We can now identify the core of the asymmetry objection. As it is usually understood, wavefunction realism is C-wavefunction realism. As such, it involves a purely haecceitistic commitment to a fact about which space is inhabited by the fundamental field and a fortiori involves a choice of one among a continuum infinity of versions of wavefunction realism that differ merely haecceitistically. From this point of view, the asymmetry objection is rather similar to an objection to spacetime substantivalism—roughly, the thesis that there is a manifold of points that are the bearers of geometric qualities. Just like substantivalism is vulnerable to the challenge that it allows for in-principle undetectable differences between possibilities that concern merely which spacetime point plays which qualitative role, wavefunction realism is vulnerable to the complaint that it allows for in-principle undetectable differences between possibilities that concern merely which space is inhabited by the fundamental field.
This suggests a roughly similar landscape of response strategies. First of all, there is the relationalist strategy. The spacetime relationalist denies the existence of the entities about which qualitatively identical substantivalist worlds disagree: spacetime points. Relationalists instead propose a theory that trades exclusively in qualitative geometric relations among concrete entities. The corresponding response to the asymmetry objection is to propose a theory which denies the existence of the kinds of entities about which unitarily equivalent versions of wavefunction realism disagree: the spaces that contain the fundamental field. This response would involve proposing a theory in terms of primitives that correspond to mathematical structure invariant under unitary equivalence: namely, the structure of an abstract Hilbert space with a privileged range of operators on that space. The quantum analogue of the relationalist response would thus result in a type of Hilbert-space fundamentalism, a view that has recently received increasing attention in the literature.Footnote 13
Hilbert-space fundamentalism may be an interesting view in its own right. But it is not a wavefunction realist response to the asymmetry objection. Wavefunction realists are committed to a fundamental ontology of a field on a high-dimensional space; Hilbert-space fundamentalist views are inconsistent with this commitment.
A second possible response strategy mirrors what has become known as sophisticated substantivalism. The sophisticated substantivalist agrees with the substantivalist as regards the existence of spacetime points, but denies that permutations of spacetime points result in new metaphysical possibilities. Sophisticated substantivalism is thus the conjunction of substantivalism with a modal constraint along the following lines: if the spacetime points at some world instantiate some qualitative profile, they instantiate this profile at every metaphysical possibility at which they exist. The corresponding response to the asymmetry objection conceives of wavefunction realism as the view that asserts, of some specific space, that it is inhabited by the fundamental field, but denies that there is a metaphysical possibility where another space is inhabited by a qualitative duplicate of this field. This sort of ‘sophisticated’ wavefunction realism would block the asymmetry objection by virtue of disallowing the pernicious modal variations in question: according to this sort of view, if C-wavefunction realism is true, W-wavefunction realism is necessarily false; and vice versa.
This response strategy can claim some independent support from certain corners of the fundamentality literature. According to some conceptions of fundamentality, what is fundamental is necessarily fundamental.Footnote 14 On such views, if C is the container of the fundamental field then C is necessarily the container of the fundamental field, and so a fortiori there is no possibility at which the fundamental field is contained in W.
Nonetheless, those in the grip of the asymmetry objection are unlikely to be satisfied with the sophisticated substantivalist response. The resulting version of wavefunction realism still involves a haecceitistic commitment; the main difference is that this haecceitistic commitment is declared metaphysically necessary. While the view is no longer open to the charge of allowing in-principle undetectable modal variations, the asymmetry objection returns in the guise of a more general worry about underdetermination. Consider two versions of the sophisticated substantivalist response: first, sophisticated C-wavefunction realism, according to which C is necessarily fundamental; and second, sophisticated W-wavefunction realism according to which it is W that is necessarily fundamental. These two theories give inconsistent accounts of the landscape of metaphysical modality: they provide inconsistent accounts of what the metaphysical possibilities are. Modal reality conforms to at most one of the two theories; but there seem to be no principled reasons for deciding which one it is. Differently put: the problem is no longer that our wavefunction realism allows for in-principle undetectable differences between possibilities as regards which space is fundamental; rather, the problem is that our total evidence (including, in particular, arguments for wavefunction realism) seems to underdetermine the choice among a continuum infinity of incompatible ways of imposing the modal constraint that is part of the sophisticated substantivalist strategy.Footnote 15
This suggests that the trouble arising from the asymmetry objection has little to do with whether or not possibilities can differ merely as regards which space contains the fundamental field. Whether there are metaphysical possibilities that differ merely haecceitistically is a question that is likely to turn on fairly general considerations that are not sensitive to local disputes about the ontology of non-relativistic quantum mechanics. Indeed, it would be surprising if it turned out that wavefunction realism would require taking sides in a dispute of this kind.
The trouble for wavefunction realism has its source elsewhere. As we have seen, both proponents and opponents of wavefunction realism have understood the view as the thesis that asserts, of some specific space, that it is inhabited by the fundamental field. But this saddles wavefunction realism with a problematic commitment to an in-principle undetectable haecceitistic fact.
This diagnosis suggests that wavefunction realism should be understood as the distinct thesis that there is some (appropriately structured) space that is inhabited by the fundamental field: some space, that is, which plays the role of container space. Wavefunction realism, construed as such a role-based thesis, is silent about which space plays this role: the theory is indifferent as regards the intrinsic nature of the space inhabited by the fundamental field. On the role-based conception, wavefunction realism is not the de re thesis that asserts, of a particular space, that it is the container of the fundamental field; rather, it is the de dicto thesis that there is some space which plays the role of containing the fundamental field.
According to the standard conception, wavefunction realism is C-wavefunction realism and thus involves a commitment to a haecceitistic fact. On the reconception of the view I propose, wavefunction realism does not involve such a commitment: its truth does not hinge on the intrinsic nature of the space that acts as the container of the fundamental field. Granted: on the assumption that there are such things as intrinsic natures, in a world where wavefunction realism is true, there is a haecceitistic fact about which space contains the fundamental field. But this is not because wavefunction realism itself dictates this haecceitistic fact: with regard to the claims that wavefunction realism makes about physical reality, any two appropriately structured spaces are perfectly on a par. On this conception of wavefunction realism, it is not our theorizing that privileges one of these spaces; it is the world that does.Footnote 16
The role-based response to the asymmetry objection is thus distinct from an anti-haecceitist doctrine known as quantifier generalism.Footnote 17 According to the latter, fundamental reality is ‘general’ in the sense that no fundamental fact implies that a particular individual has some property or stands in some relation; rather, the fundamental facts merely have the corresponding existentially quantified implications to the effect that there is something that has this property or that stands in this relation. But role-based wavefunction realism is consistent with the denial of quantifier generalism: it simply has nothing to say about whether reality is ultimately haecceitistic or not.Footnote 18
This illustrates an important respect in which our dialectical situation is different from that of spacetime substantivalism. What is at stake in the debate about the metaphysics of spacetime is precisely whether there are possibilities that differ merely as regards which spacetime points play which qualitative role. In that debate, it would not be a dialectical advance to propose a ‘role-based’ version of spacetime substantivalism that merely asserts that some spacetime points play the relevant qualitative roles: such a view would simply not be responsive to the central question of the debate. By contrast, wavefunction realism is a response to a certain specific question about the ontology of quantum mechanics: which physical entities (if any) are represented by the wavefunction? It is the central contention of the role-based response to the asymmetry objection that the wavefunction-realist answer to this ontological question does not hinge on whether there are possibilities that differ merely as regards the intrinsic nature of the space inhabited by the fundamental field. This latter question is one on which role-based wavefunction realists are silent.
From a certain perspective, this is as it should be. There is a fairly popular view according to which physics tells us about the fundamental roles that are satisfied in the world rather than about the intrinsic natures of the things that satisfy them. The general idea that physics is silent on the intrinsic natures of the realizers of fundamental roles follows from Lewis-Langton-style ‘humility’ theses according to which, roughly, although such realizers exist fundamentally, the true theory that fixes the intrinsic natures of the actual role-realizers is in-principle unknowableFootnote 19 or even inexpressible.Footnote 20 Another implementation of this idea can be found in the recent structuralist trend in the philosophy of physics according to which only the relevant roles—conceived as ‘structural’ in a way that is cashed out in terms of nomic, causal, geometric or other physically substantive relations such as relations of entanglement—but not their realizers are part of the fundamental ontological inventory of the world.Footnote 21 Both types of approach agree that physics doesn’t tell us about intrinsic natures of the things that satisfy the fundamental roles—either because such natures are in-principle unknowable or inexpressible, or because we have independent reason to think that there are no such things as fundamental role-realizers. Role-based wavefunction realism is consistent with this kind of constraint.
This move also strengthens the adaptability of wavefunction realism to more sophisticated physical theories, such as quantum field theory (QFT). As noted earlier, one aspect of the asymmetry objection is that wavefunction realism comes down on the wrong side of the unitary equivalence between the multitude of wavefunction representations: in QFT, it is the momentum representation (rather than its Fourier dual) that plays a crucial role.Footnote 22 It should be clear that this objection is ineffective against role-based wavefunction realism. This view implies merely that there is some appropriately structured fundamental space that contains a fundamental field. Precisely by virtue of its neutrality with regard to the intrinsic nature of the container of the fundamental field, this view can be transposed to settings in which this container has a different structure and plays a different physical role. The move to a role-based conception of wavefunction realism improves its compatibility with future physics.
To be sure: this is not to say that role-based wavefunction realism faces no challenges in the quantum-field-theoretic realm. On the contrary, there is reason to think that these challenges are formidable.Footnote 23 Our result is more limited: the claim that wavefunction realism faces challenges in QFT as a result of privileging the position representation is not true.
Our discussion illustrates why the standard conception of wavefunction realism as C-wavefunction realism is a liability for the wavefunction realists’ agenda. One popular argument in favor of wavefunction realism is that it preserves locality and separability (Ney 2021). But such arguments are not sensitive to the differences between C-wavefunction realism and W-wavefunction realism. To begin with, if the fundamental ontology at a C-world is separable, then so is the fundamental ontology at its Fourier dual (and also its permutation dual). Moreover, although Fourier-dual worlds do not agree as regards the locality of the fundamental dynamical laws—as evidenced by the fact that (3) but not (4) gives rise to a partial differential equation with regard to container-space variables—this is not the case for permutation dual worlds: as noted earlier, if the fundamental dynamical law at some C-world is given by (3), the fundamental dynamical law at its permutation-dual world is given by the Hamiltonian in which k plays the role that q plays in (3). While the (hitherto standard) conception of wavefunction realism as C-wavefunction realism therefore has no bearing on central wavefunction realist arguments, its commitment to a haecceitistic fact about the intrinsic nature of the space inhabited by the fundamental field makes it vulnerable to the asymmetry objection. Our result may thus be put as follows: although proponents of wavefunction realism have undoubtedly conceived of their view as (what I refer to as) C-wavefunction realism, their philosophical ends would have been better served by adopting the role-based conception developed in this paper.Footnote 24