraic approach to quantum theory; and we call the resulting algebras of determinate observables, "maximal *beable* subalgebras". The capstone of our exploration is a generalized version of Bub and Clifton's theorem that applies to arbitrary (i.e., both mixed and pure) quantum states, to Hilbert spaces of arbitrary (i.e., both finite and infinite) dimension, and to arbitrary observables (including those with a continuous spectrum). Moreover, in the special case covered by the original Bub-Clifton theorem, our theorem reproduces their result under strictly weaker assumptions. This added level of generality permits us to treat several topics that were beyond the reach of the original Bub-Clifton result. In particular: (a) We show explicitly that a (non-dynamical) version of the Bohm theory can be obtained by granting privileged status to the position observable. (b) We show that Clifton's (1995) characterization of the Kochen-Dieks modal interpretation is a corollary of our theorem in the special case when the density operator is taken as the privileged observable. (c) We show that the 'uniqueness' demonstrated by Bub and Clifton is only guaranteed in certain special cases -- viz., when the quantum state is pure, or if the privileged observable is compatible with the density operator. We also use our results to articulate a solid mathematical foundation for certain tenets of the orthodox Copenhagen interpretation of quantum theory. For example, the uncertainty principle asserts that there are strict limits on the precision with which we can know, simultaneously, the position and momentum of a quantum-mechanical particle. However, the Copenhagen interpretation of this fact is not simply that a precision momentum measurement necessarily and uncontrollably disturbs the value of position, and vice-versa; but that position and momentum can never in reality be thought of as simultaneously determinate. We provide warrant for this stronger 'indeterminacy principle' by showing that there is no quantum state that assigns a sharp value to both position and momentum; and, a fortiori, that it is mathematically impossible to construct a beable algebra that contains both the position observable and the momentum observable. We also prove a generalized version of the Bub-Clifton theorem that applies to "singular" states (i.e., states that arise from non-countably-additive probability measures, such as Dirac delta functions). This result allows us to provide a mathematically rigorous reconstruction of Bohr's response to the original EPR argument -- which makes use of a singular state. In particular, we show that if the position of the first particle is privileged (e.g., as Bohr would do in a position measuring context), the position of the second particle acquires a definite value by virtue of lying in the corresponding maximal beable subalgebra. But then (by the indeterminacy principle) the momentum of the second particle is not a beable; and EPR's argument for the simultaneous reality of both position and momentum is undercut. (shrink)

In this text, I explore the intertwined relationship between philosophy of science and theology, and the challenges they both encounter in the academic realm. While theology is struggling to maintain its relevance, philosophy of science has gained recognition and offers valuable insights to theologians. I argue that theologians can benefit from engaging with the content and methods of specific sciences, and philosophers of science can help affirm the methodological legitimacy of theology. However, it is important to note that science (...) itself is facing its own set of problems. There is a decline in interest and job satisfaction among scientists, a growing gap between science and the humanities, a crisis of conceptual understanding, and a perceived lack of meaning and purpose. In this context, I propose that theology and philosophy can play a crucial role in addressing these issues. In particular, theology can offer a supportive environment for discussing meaning, values, and the ultimate questions of human existence. Philosophy, on the other hand, can help bridge the gap between science and the humanities, fostering interdisciplinary dialogue and a more holistic understanding of our world. (shrink)

Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don’t we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. (...) We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher–Haag–Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality-inspired principle we call charge recombination. 1 Introduction2 Paraparticles in Quantum Theory3 Theoretical Equivalence3.1 Field systems in algebraic quantum field theory3.2 Equivalence of field systems4 A Brief History of the Equivalence Thesis4.1 The Green decomposition4.2 Klein transformations4.3 The argument of Drühl, Haag, and Roberts4.4 The Doplicher–Roberts reconstruction theorem5 Sharpening the Thesis6 Discussion6.1 Interpretations of Quantum Mechanics6.2 Structuralism and haecceities6.3 Paraquark theories. (shrink)

I argue against the claim -- advocated by Albert Einstein, Bernard Williams, and Ted Sider, among others -- that a description is objective only if it says how the world is in itself. Instead, I argue for the claim -- inspired by comments of Niels Bohr -- that a family of descriptions is objective only if they co-vary with their respective descriptive contexts. Moreover, I claim that "there is a shared objective reality" simply means that it is possible to (...) satisfy this kind of covariance requirement. (shrink)

Here is a fresh look at how science contributes to the bigger picture of human flourishing, through a collage of science and philosophy, richly illustrated by the authors' own experience and personal reflection. They survey the territory of fundamental physics, machine learning, philosophy of human identity, evolutionary biology, miracles, arguments from design, naturalism, the history of ideas, and more. The natural world can be appreciated not only for itself, but also as an eloquent gesture, a narrative and a pointer beyond (...) itself. Our human journey is not to a theorem or a treatise, but to a meeting which encompasses all our capacities. In this meeting, science is the way to find out about the structure of the physical world of which we are a part, not a means to reduce ourselves and our fellow human beings to mere objects of scrutiny, and still less to attempt the utterly futile exercise of trying to do that to God. We have intellectual permission to be open to the notion that God can be trusted and known. The material world encourages an open-hearted reaching out to something more, with a freedom to seek and to be received by what lies beyond the scope of purely impersonal descriptions and attitudes. (shrink)

This volume gathers together ground-breaking work on the foundations and philosophy of quantum physics, by one of the most brilliant and productive researchers in the field. Rob Clifton died tragically in 2002 at the age of 38; two of his colleagues, themselves leading philosophers of physics, present fourteen of his finest papers here, all of which combine exciting philosophical discussion with rigorous mathematical results. Quantum Entanglements offers inspiration and substantial reward to anyone working on the foundations of quantum theory, (...) with backgrounds in philosophy, physics, or mathematics. (shrink)

Everyone will find something interesting in this book, and many will find something or other that they completely disagree with. William Demopoulos was no fan o.

In recent papers Hans Halvorson has offered a critique of the semantic view of theories, showing that theories may be the same although the corresponding sets of models are different and, conversely, that theories may be different although the corresponding sets of models are the same. This critique will be assessed, first, as it pertains to issues concerning scientific models in the empirical sciences and, second, independent of any concern with empirical science.

Since the beginning of the 20th century, philosophers of science have asked, "what kind of thing is a scientific theory?" The logical positivists answered: a scientific theory is a mathematical theory, plus an empirical interpretation of that theory. Moreover, they assumed that a mathematical theory is specified by a set of axioms in a formal language. Later 20th century philosophers questioned this account, arguing instead that a scientific theory need not include a mathematical component; or that the mathematical component need (...) not be specified by a set of axioms in a formal language. We survey various accounts of scientific theories entertained in the 20th century -- removing some misconceptions, and clearing a path for future research. (shrink)

Halvorson argues that the semantic view of theories leads to absurdities. Glymour shows how to inoculate the semantic view against Halvorson's criticisms, namely by making it into a syntactic view of theories. I argue that this modified semantic-syntactic view cannot do the philosophical work that the original "language-free" semantic view was supposed to do.

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. V. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory (...) can be reduced to one in which all truths are analytic. This book dissects these and other arguments through in-depth investigation of the mathematical facts undergirding them. It presents a systematic, mathematically rigorous account of the key notions arising from such debates, including theory, equivalence, translation, reduction, and model. The result is a far-reaching reconceptualization of the role of formal methods in answering philosophical questions. (shrink)

We show that three fundamental information-theoretic constraints -- the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment -- suffice to entail that the observables and state space of a physical theory are quantum-mechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about (...) nonlocality and bit commitment. (shrink)

David Malament (1996) has recently argued that there can be no relativistic quantum theory of (localizable) particles. We consider and rebut several objections that have been made against the soundness of Malament’s argument. We then consider some further objections that might be made against the generality of Malament’s conclusion, and we supply three no‐go theorems to counter these objections. Finally, we dispel potential worries about the counterintuitive nature of these results by showing that relativistic quantum field theory itself explains the (...) appearance of “particle detections.”. (shrink)

We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the canonical commutation relations. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.

We discuss ways in which category theory might be useful in philosophy of science, in particular for articulating the structure of scientific theories. We argue, moreover, that a categorical approach transcends the syntax-semantics dichotomy in 20th century analytic philosophy of science.

Kierkegaard's Either-Or is a book about the choice between aesthetic, ethical, and religious approaches to life. I show that Either-Or also contains a proposal for philosophy of science, and in particular, about the ideal epistemic state for human beings. Whereas the Cartesian-Hegelian tradition conceived of the ideal state as one of detached deliberation -- i.e. "seeing the world as it is in itself" -- Kierkegaard envisions the ideal state as the achievement of equilibrium between the "spectator" and "actor" aspects of (...) the human being. Kierkegaard's proposal thus sets the stage for Niels Bohr's "epistemological lesson of quantum mechanics". (shrink)

How Logic Works is an introductory logic textbook that is different by design. Rather than teaching elementary symbolic logic as an abstract or rote mathematical exercise divorced from ordinary thinking, Hans Halvorson presents it as the skill of clear and rigorous reasoning, which is essential in all fields and walks of life, from the sciences to the humanities—anywhere that making good arguments, and spotting bad ones, is critical to success. Instead of teaching how to apply algorithms using “truth (...) trees,” as in the vast majority of logic textbooks, How Logic Works builds on and reinforces the innate human skills of making and evaluating arguments. It does this by introducing the methods of natural deduction, an approach that teaches students not only how to carry out a proof and solve a problem but also what the principles of valid reasoning are and how they can be applied to any subject. The book also allows students to transition smoothly to more advanced topics in logic by teaching them general techniques that apply to more complicated scenarios, such as how to formulate theories about specific subject matter. How Logic Works shows that formal logic—far from being only for mathematicians or a diversion from the really deep questions of philosophy and human life—is the best account we have of what it means to be rational. By teaching logic in a way that makes students aware of how they already use it, the book will help them to become even better thinkers. A concise, readable, and user-friendly introduction to elementary symbolic logic that primarily uses natural deduction rather than algorithmic “truth trees” Draws on more than two decades’ experience teaching introductory logic to undergraduates Provides a stepping stone to more advanced topics. (shrink)

I look at the distinction between between realist and antirealist views of the quantum state. I argue that this binary classification should be reconceived as a continuum of different views about which properties of the quantum state are representationally significant. What's more, the extreme cases -- all or none --- are simply absurd, and should be rejected by all parties. In other words, no sane person should advocate extreme realism or antirealism about the quantum state. And if we focus on (...) the reasonable views, it's no longer clear who counts as a realist, and who counts as an antirealist. Among those taking a more reasonable intermediate view, we find figures such as Bohr and Carnap -- in stark opposition to the stories we've been told. (shrink)

The hole argument purportedly shows that spacetime substantivalism implies a pernicious form of indeterminism. We show that the argument is seductive only because it mistakes a trivial claim (viz. there are isomorphic models) for a significant claim (viz. there are hole isomorphisms). We prove that the latter claim is false -- thereby closing the debate about whether substantivalism implies indeterminism.

Jill North argues that Hamiltonian mechanics provides the most spare -- and hence most accurate -- account of the structure of a classical world. We point out some difficulties for her argument, and raise some general points about attempts to minimize structural commitments.

I argue in favor of the strategy of methodological naturalism in science; that is, restricting the scope of investigations to the natural world. I argue that this strategy is most plausibly motivated by supernaturalist theism, rather than by metaphysical naturalism.

Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.

Clifton, Bub, and Halvorson (CBH) have recently argued that quantum theory is characterized by its satisfaction of three fundamental information-theoretic constraints. However, it is not difficult to construct apparent counterexamples to the CBH characterization theorem. In this paper, we discuss the limits of the characterization theorem, and we provide some technical tools for checking whether a theory (specified in terms of the convex structure of its state space) falls within these limits.

Philosophical reflection on quantum field theory has tended to focus on how it revises our conception of what a particle is. However, there has been relatively little discussion of the threat to the "reality" of particles posed by the possibility of inequivalent quantizations of a classical field theory, i.e., inequivalent representations of the algebra of observables of the field in terms of operators on a Hilbert space. The threat is that each representation embodies its own distinctive conception of what a (...) particle is, and how a "particle" will respond to a suitably operated detector. Our main goal is to clarify the subtle relationship between inequivalent representations of a field theory and their associated particle concepts. We also have a particular interest in the Minkowski versus Rindler quantizations of a free Boson field, because they respectively entail two radically different descriptions of the particle content of the field in the *very same* region of spacetime. We shall defend the idea that these representations provide *complementary descriptions* of the same state of the field against the claim that they embody completely *incommensurable theories* of the field. (shrink)

According to the premises of the fine-tuning argument, most nomologically possible universes lack intelligent life; and the fact that ours has intelligent life is best explained by supposing it was created. However, if our universe was created, then the creator chose the laws of nature, and hence chose in favor of lifeless universes. In other words, the fine-tuning argument shows that God prefers universes without intelligent life; and the fact that our universe has intelligent life provides no new evidence for (...) God's existence. (shrink)

This three-part paper comprises: (i) a critique by Halvorson of Bell’s (1973) paper ‘Subject and Object’; (ii) a comment by Butterfield; (iii) a reply by Halvorson. An Appendix gives the passage from Bell that is the focus of Halvorson’s critique.

Each many-sorted theory can be converted to an unsorted theory. But this conversion procedure is not uniquely determined, leading to a dilemma: which unsorted theory captures the content of the corresponding many-sorted theory?

Many of the "counterintuitive" features of relativistic quantum field theory have their formal root in the Reeh-Schlieder theorem, which in particular entails that local operations applied to the vacuum state can produce any state of the entire field. It is of great interest then that I.E. Segal and, more recently, G. Fleming (in a paper entitled "Reeh-Schlieder meets Newton-Wigner") have proposed an alternative "Newton-Wigner" localization scheme that avoids the Reeh-Schlieder theorem. In this paper, I reconstruct the Newton-Wigner localization scheme and (...) clarify the <em>limited</em> extent to which it avoids the counterintuitive consequences of the Reeh-Schlieder theorem. I also argue that there is no coherent interpretation of the Newton-Wigner localization scheme that renders it free from act-outcome correlations at spacelike separation. (shrink)

Although Bohr's reply to the EPR argument is supposed to be a watershed moment in the development of his philosophy of quantum theory, it is difficult to find a clear statement of the reply's philosophical point. Moreover, some have claimed that the point is simply that Bohr is a radical positivist. In this paper, we show that such claims are unfounded. In particular, we give a mathematically rigorous reconstruction of Bohr's reply to the _original_ EPR argument that clarifies its logical (...) structure, and which shows that it does not rest on questionable philosophical assumptions. Rather, Bohr's reply is dictated by his commitment to provide "classical" and "objective" descriptions of experimental phenomena. (shrink)

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyse entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter's ability to (...) perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard non-relativistic quantum theory and classical relativistic field theory. (shrink)

The thesis of quantifier variance is consistent and cannot be refuted via a collapse argument.

I discuss motivations for methodological naturalism in science. I argue that methodological naturalism neither needs nor supports metaphysical naturalism.

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

The nature of antimatter is examined in the context of algebraic quantum field theory. It is shown that the notion of antimatter is more general than that of antiparticles. Properly speaking, then, antimatter is not matter made up of antiparticles—rather, antiparticles are particles made up of antimatter. We go on to discuss whether the notion of antimatter is itself completely general in quantum field theory. Does the matter–antimatter distinction apply to all field theoretic systems? The answer depends on which of (...) several possible criteria we should impose on the space of physical states. (shrink)

Max Jammer claimed that, "There can be no doubt that the Danish precursor of modern existentialism and neo-orthodox theology, Soren Kierkegaard, through his influence on Bohr, affected also the course of modern physics to some extent." Despite Jammer's failure to supply sufficient evidence for this claim, I argue that it is not completely off base. In particular, I argue that Kierkegaard and Bohr belong to a common philosophical tradition, and I begin to investigate some of the themes that characterize this (...) tradition. (shrink)

The Univalent Foundations of mathematics take the point of view that all of mathematics can be encoded in terms of spatial notions like "point" and "path". We will argue that this new point of view has important implications for philosophy, and especially for those parts of analytic philosophy that take set theory and first-order logic as their benchmark of rigor. To do so, we will explore the connection between foundations and philosophy, outline what is distinctive about the logic of the (...) Univalent Foundations, and then describe new philosophical theses one can express in terms of this new logic. (shrink)