Empirical Underdetermination for Physical Theories in C* Algebraic Setting: Comments to an Arageorgis's Argument
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Empirical Underdetermination for Physical Theories in C* Algebraic Setting: Comments to an Arageorgis’s Argument Chrysovalantis Stergiou1 Received: 27 December 2019 / Accepted: 8 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, I reconstruct an argument of Aristidis Arageorgis against empirical underdetermination of the state of a physical system in a C*-algebraic setting and explore its soundness. The argument, aiming against algebraic imperialism, the operationalist attitude which characterized the first steps of Algebraic Quantum Field Theory, is based on two topological properties of the state space: being T1 and being first countable in the weak*-topology. The first property is possessed trivially by the state space while the latter is highly non-trivial, and it can be derived from the assumption of the algebra of observables’ separability. I present some cases of classical and of quantum systems which satisfy the separability condition, and others which do not, and relate these facts to the dimension of the algebra and to whether it is a von Neumann algebra. Namely, I show that while in the case of finitedimensional algebras of observables the argument is conclusive, in the case of infinite-dimensional von Neumann algebras it is not. In addition, there are cases of infinite-dimensional quasilocal algebras in which the argument is conclusive. Finally, I discuss Porrmann’s construction of a net of local separable algebras in Minkowski spacetime which satisfies the basic postulates of Algebraic Quantum Field Theory. Keywords Empirical underdetermination · Algebraic formulation of physical theories · State space · First countable · Separability
1 Introduction In this paper, I explore the possibility of distinguishing and completely determining the actual state of a physical system from its possible states on the basis of empirical evidence. In particular, I discuss an argument in favor of the view that in the limit of To the Memory of Aristidis Arageorgis. * Chrysovalantis Stergiou [email protected] 1
The American College of Greece – Deree, Athens, Greece
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Foundations of Physics
empirical research, when all possible empirical evidence is taken into account, the state is both distinguishable from any other state and completely determined by the evidence. The soundness of such an argument would defeat the thesis of empirical underdetermination of the state of a physical system. The argument is due to Arageorgis [1] and it concerns classical and quantum theories formulated in the C*-algebraic framework. It shows that the distinguishability of the actual state follows naturally from general topological assumptions related to the C*-algebraic framework. Nevertheless, the complete determination of the state of the system rests on the controversial assumptions that the state space is first countable and that the algebra of observables is separable. In an attempt to probe the nature of Arageorgis’s argument, I present different cases
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