A standard move in contemporary philosophy of spacetime is to model a spacetime theory as consisting of a smooth (i.e. infinitely differentiable), second countable, Hausdorff manifold on which are defined some tensor fields which encode spatial and temporal relations (in relativity theory, a Lorentzian metric tensor field) and some other tensor (or spinorial tensor) fields representing a matter distribution. The elements of the smooth manifold are typically treated as constituting the domain of discourse, call it M, of the theory; these elements, (commonly referred to as the ‘spacetime points’), are considered to be part of the fundamental, non-derived, non-emergent ontology of the theory.
We take such a structure as the starting point for our discussion, using it to define a notion of localisability in Sect. 2.1. However, as we show in Sect. 2.2, such localisability is undefinable below a certain distance in the noncommutative space proposed by Alain Connes. In this sense, such distances are unphysical, and with them point regions.
Meaning and definition
In this paper we are concerned with physical, hence metrical space; our conception of localisation is correspondingly metrical: localisation within some region of determinate size. Thus our arguments will be to the effect that nothing can be localised in regions smaller than a certain size in our quantum spaces: that there are no smaller regions. (Insofar as our interest is in the status of points, which are closed intervals, our definition of localisation invokes boundaries of objects.) Of course metrical notions of localisation are familiar from the mereology literature: for example,‘exact location’, according to which ‘entity x is exactly located at a region y if and only if it has the same shape and size as y’ (Gilmore 2018). Metrical localisation is to be contrasted with thinner topological conceptions, say of proper or improper set inclusion. But since the spaces, classical and quantum, which we consider are metrical—because they purport to represent physical space—a metrical notion is appropriate in this context.
Therefore, in one dimension, an entity is exactly localised within some finite interval of length, call it \(\delta \) (from which one can straightforwardly define the more useful notion of localisation to a finite area) iff the coordinate functions, \(x^i\) associated with the boundaries of that object satisfy the following constraint:
$$\begin{aligned} \delta =|x^i(p)-x^i(q)|\ge 0 \end{aligned}$$
(1)
where \(|\cdot |\) is a given norm in \({\mathbb {R}}^n\), \(\delta < \infty \) and \(p,q\in M\) are its boundary points.Call the claim that it is possible to localise a body to an arbitrarily small interval the localisability thesis.
The domain has metric structure, which is to say, enough structure to allow us to define a distance between any two of its elements. We begin by looking at how the story about this structure might be told in a mathematical textbook. We can define a property like location in the manner above because a coordinate function can be stipulated to be an isometry from the domain, M, to \({\mathbb {R}}\). If, in addition, we do not wish to privilege a particular position, and instead care only about distances between pairs of points (e.g. boundary points of objects), we need to associate with M an entire equivalence class of coordinate functions, each of which agrees on the length between end points.
In this way, our preferred coordinatisations can be thought of as inducing metric structure on M, inherited from the primitive metric structure of \({\mathbb {R}}\). And something similar is true of topological, smooth, linear or any other form of geometric or algebraic structure—more exotic structures can be imposed on M by choosing different mathematical spaces as target of coordinate functions. Let us call such a space a structured domain, and denote it as \(M_s\). This induced metric structure on the structured domain allows us talk of separations between elements of \(M_s\) in terms of separations of the images in \({\mathbb {R}}\) of those elements under the coordinate functions.
On this set up, the space of coordinate functions encodes certain facts about some of the structure that our theory recognises. For example, the fact that we associated M with an equivalence class of coordinate functions, each element of which disagrees over the precise coordinate value to which a particular element of M is mapped, means that we cannot identify absolute positions in M.Footnote 3 To put this another way, the labels of absolute positions are not invariant under the automorphisms of the space of privileged coordinate functions. Here, automorphisms of the structured domain should be understood as bijections from the space of coordinates to itself that preserve the space of privileged coordinate functions.
Adapting terminology from model-theory, we refer to this condition as undefinability (cf. Enderton 2001), and where definability is characterised thus:
A piece of mathematical structure is definable (relative to a structured domain) if and only if it is invariant under the automorphisms of that structured domain,
As is demonstrated in the example below, for a putatively spatial or spatiotemporal theory (i.e. a theory in which there are no ‘internal’ degrees of freedom) we can always characterise a structured domain by restricting the class of privileged coordinate systems.
Consider a structured domain \(M_s\) of uncountable cardinality whose automorphisms characterise it as a topological manifold (i.e. whose coordinate functions are stipulated to be homeomorphisms into \({\mathbb {R}}\)). Let us structure this domain further by defining a set of maps, \(g:N\rightarrow {\mathbb {Z}}\) from a countable proper subset of \(M_s\) to the integers, \({\mathbb {Z}}\). Let us stipulate that these coordinate functions are isometries, thus imposing on N a metric structure. Denote the smallest distance between any two points in N as \(\mu \).
Now consider a topology-preserving map from M to itself (i.e. a homeomorphism) which, in addition, preserves the distances between the elements of N as determined by the discrete coordinate function g. Our structured domain \(M_s\), is now characterised in terms of this more restricted class of coordinate systems.
Consider the following set of pairs of elements \(p,q\in M\), call it \(R_{\mu }\):\(\{\langle p,q\rangle | d(p,q)<\mu \}\). One characteristic feature of this set is that if one element of a pair that constitutes \(R_\mu \) is contained in N, then the other is not. So this set is not invariant under all automorphisms of \(M_s\): for example, a homeomorphism on M which is the identity on N unchanged, but permutes every other element of M. In this case, \(R_{\mu }\) is not invariant under the automorphisms of \(M_s\), so is not definable relative to \(M_s\). Here, we simply cannot model indeterminacy of location as mere underdetermination between models each of which specifies an arbitrarily precise location, given our understanding of exact localisation, because we cannot associate a real number \(\delta < \mu \) in accordance with Eq. (1).
Our restriction to a domain with this privileged discrete structure was contrived. But it was contrived in order to make an important point about representation and indeterminacy; the examples we discuss in the rest of the paper will be more physically motivated. Quantum indeterminacy in position of a particle on the de Broglie-Bohm interpretation, for example, can be unproblematically represented by an underdetermination between different models, where each model has associated with it a perfectly determinate notion of location, derived from the definability of arbitrarily short lengths (more precisely, a family of two-place relations on M corresponding to arbitrarily small separations). And this is perfectly permissible (indeed, encouraged) in a formalism which treats the structured domain as a metric space. In what follows, however, we will argue that a quantum mechanical approach to geometry mandates a noncommutative, discrete structure on the domain of discourse. As a result, separations below certain scales (again, more precisely, a family of 2-place relations on M corresponding to separations below a certain scale) will not be definable, so indeterminacy in position cannot be modelled by underdetermination between models with determinate notions of location.
The spectral approach
It appears, therefore, that we have a problem. It is all very well to say that a metric space can be characterised by assigning a real number to each pair of its elements, but how does one do this systematically for a domain M with uncountably many objects? We cannot simply list numbers associated with all pairs of elements in M. The standard move is to define a line element on M, \(ds^2=g_{ij}dx^i dx^j\), which represents, roughly, an infinitesimal displacement, and then integrate this line element along an arc that connects any two points in M in order to assign to that pair some real number. This ‘arc-connectedness’ is the basis of all differential geometry, and is an extremely powerful piece of mathematical technology.
Unfortunately, this move brings with it a problem: this definition of length requires that we can define arbitrarily short lengths, i.e. that the space is arc-connected. This won’t do–recall from the previous section that we had good reasons to believe that lengths above a certain minimal scale ought to be definable even if lengths were not definable below that scale. In other words, our desideratum is the ability to define some lengths in a space that is not arc-connected, so our definition of length cannot require that a space be arc-connected. How ought one proceed in light of this demand?
Alain Connes’ formalism of spectral triples promises to solve this problem. In order to assess the plausibility of this claim, we need to understand both the motivations and the mechanics behind this proposal. We begin by making precise the questions of interest to which the spectral triple formalism provides the answer:
Question 1: What is the minimal structure of the domain of discourse required to make sense of locations?
As discussed earlier, in physical space a natural answer to this question is ‘the structure of a metric space, \(\langle M, d\rangle \)’, where d is the geodesic distance between any two elements in the domain of discourse. Differential geometry demonstrates that arc-connectedness (together with a few other assumptions) is sufficient to characterise a metric space. Our aim in this section is to explicate a generalisable alternative characterisation, thus demonstrating that arc-connectedness is not necessary. In other words, we seek to answer in the affirmative the following question:
Question 2: Can we turn a domain of discourse of uncountable cardinality into a metric space even if it is not assumed to be arc-connected?
The goal, then, is to recover a metric space \(\langle M,d \rangle \) algebraically, from a starting point that does not assume any metric or topological structure on the structured domain. Connes (1995) proposes using spectral triples, \(\langle {\mathscr {A}}, {\mathscr {H}}, D\rangle \), where \({\mathscr {A}}\) is a particular kind of algebra over a field, known as a \(C^{\star }\)-algebraFootnote 4, \({\mathscr {H}}\) is a Hilbert space, and D is a particular self-adjoint operator over that Hilbert space, known as a Dirac operator. The idea is simple: define, in algebraic terms, the structures that we know and love from differential geometry as special cases of these triples, check that we can still do differential geometry after this step, and then modify and deform these structures in such a way as to generalise differential geometry to unfamiliar domains. In particular, this will allow us to ascribe to a structured domain enough structure to define all and only lengths above a certain scale, satisfying our earlier desideratum.
We split the task of recasting differential geometry in the language of spectral triples into two steps:
- Step 1::
-
Recover a topological manifold, M, from a spectral triple.
- Step 2::
-
Recover a geodesic distance function, d on M, from a spectral triple.
Step 1 is completely straightforward and it builds on mathematical work in the forties, mainly a well-known representation theorem due to Gelfand and Naimark (1943), which we describe briefly. Consider a Hausdorff topological space: for simplicity we may think of a manifold, M, for instance the (2-dimensional) plane or sphere. Defined on it are the scalar fields, continuous functions that assign a complex number to each point: the set of such functionsFootnote 5 is known as C(M). To understand the following, it is important to distinguish between such fields, which are functions over all space, from their values at a point: the former are complete ‘configurations’ of individual point-values.Footnote 6
Two scalar fields, \(\phi \) and \(\psi \in C(M)\), can be multiplied together in an obvious way to obtain a third, \(\chi \in C(M)\)—the value of \(\chi \) at any point p, is just the ordinary product of the values of \(\phi \) and \(\psi \) at p: \(\chi (p) = \phi (p)\cdot \psi (p)\). Such ‘pointwise’ multiplication of fields is in fact so obvious as to almost be invisible: how could there be an alternative? In Sect. 5.1, we shall see that there are alternative rules for multiplying fields, and indeed, they may even be more physical than pointwise multiplication.
Because ordinary multiplication is commutative, \(a\cdot b=b\cdot a\), so is pointwise multiplication for elements of C(M): \(\phi \cdot \psi =\psi \cdot \phi \).The algebra contains a great deal of information about the space on which the fields live. In fact, the algebra contains all the information that we typically take to characterize a topological space. Topology, understood as characterizing relations between points, can be reconstructed from purely algebraic data as maximal idealsFootnote 7, the neighborood of a point can be likewise inferred from the relations among ideals. Global characteristics are also encoded in the algebra; for example a closed compact space (such as a circle) is described by an algebra which contains a multiplicative identity element. By contrast open spaces such as the real line are described by algebras which lack such an element. In short, this representation theorem states the logical equivalence of a space topology and its algebra C(M).
It’s worth emphasizing the strength of this point, by reflecting on what is meant by an ‘algebra’: nothing but a pattern of relations—a structure—with respect to some abstract operations. One might, for instance fully characterize an algebra by saying that there are two elements, \(\{a,b\}\), and an operation \(\circ \), such that \(a\circ a=b\circ b=b\) and \(a\circ b=b\circ a=a\) (and specifying that the operation is associative). What the elements are is not relevant, neither is the meaning of \(\circ \); all that matters is how many elements, and what function on pairs of elements \(\circ \) is. Of course, an algebra can have different concrete representations: concretely, a might be represented by the set of true propositions, and b by the set of false propositions, in which case \(\circ \) is represented by the boolean not-biconditional connective. But there are other representations: addition mod-2 for instance (and perhaps a could be represented by the presence of a 30 kg hemisphere of uranium 235, b by the absence, and \(\circ \) by the operation of putting together—the critical mass of \(\hbox {U}^{235}\) is 52 kg! ). These are different representations of a single algebra, which captures their common structure.
It is not relevant that the concrete elements of the algebra are fields over the manifold, all that need be specified are their relations with respect to a binary operation. However, the scalar fields on a particular manifold define a specific algebra, and, according to the representation theorem, no other manifold has scalar fields with the same algebra. The point is that the algebra does all the work: there is nothing smuggled in about the manifold simply because we realize the algebra with fields over it. Additionally, every abstract commutative \(C^\star \)-algebra can be represented as an algebra of continuous functions C(M) over some Hausdorff space M. So that settles our choice of algebra in the spectral triple: let \({\mathscr {A}}\) be C(M), the algebra of continuous complex-valued functions over M. In general, \({\mathscr {A}}\) determines a Hilbert space (the \({\mathscr {H}}\) in the spectral triple) as well via a procedure known as the GNS construction.Footnote 8
Step 2 is a little more involved, and it follows step 1 by nearly half a century, thanks to the work of Alain Connes and others. As we noted above, we cannot immediately recover, from the algebra of continuous functions, a metric manifold \(\langle M, d\rangle \) in the way that we could recover a compact topological manifold M. One might wonder, however, if there is some subalgebra of C(M) that encodes metric facts about M. And there is, but this algebra can only be picked out if we allow ourselves the third piece of structure in a spectral triple, the Dirac operator, D. If we stipulate that our domain of discourse, \(\langle M, d\rangle \), has enough structure for us to define a notion of spacetime spinors, roughly speaking, we can then define a differential operator, \(D:=i\gamma ^{\mu }\partial _{\mu }\) on the vector space of these spinors.Footnote 9 With this extra structure, we have enough machinery to isolate a subalgebra of \({\mathscr {A}}\) that will allow us to recover d, the geodesic distance on M.
Consider the subalgebra of C(M) known as the (algebra of) Lipschitz functions, defined as follows:
Given two metric spaces, \(\langle M, d_M\rangle \) and the real line \({\mathbb {R}}\), the function \(f:M\rightarrow {\mathbb {R}}\) is a real-valued Lipschitz function if and only if for all \(x_1,x_2 \in M\) there exists a real-valued constant, K such that:
$$\begin{aligned} |f(x_1)-f(x_2)|\le K d_M(x_1,x_2) \end{aligned}$$
(2)
Since Lipschitz functions can only be defined on metric spaces, the idea is that if we can find a subalgebra of \({\mathscr {A}}\) of Lipschitz functions, \({\mathscr {A}}_L\), we could use that algebra to reconstruct the geodesic distance on M. The problem thus splits into two parts:
- Part 1::
-
Identify the subalgebra of C(M) that constitutes the algebra of Lipschitz functions that we denote as \(C_L(M)\).
- Part 2::
-
Identify the appropriate Lipschitz function such that for every pair of points on M one can identify it with the geodesic distance between those two points using equation (2).
For part 1, we start by considering a bounded measurable function, \(a \in C(M)\), on M. From the GNS construction associated with this algebra, we know that it can be represented as an operator on a Hilbert space.Footnote 10 There is a theorem that states that this function will be almost everywhereFootnote 11 equal to a Lipschitz function if and only if the commutator [D, a] is bounded Connes (1995), where D is the Dirac operator defined on the Hilbert space \({\mathscr {H}}\). So if we specify a Dirac operator, we can identify the algebra of Lipschitz functions, \(C_L(M)\), a proper subalgebra of C(M).
For part 2, we see directly from the definition of a Lipschitz function, that, for the value \(K=1\), the supremum of the norm of the difference of image points is the geodesic distance. We are thus led to the following suggestion for the geodesic distance function on M:
$$\begin{aligned} d(p,q)=\mathrm {sup}\{|a(p)-a(q)|; a \in C(M),\Vert [D,a]\Vert \le 1\} \end{aligned}$$
(3)
where \(|\cdot |\) is the \(L^2\)-norm on \({\mathbb {C}}\) and \(\Vert \cdot \Vert \) is the norm on the Lipschitz algebra \(C_L(M)\). Note that we are defining a distance function on M indirectly—by appealing to structure in the algebra \(C_L(M)\). We need to establish that this, in fact, gives us the correct expression for the geodesic distance. The rigorous mathematical demonstration, detailed in Connes (1995, Ch. 6), requires the introduction of more technical machinery than we have introduced here. The upshot is that, with the help of some mathematical footwork, when the algebra is commutative, one can map each path between two points in the manifold to a norm in the Lipschitz algebra in such a way that the shortest path is mapped to the largest norm and the longest path the smallest norm. The geodesic distance in an arc connected space is then mapped to a supremum norm in the algebra. Now this link no longer exists when the algebra is noncommutative, but we can, nonetheless still speak of a geodesic distance expressed in terms of (only) the Lipschitz algebra norm.
The advantage of the use of spectral triples, in the context of the discussion in the previous section, is clear—we can, using equation ((3)), define a notion of distance between two elements of a domain of discourse even when that domain is not assumed to be arc-connected. We thus have the construction that we required in order to answer question 2 in the affirmative. Crucially, since we no longer need to assume M is arc-connected, we can generalise the algebra, from the commutative C(M) to a noncommutative \({\mathscr {A}}\). We do not need to worry that there is no longer a sensible notion of infinitesimal distance. All we need is a determinate specification of geodesic distance between elements of the subset of the domain of discourse for which the notion of separation is definable. This specification does not need to piggyback on a specification of infinitesimal distance, and is thus still available to us in spaces characterised by having a noncommutative algebra of functions.
Denote the convex subsets of a generic algebra \({\mathscr {A}}\) as \(S({\mathscr {A}})\); and the ‘extreme boundary’ of a convex set K, as \(\partial _E (K)\).Footnote 12 Elements of the extreme boundary of \(S({\mathscr {A}})\) are also known as pure states. It turns out that the space of pure states is also homeomorphic to M when \({\mathscr {A}}\) is commutative. So now we have two spaces, M and \(\partial _E S(C(M))\), arrived at by different constructions, that are homeomorphic to each other. Let us focus on one important pair, \(\partial _E S(C(M))\) and M. This isomorphism means that points in M stand in a one-one correspondence with pure states. Consider the appropriate expression for d on \(\partial _E S(C(M))\):Footnote 13
$$\begin{aligned} d_{\partial _E S(C(M))}(\alpha ,\beta )=\mathrm {sup}\{|\alpha (a)-\beta (a)|; a \in {\mathscr {A}},\Vert [D,a]\Vert \le 1\} \end{aligned}$$
(4)
All of this demonstrates that we should not be fooled into thinking that a geodesic distance can only be defined when M and \(\partial _E S({\mathscr {A}})\) are homeomorphic—once this isomorphism is broken by replacing a commutative algebra with a noncommutative algebra of functions on M, \(\partial _E S({\mathscr {A}})\) is still a metric space with a metric given by equation (3). But now, this metric space is no longer isometric to the space \(\langle M, d_M\rangle \) (unsurprisingly, given that they are no longer even homeomorphic). Consequently, and crucially, the pure states are no longer identified with points in M (the choice of terminology is not accidental—in quantum mechanics, these are the standard pure states that can be identified with rays in the system’s Hilbert space).
To reiterate, when \({\mathscr {A}}=C(M)\), which is commutative, the space of pure states, \(\partial _E S(C(M))\) is homeomorphic (and can be made isometric) to the space M, so it does not matter which space we begin with. This is no longer true when \({\mathscr {A}}\) is noncommutative. In this case, the geodesic distance function on \(\partial _E S({\mathscr {A}})\) still maps pairs of pure states to real numbers, but the space of pure states, \(\partial _E S({\mathscr {A}})\) is no longer homeomorphic to the domain of the functions that constitute \({\mathscr {A}}\). The function define in equation (4) now identifies distances between pure states which cannot, in general, be interpreted as points of space.
To relate this construction to the discussion in Sect. 2.1, let us consider the algebra of coordinate functions. The algebraic structure on this space privileges certain coordinate functions, and automorphisms of the algebra preserve this privileging, thus allowing us to structure the structured domain, \(M_s\). For certain noncommutative algebras (for example, the noncommutative algebra of coordinate functions that we discuss in the second half of this paper), the set of pure states is not a topological manifold, a fortiori cannot be interpreted as a set of points. This is what underlies the ‘pointlessness’ of NCG, as alluded to in several discussions of NCG, for example:
The concept of a point becomes evanescent, and in some cases one is forced to abandon it altogether. (Lizzi 2009, p. 95)
The conceptual shift in NCG is to treat the algebra \({\mathscr {A}}\) as fundamental, and the structure on the space M as derived (we will explore some of the metaphysical consequences of this move in Sect. 4). In regimes where we need only focus on commuting algebras of observables, distances between pure states can, to whatever the appropriate degree of approximation, be identified with distances between locations.
We are now in a position to understand how this construction allows us to determine under what circumstances the 2-place ‘spatial separation’ relation is definable. Define an automorphism, h of a spectral triple as an automorphism of \({\mathscr {A}}\) such that leaves the Dirac operator invariant. To reiterate, the spectral triple is playing the same role—structuring the structured domain—as the privileged coordinate systems were in the simple example in Sect. 2.1.
If \({\mathscr {A}}=C(M)\), then D picks out the same distance function before and after the transformation. In other words, h is an automorphism of \(\langle C(M), {\mathscr {H}}, D\rangle \) iff
$$\begin{aligned} d(p,q)=\mathrm {sup}\{|h(a(p))-h(a(q))|; a \in {\mathscr {A}},\Vert [h(D),h(a)]\Vert \le 1\} \end{aligned}$$
(5)
where d(p, q) is defined as in Eq. (3).
This d(p, q) is equal to the standard Riemannian distance defined directly on M:
$$\begin{aligned} d_M(p,q)=\mathrm {inf}\int _\gamma \sqrt{ g_{ij}dx^i dx^j} \end{aligned}$$
(6)
Consider a set of pairs of elements of \(\langle M,d_M\rangle \):
$$\begin{aligned} R_\mu := \{\langle p,q\rangle \in M|d_M(p,q)=\mu \} \ \end{aligned}$$
(7)
where \(d_M(p,q)=\mathrm {inf}\int _\gamma g_{ij}dx^i dx^j=\mu \) and \(\mu \in {\mathbb {R}}\). This relation is invariant under an automorphism h on \(\langle M,d_M\rangle \). This automorphism will induce an automorphism \(h_A\) on \(\langle C(M),{\mathscr {H}},D\rangle \) such that:
$$\begin{aligned} \mathrm {sup}\{|h_A(a(p))-h_A(a(q))|; a \in \partial _E S(C(M)),\Vert [h_A(D),h_A(a)]\Vert \le 1\}=\mu . \end{aligned}$$
(8)
We can now define a relation \(R_{\mu }^{C(M)}\) on \(\partial _E S(C(M))\):
$$\begin{aligned} R_{\mu }^{C(M)}:=\{\langle \alpha ,\beta \rangle \in \partial _E S(C(M))| d_{\partial _E S(C(M))}(\alpha ,\beta )=\mu \} \end{aligned}$$
(9)
This relation is invariant under all and only the automorphisms of \(\partial _E S(C(M))\) induced by automorphisms that preserve \(R_\mu \). So \(R_{\mu }\) and \(R_{\mu }^{C(M)}\) are equivalent, and it does not matter whether we use an algebraic or geometric description. They agree on the magnitudes that we are interested in here—lengths and areas, and consequently, as \(\mu \) is made arbitrarily small, both \(R_{\mu }^{C(M)}\) and \(R_\mu \) can be thought of as picking out the same relation even though they are defined on different sets. We propose that a necessary condition for the equivalence of a relation on a normed space (like \(\partial _E S(C(M))\)) to a relation of spatial separation on a manifold M is that there exist a homeomorphism between the two spaces (given our understanding of localisability, this fails to be a sufficient condition).
Of course, the point of characterising the same structure in two different ways is that the new characterisation still applies when we leave the classical regime and instead consider noncommutative algebras of observables. Let \(\theta \) quantify the magnitude of the noncommutativity of \({\mathscr {A}}\), i.e. \(\forall {\hat{x}},{\hat{y}} \in {\mathscr {A}}, [{\hat{x}},{\hat{y}}]= i\theta \). Since \(\theta \) is the commutator of distances, it is an area, which, for example, one could identify with the square of Planck’s length \(\sim 10^{-70}~\hbox {m}^2\).
Let \({\mathscr {A}}\) be noncommutative, and at the same time, let us structure M as a metric space, \(\langle M,d_M\rangle \). In this case, as mentioned above, M will no longer be isometric (or even homeomorphic) to the space of pure states, \( \partial _E S({\mathscr {A}})\). We can, of course, still define a relation \(R_\mu \) on \(\langle M,d_M\rangle \), because it remains invariant under automorphisms, h of \(\langle M,d_M\rangle \). But things change when we take the algebra \({\mathscr {A}}\) as fundamental. For commutative algebras, we could exploit the assumed homeomorphism between \(\partial _E S({\mathscr {A}})\) and M to speak of two ‘equivalent’ relations, one defined on \( \partial _E S({\mathscr {A}})\), the other on M.
Consider relation \(R^{{\mathscr {A}}}_{\mu }\), where \({\mathscr {A}}\) is noncommutative:
$$\begin{aligned} R^{{\mathscr {A}}}_{\mu }:= \{\phi ,\psi \in \partial _E S({\mathscr {A}})|d_{\partial _E S({\mathscr {A}})}(\phi ,\psi )=\mu \} \ \end{aligned}$$
(10)
where \( d_{\partial _E S({\mathscr {A}})}(\phi ,\psi )\) is given by equation (3).
While this relation is, indeed, invariant under automorphisms of the noncommutative spectral triple, \(\langle {\mathscr {A}}, {\mathscr {H}}, D \rangle \), therefore definable, it is no longer equivalent to the relation \(R_\mu \). This is because although the metric space of pure states \(\langle \partial _E S({\mathscr {A}}), d_{\partial _E S({\mathscr {A}})}\rangle \) is invariant under spectral triple automorphisms, it is no longer homeomorphic to \(\langle M,d_M\rangle \). Therefore it is no longer possible to assess whether the two relations are co-extensive: \(R_\mu \) and \(R^{{\mathscr {A}}}_{\mu }\) are incommensurable relations. This clashes with what we had identified as a necessary condition for \(R^{{\mathscr {A}}}_{\mu }\) to represent a spatial separation of magnitude \(\mu \): that \(\partial _E S({\mathscr {A}})\) is homeomorphic to M.
We can, however, restrict our attention either to commutative algebras, or to regimes in which the algebra of relevant observables can be treated as being commutative (i.e. the scale \(\mu \) is much larger than the noncommutation factor \(\theta \)). In these scenarios, \(R^{{\mathscr {A}}}_{\mu }\) can be seen to be equivalent to (or nearly equivalent to) \(R_\mu \). We can therefore have localization within sufficiently large regions, but not within regions below a certain scale. The upshot of this discussion is that \(R_\mu \), and hence our concept of localisation, is definable in a theory whose domain of discourse is a spectral triple only if the algebra is commutative. Since we can express all of the dynamically meaningful components of the noncommutative theory without making any reference to separations below the scale \(\mu \), on Occamist grounds, we excise these putative regions—including points—of spacetime from our ontology.