Abstract
Ruetsche (Interpreting quantum theories, 2011) argues that a problem of unitarily inequivalent representations arises in quantum theories with infinitely many degrees of freedom. I provide an algebraic formulation of classical field theories and show that unitarily inequivalent representations arise there as well. I argue that the classical case helps us rule out one possible response to the problem of unitarily inequivalent representations called Hilbert Space Conservatism.
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Notes
This assumes that a quantum theory is a concrete Hilbert space representation of canonical commutation or anti-commutation relations. This assumption is not universally shared and we will see later on how it is challenged.
While this paper provides an argument against Hilbert Space Conservatism about classical field theories, I do not claim to provide a comprehensive argument for Algebraic Imperialism about classical field theories. I will lay out some possible objections to Algebraic Imperialism about classical field theories and some possible responses in Sect. 5. But whatever the outcome of these debates about Algebraic Imperialism, my central claim is only that Hilbert Space Conservatism about classical field theories is untenable, and this teaches us about Hilbert Space Conservatism even in the quantum case.
For more on operator algebras, see Kadison and Ringrose (1997), Sakai (1971), and Landsman (1998). For more on the algebraic formalism and axioms of algebraic quantum theory, see Haag (1992), Bratteli and Robinson (1987) and Emch (1972). For philosophical introductions, see Halvorson (2006) and Ruetsche (2011, Chap. 4).
For example, one might restrict attention to only open double cones (see, e.g. Halvorson 2006, p. 740).
Although translation-invariance is typically taken as only a necessary condition for being a vacuum state, for the purposes of this paper I take it as a definition of vacuum states so that what I say later on will apply to any state that is even a candidate for being a vacuum state.
The Stone–von Neumann theorem carries additional assumptions as well. It assumes that the phase space of the classical theory is symplectic and that the representation is continuous in an appropriate sense.
The motivation for this standard practice is that in the weak operator topology, a sequence of observables well approximates (i.e. converges to) another observable just in case the sequence approximates the observable with respect to all possible expectation values and transition probabilities, and hence with respect to the empirical predictions of the theory. One can question already whether this motivation succeeds in justifying the weak operator topology (Feintzeig 2015), but one need not do so for the argument against Hilbert Space Conservatism that follows.
One could generalize by repeating these constructions for smooth sections of an arbitrary vector bundle over \({\mathcal {M}}\).
Some technical caveats about “appropriate” configuration spaces: first, we require that \({\mathcal {U}}\) be closed under translations (see Footnote 17). Second, we also would like to put a topology on \({\mathcal {U}}\) so that we have the option of restricting attention to observables that are continuous functions on \({\mathcal {U}}\).
For this definition to make sense, we must require that \({\mathcal {U}}\) is closed under translations, i.e. if \(\varphi \in {\mathcal {U}}\), then \(\beta _x(\varphi )\in {\mathcal {U}}\).
There is another notion of representation one can use for abelian algebras (see Landsman 1998, p. 76), according to which observables are represented as functions on a manifold in much the same way we started this section. One can prove some uniqueness results about such representations (Landsman 1998, p. 80). While that notion of representation is fruitful for many purposes, I will not use it in this paper. The purpose of this paper is specifically to investigate the significance of Hilbert space representations of an algebra of observables.
Recall that each determinate field configuration in \({\mathcal {U}}\) defines one of these pure states, so that \({\mathcal {P}}({\mathfrak {A}})\) (with the weak* topology) may be understood as a kind of compactification of \({\mathcal {U}}\).
Here, the relevant cyclic vector \(\varOmega _\omega \) is the constant unit function.
Admittedly, this argument works only for irreducible representations in the classical case. See Feintzeig (2015) for a general solution.
Thanks to an anonymous referee for bringing this objection to my attention.
Superselection sectors are also sometimes thought to have some extra dynamical significance. In the classical case, unitarily inequivalent representations are only significant for the notion of superposition and not for dynamics.
A two-sided ideal is a subalgebra \(\mathfrak {I}\subseteq {\mathfrak {A}}\) such that for all \(A\in \mathfrak {I}\) and \(B\in {\mathfrak {A}}\), we have \(AB\in \mathfrak {I}\) and \(BA\in \mathfrak {I}\).
Thanks to an anonymous referee for bringing this to my attention.
As noted in Sect. 5, there may be real issues involved in choosing the abstract algebra to use in constructing the theory. For simpler classical systems with finitely many degrees of freedom (i.e., for which the configuration space \({\mathcal {U}}\) is locally compact), the choice of algebra is more obvious. One standardly uses \(C_0({\mathcal {U}})\), the continuous functions on \({\mathcal {U}}\) that vanish at infinity, because the pure state space of this algebra corresponds exactly to the collection of field configurations \({\mathcal {U}}\). However, for the case we have considered of classical field theory with infinitely many degrees of freedom, things are not so simple because \({\mathcal {U}}\) is not locally compact. I have restricted attention to two choices: the algebra of bounded continuous functions (in some appropriate topology) or the algebra of bounded (not necessarily continuous functions) on \({\mathcal {U}}\). I do not claim that these are the only choices one could make. For now, I only claim that using one or the other of the algebras listed above allows us to represent at least as many states as we would like. This suffices for the conclusions I wish to draw here.
Moreover, I think it is important to investigate the possible algebraic options further for a complete understanding of the relationship between classical field theory and quantum field theory. For example, one might be able to apply the methods of deformation quantization to an algebraic formulation of the field theoretic case to illuminate conceptual issues in both classical and quantum field theory.
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Acknowledgments
I would like to thank David Baker, Jeff Barrett, Hans Halvorson, David Malament, John Manchak, Laura Ruetsche, and especially Jim Weatherall for extremely helpful comments and discussions concerning this paper that lead to many improvements. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under NSF Grant No. DGE-1321846. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
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Feintzeig, B. Unitary inequivalence in classical systems. Synthese 193, 2685–2705 (2016). https://doi.org/10.1007/s11229-015-0875-1
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DOI: https://doi.org/10.1007/s11229-015-0875-1