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 finite-dimensional 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.
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Notes
I would like to thank the anonymous referee for prompting a further clarification of the notion of state as intended in this paper.
The most common interpretation of the observables is in terms of operations performed in the laboratory. However, I do not want to openly commit myself to an operationalist interpretation.
For a detailed analysis of the concept of algebraic imperialism and its variants, consult [20].
Even if \(Q\) were not a self-adjoint element, it could be analyzed in terms of two self-adjoint elements \(X,Y\): \(Q = X + iY\). Since \(\omega_{1} \left( Q \right) \ne \omega_{2} \left( Q \right)\), then either \(\omega_{1} \left( X \right) \ne \omega_{2} \left( X \right)\) or \(\omega_{1} \left( Y \right) \ne \omega_{2} \left( Y \right)\). Hence, there is always a self-adjoint element distinguishing to distinct states.
I thank Ben Feintzeig for stirring up this part of the discussion.
The Euclidean space \({\mathbb{R}}^{n}\) is second countable since the set \({\mathcal{B}} = \left\{ {{\rm B}_{r} \left( x \right):x \in {\mathbb{Q}}^{n} , r \in {\mathbb{Q}},r > 0} \right\}\) of open balls centered at points having rational coordinates is a countable base for the topology.
For a philosophical discussion of symmetry breaking in quantum spin systems, consult [19].
Porrmann ([17], Appendix A) delivers a proof of a slightly more general fact about the existence a separable C*-subalgebra \({\mathcal{A}}_{sep}\) of any unital subalgebra \({\mathcal{A}}\) of \({\mathcal{B}}\left( {\mathcal{H}} \right)\), where \({\mathcal{H}}\) is a separable Hilbert space, which is dense in \({\mathcal{A}}\) in the strong operator topology: \(\overline{{{\mathcal{A}}_{sep} }}^{SOT} = {\mathcal{A}}\).
For a recent review of the relevant facts, consult (Halvorson and Mueger [12], pp. 749–752).
Porrmann mentions “standard diamonds” but I believe he refers to double cones. A diamond is any (open) region of Minkowski space time, bounded or unbounded, that satisfies the relation: \(O = O^{\prime\prime}\), where \(O^{\prime}\) is the subset of \({\mathbb{R}}^{4}\) which contains all points at spacelike distance with every point of \(O\) ([13], p. 24). A double cone is a well-known case of a bounded diamond region defined as the interior of the intersection of the forward and the backward light cones of two timelike distant spacetime points. Alexandrov has observed that double cones provide a base for the topology of Minkowski spacetime, \({\mathbb{R}}^{4}\) ([5], p. 5).
For more details, especially about relativistic covariance, consult ([16], p. 55).
Porrmann refers to the Fredenhagen–Hertel compactness condition which restricts the number of states of finite total energy on a given local algebra.
See, https://mathoverflow.net/questions/311534/topology-of-state-space-in-von-neumann-algebras/311653?noredirect=1#comment777613_311653 and https://math.stackexchange.com/questions/2929380/sufficient-conditions-for-a-c-algebra-to-be-separable I would like to thank especially Robert Furber and Martin Argerami for their contribution to this discussion.
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Acknowledgements
The author would like to thank very much L. Ruetsche for her comments, her overall help and encouragement. Also, he thank B. Feintzeig for his comments.
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To the Memory of Aristidis Arageorgis.
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Stergiou, C. Empirical Underdetermination for Physical Theories in C* Algebraic Setting: Comments to an Arageorgis's Argument. Found Phys 50, 877–892 (2020). https://doi.org/10.1007/s10701-020-00358-0
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DOI: https://doi.org/10.1007/s10701-020-00358-0