Abstract The fundamental entity in the algebraic approach to quantum theory is an abstract C*-algebra đ together with its set of states âą, that is, the set of all positive linear functionals on đ. I shall call such a pair an (abstract) quantum system. The theory of C*-algebras and their representations provides the most powerful abstract approach to quantum theory; unlike the propositional lattice approach it also has direct applications in mainstream physics, and in a certain sense is even forced upon us in quantum field theory (OFT) (sec sections (2.11), (2.16)-(2.18)). Mathematically, the subject is a complex of analysis and algebra; it is a rather wild area involving high degrees of infinity. Philosophically it is most strongly associated with opera tionalism; in part this rellects the predilections of its originators, and in part the continuing philosophical appeal of positivism within the mathematics community. (I do not know why this is so.)
Abstract The fundamental entity in the algebraic approach to quantum theory is an abstract C*-algebra đ together with its set of states âą, that is, the set of all positive linear functionals on đ. I shall call such a pair an (abstract) quantum system. The theory of C*-algebras and their representations provides the most powerful abstract approach to quantum theory; unlike the propositional lattice approach it also has direct applications in mainstream physics, and in a certain sense is even forced upon us in quantum field theory (OFT) (sec sections (2.11), (2.16)-(2.18)). Mathematically, the subject is a complex of analysis and algebra; it is a rather wild area involving high degrees of infinity. Philosophically it is most strongly associated with opera tionalism; in part this rellects the predilections of its originators, and in part the continuing philosophical appeal of positivism within the mathematics community. (I do not know why this is so.)
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observablesâmeasurable quantities of the systemâgiven by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.
In this introductory chapter, which actually is the core of the entire book, we make a serious effort to bring together a variety of ideas from philosophy and physics. We first lay the epistemological foundations for our approach to particle physics. These philosophical foundations, combined with tools borrowed from the amazingly sophisticated, but still not entirely satisfactory, mathematical apparatus of present-day quantum field theory and augmented by some new ideas, are then employed to develop a mathematical representation of fundamental particles and their interactions. In this process, also the relation between âfundamental particle physicsâ and âquantum field theoryâ is going to be clarified. This first chapter consists of two voluminous sections: a number of philosophical contemplations followed by a discussion of mathematical and physical elements . The reader might wonder why these massive sections are not presented as two separate chapters. The reason for keeping them together is the desire to emphasize the intimate relation between these two sections: the selection and construction of the mathematical and physical elements is directly based on our philosophical contemplations. The beauty of the entire approach is a fruit of this intimate relationship. The presentation of the material in this chapter is based on the assumption that the reader has a basic working knowledge of linear algebra and quantum mechanics. If that is not the case, the reader might want to consult the equally entertaining and serious introduction Quantum Mechanics: The Theoretical Minimum by Susskind and Friedman [4]. The book Quantum Mechanics and Experience by Albert [5] offers a fascinating introduction with a strong focus on the measurement problem. An almost equation-free discussion of the history and foundations of quantum mechanics can be found in the book Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information by Whitaker [6]. Complex vector spaces, Hilbert space vectors and density matrices for describing states of quantum systems, bosons and fermions, canonical commutation relations, Heisenberg's uncertainty relation, the Schrodinger and Heisenberg pictures for the time evolution of quantum systems, as well as a basic idea of the measurement process are all referred or alluded to in Section 1.1; these basics of quantum mechanics are recapped only very briefly in Section 1.2. The crucial construct of Fock spaces is explained in a loose way in Section 1.1.3 and later elaborated in full detail in Section 1.2.1.
This systematic algebraic approach concerns problems involving a large number of degrees of freedom. It extends the traditional formalism of quantum mechanics, and it eliminates conceptual and mathematical difficulties common to the development of statistical mechanics and quantum field theory. Further, the approach is linked to research in applied and pure mathematics, offering a reflection of the interplay between formulation of physical motivations and self-contained descriptions of the mathematical methods.The four-part treatment begins with a survey of algebraic approaches to certain phys
Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, GĂŠrard Emch argued forcefully for the new approach [6].
Quantum field theory is the ultimate expression of quantum mechanics. Its formalism provides the framework for all modern fundamental theories about the nature of matter. However, it involves difficult and intricate mathematics, which is why physics students donât generally come across it until postgraduate level, and then only if their subject specialism requires the insights that it provides. In ordinary quantum theory, observable properties are represented by Hermitian operators, with eigenstates that form a basis for the expansion of a systemâs states. Philosophers have been debating the correctness of such arguments, but now it seems that experimental science has come along and shown that, at the quantum level at least, the objects we study have no substance to them, independent of their properties. In classical physics, interactions between particles can be considered by either an examination of forces or from the potential energies involved.
Abstract According to a regnant criterion of physical equivalence for quantum theories, a quantum field theory (QFT) typically admits continuously many physically inequivalent realizations. This, the second of a twoâpart introduction to topics in the philosophy of QFT, continues the investigation of this alarming circumstance. It begins with a brief catalog of quantum field theoretic examples of this nonâuniqueness, then presents the basics of the algebraic approach to quantum theories, which discloses a structure common even to âphysically inequivalentâ realizations of a QFT. Finally, it introduces and evaluates a handful of strategies for interpreting quantum theories in the face of the nonâuniqueness of their Hilbert space representations.