In this section, I develop a symmetry-fundamentalist ontology for non-relativistic N-particle quantum mechanics. The mathematical framework I take as the starting point consists of two things: first, a Hilbert space used to characterize the quantum states of the world. Second, a range of Hermitian operators that encode the physical structure among quantum states; more on this in a moment. Call this formalism the Hilbert space framework.Footnote 13 Some version of it is part of the formalism of virtually every existing theory of non-relativistic quantum mechanics, including Bohmian mechanics, Everettian quantum mechanics, and dynamical-collapse theories.
A Hilbert-space automorphism can be thought of as a relation among Hilbert space vectors: the relation that obtains between vectors x and y iff the automorphism maps x to y. A natural way to implement fundamentalism about state-space symmetries is therefore to posit fundamental relations that correspond to these automorphisms.
This suggests that the relata of those relations should be regarded as fundamental entities as well. On a straightforward understanding of fundamentality, an entity is fundamental if and because it figures in some fundamental proposition.Footnote 14 Thus, if the physical counterpart of a state-space symmetry is a fundamental relation, then this is so because it occurs in some fundamental proposition. But the sorts of propositions we express in terms of state-space symmetries also mention the relata of those relations: whatever physical entities correspond to Hilbert space vectors. On this line of thought, the fundamentality of the relations corresponding to state-space symmetries implies the existence of a fundamental space whose points are the relata of those relations.
It is natural to think of this fundamental space as state space, the space of quantum states of the world. In the context of non-relativistic N-particle quantum mechanics, this raises an important choice point. It is standard to assume that quantum states correspond to Hilbert space rays rather than to Hilbert space vectors. On this view, quantum state space corresponds not to linear Hilbert space but to projective Hilbert space.
Notwithstanding this standard assumption, physicists and philosophers alike routinely appeal to linear Hilbert space structure in their theoretical inferences; indeed, reformulating the entire technical machinery of quantum mechanics in the language of projective Hilbert spaces is a challenging task in its own right.Footnote 15 Articulating an ontology that is fundamentalist about projective Hilbert space therefore faces obstacles that go beyond the usual difficulties of metaphysical theorizing about some theory of mathematical physics: it needs to engage with (and perhaps even develop new aspects of) a mathematical machinery that is non-standard in both physics and philosophy. In light of these considerations, some may be moved to more seriously entertain a seemingly naïve ontology that breaks with orthodoxy and posits states that stand in one-to-one correspondence to Hilbert space vectors.
Although the project of developing a state-space fundamentalist ontology for projective Hilbert space is beyond the scope of this paper, the view I propose strikes a compromise between this ambitious project and the unorthodox view. According to the ontology I call ray substantivalism, there is a fundamental space that has the structure of linear Hilbert space. But physical states are identified not with points in this space but rather with certain mereological constructions out of these points—entities that are in one-to-one correspondence with rays. To develop this view, I will first give an explicit statement of the more simple-minded ontology, a view I call vector substantivalism. This will serve as the basis for the subsequent articulation of ray substantivalism.
It is worth stressing right at the start that vector and ray substantivalism are importantly different from proposals that go by the label of wavefunction realism. According to these proposals, the main role of Hilbert space is to characterize the physical fields on some underlying fundamental space, fields that are represented by the wavefunction. Within the Hilbert space framework, the wavefunction of an abstract Hilbert space ‘ket’ \(|\varphi \rangle\) is defined as \(\varphi (q_1,...,q_N) = \langle q_1,...,q_N|\varphi \rangle\), where \(|q_1,...,q_N\rangle\) is a simultaneous eigenvector of the position operators and N is the number of particles.Footnote 16 Vector and ray substantivalists take a different view of the role of Hilbert space: instead of being a vehicle for characterizing physical fields on some underlying fundamental space, Hilbert space corresponds to a fundamental space in its own right—the space whose points are in one-to-one correspondence with abstract Hilbert space vectors.
Vector substantivalism
With those clarifications out of the way, we can move on to the first step: a precise statement of vector substantivalism. I will proceed in analogy to the symmetry-fundamentalist ontology for classical particle mechanics developed in (Schroeren 2020).
The classical account has two main elements: first, the specification of a fundamental space with the structure of a classical phase space; and second, the specification of a sparse collection of fundamental properties and relations such that every relevant materialistic proposition (i.e. every proposition about the positions and momenta of classical particles) can be defined in terms of those fundamental properties and relations, together with the structure of the fundamental space. The key elements in this collection are two dyadic relations that correspond to the classical state-space symmetries of shifting the positions and momenta of individual particles:
Classical position-almost-sameness. A dyadic relation. Relates two points iff they agree about the momenta of all particles and about the positions of all but one particle.
Classical momentum-almost-sameness. A dyadic relation. Relates two points iff they agree about the positions of all particles and about the momenta of all but one particle.
The scheme by which every materialistic proposition can be defined in terms of these primitives exploits the mathematical correspondence, within Hamiltonian mechanics, between position- and momentum phase-space coordinate functions of individual particles on the one hand and the generators of momentum- and position-shift phase-space symmetries on the other. The specifics of this scheme won’t matter in what follows;Footnote 17 for present purposes, I merely want to point out that the central ontological primitives I’ll introduce in this paper are quite similar to classical position- and momentum-almost sameness. Moreover, the scheme by which the primitives of vector and ray substantivalism serve to define every relevant materialistic proposition exploits an analogous mathematical correspondence between the position- and momentum-variables of quantum systems (usually encoded in terms of position- and momentum-operators) on the one hand and unitary momentum- and position-shift symmetries on the other.
Just like its classical counterpart, vector substantivalism has two main elements: first, a fundamental space with the structure of state space; and second, a sparse collection of primitives such that every relevant materialistic proposition can be defined in terms of them, together with the structure of the fundamental space.
To begin with, the vector substantivalist posits a range of Points—first-order individuals that stand in one-to-one correspondence with Hilbert-space vectors—as well as instants of time. Points are conceived of as quantum states; that is, they are the sorts of entities that can be ‘actualized’ at various instants of time. This is captured by the following posit:
Point actuality. A dyadic relation. Relates a Point x and an instant of time t iff x is actualized at t.
For Points to behave like states, vector substantivalism includes the following basic law: necessarily, for every instant of time there is exactly one Point that is actualized at it.Footnote 18,Footnote 19
Over and above Points, times, and Point actuality, the vector substantivalist requires primitives that confer on Points the structure of a Hilbert space. A nominalistic (or intrinsic) account of this structure is beyond the scope of this paper.Footnote 20 Here I will assume that, as far as state-space structure on Points is concerned, we can help ourselves to ontological primitives that take complex numbers as arguments. Together with standard Hilbert space axioms, the following primitives confer Hilbert space structure on states:
1. Sum. A triadic relation. Relates Points x, y, and z iff x is the sum of y and z. Write \(x=y+z\).
2. Multiplication. A triadic relation. Relates Points x and y and a complex number c iff x is a multiple by c of y. Write \(x=cy\).
3. Inner Product. A triadic relation. Relates Points x, y and a complex number c iff c is the inner product of x and y. Write \(c=(x,y)\).
The main challenge for the vector substantivalist is to show that every relevant materialistic proposition can be defined by or expressed in terms of some fundamental proposition about Points. The fundamental properties and relations that capture Hilbert space structure among Points are insufficient for this task, since any given pattern in these primitives is compatible with mutually exclusive materialistic propositions. For example, a Hilbert space of appropriate dimensionality can be used both to describe a world that contains physical systems with spin and a world that contains no such systems.Footnote 21 To define every relevant materialistic proposition in fundamental terms, the vector substantivalist therefore requires further ontological primitives. This is the second main element of vector substantivalism.
I will introduce these primitives in terms of their mathematical characterizations; but it is important to emphasize that these characterizations are not metaphysical definitions. It should go without saying that ontological primitives cannot be defined in more fundamental terms, let alone in terms of their mathematical description. As is common in metaphysical theorizing, novel fundamental posits must be justified by their role in an account that highlights their explanatory and inferential connections to more familiar notions. This is precisely my goal here: it will shortly become clear that our primitives figure in an account according to which every relevant proposition about the physical counterparts of wavefunctions is definable by some proposition about these primitives.
To begin with, let me revisit a remark I made earlier: that Hermitian operators on Hilbert space are used to encode the physical structure on quantum states. This idea is most transparent from the wavefunction realist perspective. We already saw how the wavefunction is defined in terms of the eigenvectors of the position operators: the wavefunction of any ket \(|\varphi \rangle\) is \(\varphi (q_1,...,q_N) = \langle q_1,...,q_N|\varphi \rangle\), where \(|q_1,...,q_N\rangle\) is a simultaneous eigenvector of the position operators. This procedure doesn’t just define some particular wavefunction; it defines all possible wavefunctions.Footnote 22 In other words, it yields a range of wavefunctions that characterize the possible instantaneous arrangements of the physical field over the fundamental space, arrangements that possibly differ in the property that the field assigns to a given location in this space. In this sense, Hermitian operators designated as ‘position operators’ encode the physical structure of the fundamental field posited by the wavefunction realist.
In the vector substantivalist setting, the thought is quite similar. Every Point can be mathematically characterized in terms of a linear combination of eigenvectors of the relevant Hermitian operators. This can be illustrated at the level of mathematical description: a ket \(|\varphi \rangle\) can be specified in terms of position eigenvectors as
$$\begin{aligned} |\varphi \rangle = \int _{-\infty }^{\infty }|q_1,...,q_N\rangle \langle q_1,...,q_N|\varphi \rangle dq_1...dq_N. \end{aligned}$$
(1)
Every difference between distinct kets therefore corresponds to a difference in some complex coefficients that figure in the expansions of those kets in terms of the simultaneous position eigenvectors. This illustrates how Hermitian operators encode physical relations among distinct Points: respects in which Points possibly differ from each other.
According to vector substantivalism, these physical relations correspond to state-space symmetries. To substantiate this thesis, we exploit a mathematical duality, known as Stone’s theorem, between state-space symmetries and Hermitian operators on Hilbert space.Footnote 23,Footnote 24 For ease of exposition, let me introduce some terminology. Say that each of the N real numbers that figure in simultaneous position eigenvectors such as \(|q_1,...,q_N\rangle\) are values of the position variables. Correspondingly, for any \(|q_1,q_2,...,q_N\rangle\) and \(|q'_1,q_2,...,q_N\rangle\) such that \(q_1\ne q_1'\), I will say that the two vectors differ in the first position variable. This extends to arbitrary linear combinations of eigenvectors. For example, the following vectors differ only in the first position variable:
$$\begin{aligned} |\varphi \rangle&= \int _{-\infty }^{\infty }|q_1,...,q_N\rangle \langle q_1,...,q_N|\varphi \rangle dq_1...dq_N \end{aligned}$$
(2)
$$\begin{aligned} |\phi \rangle&= \int _{-\infty }^{\infty }|q_1+a,...,q_N\rangle \langle q_1,...,q_N|\varphi \rangle dq_1...dq_N. \end{aligned}$$
(3)
Stone’s theorem says that there is a one-to-one correspondence between one-parameter unitary transformations and Hermitian operators. The mathematical details do not matter for our purposes; what’s important is that, if \({\hat{p}}\) is a momentum operator then
$$\begin{aligned} U(a) = e^{ia{\hat{p}}} \end{aligned}$$
(4)
is the unitary operator that implements a shift by amount a in the corresponding position variable. Similarly, if \({\hat{q}}\) is a position operator then
$$\begin{aligned} V(k) = e^{ik{\hat{q}}} \end{aligned}$$
(5)
is the unitary operator that implements a shift by amount k in the corresponding momentum variable. For example, the unitary operator \(U_1(a)\) that acts on the simultaneous position eigenvector \(|q_1,q_2,...,q_N\rangle\) as \(U_1(a)|q_1,q_2,...,q_N\rangle = |q_1+a,q_2,...,q_N\rangle\) implements a shift in the first position variable. The action of shift operators on eigenvectors determines their action on arbitrary linear combinations of eigenvectors. For example, it is immediate from equations (2) and (3) that
$$\begin{aligned} |\phi \rangle = U_1(a)|\varphi \rangle . \end{aligned}$$
(6)
This suggests two relational primitives for vector substantivalism that are quite similar to the primitives of classical symmetry fundamentalism reviewed earlier. The idea is that we posit one fundamental relation that obtains between two Points when they are related by a shift in some position variable and one fundamental relation that obtains between two Points when they are related by a shift in some momentum variable:
4. Position-almost-sameness. A dyadic relation. Relates two Points just in case they agree about all position variables except for a shift in one position variable.
5. Momentum-almost-sameness. A dyadic relation. Relates two Points just in case they agree about all momentum variables except for a shift in one momentum variable.
I will now show that, for systems without spin, these two relations (together with the relations that confer Hilbert space structure on Points) are sufficient for specifying a range of fundamental propositions such that, for every proposition about the ontological counterparts of wavefunctions, one of these fundamental propositions defines it.
My strategy takes inspiration from the above-mentioned mathematical definition of wavefunctions in terms of kets: \(\phi (q_1,...,q_N)= \langle q_1,...,q_N|\phi \rangle\). The main idea is that, for any fixed ket, this expression can be thought of as capturing a relation between simultaneous eigenvectors and the complex numbers. At the ontological level, the corresponding idea is that the physical counterpart of the wavefunction is some specific pattern in the inner product relation. To implement this idea, we need to define two pieces of derivative ontology: first, a property instantiated by a Point just in case its mathematical representative is a simultaneous eigenvector; second, physical entities whose surrogates are points of configuration space and points of momentum space, respectively.
I start with the ontological counterpart of a simultaneous eigenvector. Let a position-almost-same subspace be a space of Points such that any two Points in it are position-almost-same; and let a momentum-almost-same-subspace be a space such that any two Points in it are momentum-almost-same. Points in a given position- or momentum-almost-same subspace differ about the same position or momentum variables. Let y, z be Points that are position-almost same to some Point x, and suppose that the differences between x, y and the differences between x, z concern different position variables. It follows that y, z differ about more than one position variable and so at most one of y and z can be in the same position-almost-same subspace as x. (Similarly for momentum-almost-same subspaces.)
Next, we observe that, if \(|p\rangle\) is an eigenvector of a momentum operator \({\hat{p}}\) then it is also an eigenvector of the unitary operator U(a) that implements shifts in the corresponding position variable:
$$\begin{aligned} U(a)|p\rangle = e^{iap}|p\rangle \end{aligned}$$
(7)
A similar fact holds for position eigenvectors: if \(|q\rangle\) is an eigenvector of a position operator \({\hat{q}}\) then it is also an eigenvector of the unitary operator V(k) that implements shifts in the corresponding momentum variable:
$$\begin{aligned} V(k)|q\rangle = e^{ikq}|q\rangle \end{aligned}$$
(8)
In other words: kets that are eigenvectors of some momentum operator are such that the corresponding spatial shift operator maps them to complex scalar multiples of themselves; and momentum eigenvectors are the only kets with this property. Similarly, kets that are eigenvectors of some position operator are such that the corresponding momentum shift operator maps them to complex scalar multiples of themselves; and position eigenvectors are the only kets with this property. We can use this fact to define the following physical properties that correspond to the notion of an eigenvector:
A. Position-eigenpoint. A monadic property. Instantiated by a Point x iff for some momentum-almost-same subspace containing x, every Point in that subspace is a multiple of x.
B. Momentum-eigenpoint. A monadic property. Instantiated by a Point x iff for some position-almost-same subspace containing x, every Point in that subspace is a multiple of x.
The properties of being a position- or momentum-eigenpoint are instantiated by Points whose mathematical surrogates are eigenvectors of at least one position or momentum operator: for example,
$$\begin{aligned} |q_1,\varphi _2,...,\varphi _N\rangle = \int _{-\infty }^{\infty }|q_1,q_2,...,q_N\rangle \varphi _2(q_2)...\varphi _N(q_N)dq_2...dq_N \end{aligned}$$
(9)
is an eigenvector in the first position variable but not necessarily an eigenvector in other position variables. But to define the intrinsic counterparts of position and momentum wavefunctions, we also need properties instantiated by Points when their mathematical surrogates are simultaneous eigenvectors of all position or all momentum operators. These can be defined as follows:
C. Simultaneous position-eigenpoint. A monadic property. Instantiated by a Point x iff every Point momentum-almost-same to x is a multiple of x.
D. Simultaneous momentum-eigenpoint. A monadic property. Instantiated by a Point x iff every Point position-almost-same to x is a multiple of x.
The second task is to define entities whose mathematical surrogates are points of configuration space and points of momentum space, respectively. It may be tempting to think that simultaneous position- and momentum-eigenpoints are good candidates for this role. This temptation should be resisted: there are too many simultaneous position and momentum eigenpoints. This is most transparent at the level of mathematical description. Given some fixed coordinate system on configuration space, there is a one-to-one correspondence between points of configuration space and the eigenvalues of the simultaneous position-eigenvectors of position operators along the axes of this coordinate system. But there are many more distinct simultaneous position eigenvectors than there are distinct position eigenvalues: for any complex number c, \(|q_1,q_2,...,q_N\rangle\) and \(c|q_1,q_2,...,q_N\rangle\) are not just both simultaneous position eigenvectors; they are eigenvectors with identical position eigenvalues. This problem can be sidestepped using tools from mereology:
E. Configuration space point. A monadic property. Instantiated by q iff q is a fusion of Points and every atomic part of q is a simultaneous position eigenpoint and any two atomic parts of q are related by a complex scalar multiple.
F. Momentum space point. A monadic property. Instantiated by p iff p is a fusion of Points and every atomic part of p is a simultaneous momentum eigenpoint and any two atomic parts of p are related by a complex scalar multiple.
This gives the right result: for every position/momentum eigenvalue, there is a unique configuration/momentum space point: the point such that any two of its atomic parts have this eigenvalue.
We are almost ready to define the ontological counterparts of wavefunctions. The last observation we need to make is that wavefunctions are defined in terms of normalized simultaneous eigenvectors, rather than in terms of their complex scalar multiples. For example, we define the wavefunction of \(|\phi \rangle\) in terms of vectors like \(|q_1,q_2,...,q_N\rangle\) rather than in terms of \(c|q_1,q_2,...,q_N\rangle\) where \(c\ne 1\).
It is important that we take note of this fact. Recall: our strategy is to identify wavefunctions with certain specific patterns in the inner product relation. But whereas wavefunctions assign a complex number to every point of configuration or momentum space, the inner product takes Points as arguments rather than the fusions of Points. This means that, for any given Point x and configuration space point q, the atomic parts of q differ in the complex number assigned to them by the inner product. This raises the question as to which complex number the putative ontological counterpart of a wavefunction should assign to q. The answer is suggested by the mathematical observation in the previous paragraph: it is the number that bears the inner product relation to every normalized atomic part of q.
More precisely, let a Point x be normalized iff there is some complex number c of unit modulus such that \(c=(x,x)\). The ontological counterparts of wavefunctions are then defined as follows:
G. Configuration-space field of x. A dyadic relation. Relates a complex number c and a configuration space point q just in case, for every normalized atomic part y of q, \(c=(y,x)\).
H. Momentum-space field of x. A dyadic relation. Relates a complex number c and a momentum space point p just in case, for every normalized atomic part y of p, \(c=(y,x)\).
According to vector substantivalism, the non-fundamental proposition we express when we specify the wavefunction of some Point x is the proposition that details the pattern in the configuration-space field of x: the proposition that specifies, for every configuration space point q, the complex number that is the value of the configuration-space field of x at q. (Similarly for momentum.) Since the configuration-space field of x is defined in terms of the primitives (1-5), it follows that every non-fundamental proposition about the ontological counterparts of wavefunctions statable in the Hilbert space framework can be defined by (or expressed in terms of) some fundamental proposition about relations whose mathematical surrogates are state-space symmetries. Vector substantivalism is therefore an instance of symmetry fundamentalism.
Ray substantivalism
Vector substantivalism entails that Points are quantum states and is therefore at odds with the orthodox view that quantum states correspond to Hilbert space rays rather than to Hilbert space vectors. In this section, I sketch ray substantivalism, an ontology that respects this consensus. This view is not an attempt at the more ambitious project of fundamentalism about projective Hilbert space. Instead, it is the view that quantum states are identified with certain mereological fusions of Points.
Ray substantivalism is a fairly straightforward modification of vector substantivalism. The fundamental first-order posits are identical: they consist in Points and instants of time. Moreover, the ray substantivalist follows the vector substantivalist in positing Hilbert space structure (1–3) among Points as well as the relations of position- and momentum-almost-sameness (4,5). The central difference between the two views is the identification of the sorts of entities that can be actualized at various instants of time; that is, the entities that are identified with quantum states. Let a fusion s of Points be a State iff for any two atomic parts x, y of s, x is a multiple of y. According to the ray substantivalist, actuality is a fundamental relation between instants of time and States:
State actuality. A dyadic relation. Relates a State s and an instant of time t iff s is actualized at t.
In addition, the ray-substantivalist version of the basic law governing the actuality relation says that necessarily, for every instant of time there is exactly one State that is actualized at it.
The only other feature of vector substantivalism that the ray substantivalist needs to modify is the definition of configuration- and momentum-space fields. Definitions G and H associate a field with every Point. But for the ray substantivalist, this is too fine-grained: the physical entity that corresponds to wavefunctions should be such that there is one such entity for every State rather than one for every Point. The idea behind the following modified definitions is that the ray-substantivalist field of a State s is the vector-substantivalist field of the normalized atomic part of s.
G′. Normalized configuration-space field of s. A dyadic relation. Relates a complex number c and a point of configuration space q just in case c is assigned to q by the configuration-space field of the normalized atomic part of s.
H′. Normalized momentum-space field of s. A dyadic relation. Relates a complex number c and a point of momentum space p just in case c is assigned to p by the momentum-space field of the normalized atomic part of s.
This completes the outline of ray substantivalism. According to this view, at every instant of time, the propositions about the normalized configuration- and momentum-space fields statable in the Hilbert space framework are entailed by which State is actual at that instant, together with the proposition that details the pattern of properties of and relations among the atomic parts of the actualized State and other Points. Ray substantivalism agrees with vector substantivalism about the fundamental first-order posits, as well as about the fundamental structure on Points. But the two views disagree about the nature of quantum states as well as about the physical counterparts of wavefunctions.
Like vector substantivalism, ray substantivalism is an instance of symmetry fundamentalism: almost-sameness relations are fundamental relations that correspond to state-space symmetries; and every relevant proposition about the physical counterparts of wavefunctions is defined by some fundamental proposition about these relations.
Some readers might object that ray substantivalism does not seem like a genuine advance over vector substantivalism. Recall that the central motivation for the orthodox view (according to which quantum states correspond to rays rather than to vectors) is to avoid a problematic underdetermination implied by the competing view: since no experiment can distinguish between states of a quantum system that differ only by a global phase, the thesis that quantum states correspond to Hilbert space vectors implies that it is in-principle empirically inaccessible which quantum state is actual. However, it seems as if ray substantivalism achieves consistency with the standard view only at the cost of reproducing the underdetermination at the level of the atomic parts of States. On ray substantivalism, the actualized State contains a rich structure among its parts, a structure which seems in-principle empirically inaccessible. But this worry is misplaced. According to ray substantivalism, there is no in-principle experimental underdetermination of which State the world is in at any given instant of time. And given that the State of the world at some instant of time is known, the structure among its atomic parts can be inferred simply from the linear Hilbert space structure among Points. Ray substantivalism does not feature the sort of underdetermination that spells trouble for vector substantivalism.