Abstract
Recent work on quantum gravity (QG) suggests that neither spacetime nor spatiotemporally located entites exist at a fundamental level. The loss of both brings with it the threat of empirical incoherence. A theory is empirically incoherent when the truth of that theory undermines the empirical justification for believing it. If neither spacetime nor spatiotemporally located entities exist as a part of a fundamental theory of QG, then such a theory seems to imply that there are no observables and so no way that the theory can be confirmed. The threat of empirical incoherence can be addressed by treating spacetime and spatiotemporally located entities as emergent. The question then arises as to what the metaphysical nature of this emergence might be. In this paper, I explore a functionalist approach to this kind of emergence in the context of loop quantum gravity. I begin by rehearsing the spacetime functionalist’s account of emergence, clarifying the view along the way. I proceed to sketch out a functionalist treatment of spatiotemporally located entities and combine the two forms of functionalism into a double functionalism, according to which both spacetime and matter have the same functional realisers.
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Notes
This notion of ‘local’ is very weak: something that is distributed across a region of spacetime that spans several galaxies is a local beable in this sense. Thus, ‘local’ means something like located rather than locality in the technical sense.
While Huggett and Wüthrich consider the problem of empirical incoherence as it arises for LQG, it is reasonable to suppose that what they have in mind is an extended theory—LQG+—that contains matter fields.
I am grateful to a referee for suggesting this line of thought.
I take the following distinction between the two types of emergence from Wüthrich (2019, p. 4).
Lam and Esfeld (2013, p. 291) call this ‘ontological emergence’, though this is not to be confused with the notion of ‘strong emergence’ that one sometimes finds in metaphysics of mind, which implies the failure of deducibility and thus would imply the failure of the first kind of emergence.
On one interpretation, Maudlin (2007) puts pressure on the need to provide some answer to this question.
The two kinds of emergence are no doubt connected. Exactly what the relationship between the two kinds of emergence might be is unclear, however. One possibility, and the option that I will assume in what follows, is to treat the second kind of emergence as the metaphysical corollary of the first kind of emergence. Thus the first kind of emergence tells us about the relationship between two different theories which have distinct ontologies. The second kind of emergence is the relation in the world that connects the two ontologies of the two different theories, and is thus the physical structure that the first kind of emergence represents.
It may be possible to develop a viable version of the building blocks approach, see fn. 16.
In philosophy, LQG is usually treated as a theory of QG in a more restricted sense than the one just described: as a quantum theory of the gravitational field that can be used to derive GR but not necessarily QFT. In this paper, I will focus on a more ambitious version of LQG that aims at QFT as well. This more ambitious version of the view appears to carry the hopes of at least some physicists working in this area.
Note that there appear to be a number of viable interpretations of LQG available (some of which seem more friendly to the fundamental existence of spacetime than others, see Norton (ms.)). Because I will work with a particular account of LQG, what I say may not hold generally for all versions of LQG. Still, this interpretation-bound discussion remains useful as a way of modelling a particular case in which spacetime goes missing. So even if the interpretation is not the ‘correct’ one, the modelling itself has merit as a basis for understanding emergence. The methodology here is thus broadly in line with the naturalistic methodology for metaphysics described by Paul (2012b).
I have framed the importance of these metaphysical questions in terms of the empirical coherence of a theory of QG. It is worth noting, however, that the questions may arise independently of concerns about empirical coherence. On one conception of what it is to provide a theory of QG, such a theory is simply a more fundamental theory that is applicable at high energy scales, from which both GR and QFT emerge as effective theories. Assuming that GR and QFT are correct at low energy scales, and that a theory of QG is correct at high energy scales and that all three theories have different ontologies, the question then arises as to what the metaphysical relationship might be between the entities posited by the various theories. If the fundamental ontology of the relevant theory of QG is not spatiotemporal, then we must explain how both spacetime and particles emerge from a non-spatiotemporal foundation. Thus, even if there is a way to show that a theory of QG is empirically coherent without saying much about the metaphysics of QFT, the relationship between QG and QFT forces us to do so eventually.
The networks are called ‘spin’ networks because of the way that each edge is labelled by a number that is, mathematically speaking, related to the spin-value from particle physics (spin-networks were originally used to represent the splitting and joining of particles, and thus the edges represented ‘spin states’ of particles, but the spin-networks of LQG should not be taken to represent particle world-lines in the same way).
The first interpretation corresponds to one of the ‘naïve’ interpretations outlined by Norton (ms.) in which there is a pre-spatiotemporal structure in which spin-networks are embedded. The second interpretation corresponds to the Rovellian interpretation that Norton (ms.) specifies.
Note that while it is relatively easy to see how to recover space from spin-networks—spin-networks do, after all, represent volumes of space—it is harder to see how spacetime might arise. This is partly because the dynamics of LQG are not as well understood.
See Ney (2015) for a criticism of a standard mereological approach in the context of wave function realist approaches to quantum mechanics. Although Ney is not considering gravity, some of her criticisms may carry over to the case of emergent spacetime in QG. See Le Bihan (2018a, b) for a more promising mereological approach to the emergence of spacetime in the context of quantum gravity. Le Bihan uses the mereological theory of Paul (2002, 2012a) as opposed to the more standard mereology that Ney criticises.
This assumes a particular approach to the classical limit of the theory, see Oriti (2014) for an alternative.
The equation below is the end-point of the analysis. The demonstration that spacetime is emergent in LQG is much more detailed, and involves a good deal more mathematics. However, since the details of this aspect of LQG have already been outlined elsewhere [see Wüthrich (2017)] I have omitted them to keep the discussion short.
Huggett and Wüthrich outline these notions of physical salience in response to a concern identified by Maudlin (2007). As a referee has pointed out to me, it is not obvious that the worry about physical salience that Huggett and Wüthrich attribute to Maudlin is in fact the worry that Maudlin raises. Maudlin’s concern is an under-determination worry. Local beables seem to be underdetermined by the mathematical structure that one can derive. Huggett and Wüthrich don’t seem to frame Maudlin’s worry in terms of underdetermination.
See Yates (forthcoming) for further discussion of these two versions of spacetime functionalism.
These are based on Norton’s (ms.) interpretations, but I don’t wish to attribute them directly to him, as he may disagree with the way I divide up the terrain. The distinction is drawn explicitly in Le Bihan (2018a) who provides an excellent discussion of the distinction between flat and stratified views.
I am grateful to a referee for pressing me on this point, and for suggesting this interpretation of spacetime emergence.
Bilson-Thompson’s model builds on the Rishon model of leptons, see Harari and Seiberg (1981).
Imagine taking a ribbon and ‘winding it’ by twisting the ribbon through a full rotation of the circle. A twist through \(+\pi \) is, roughly, a full rotation in one direction and a twist through \(-\pi \) is a full rotation in other direction. \(2\pi \) means the ribbon is twice wound.
Differently ordered twist patterns are not distinct helons because they yield the same total twist.
Note that to say a helon is composed of two twists does not mean that the ribbon has a link of 2. Rather, what this means is that the rotation of the twist can be characterised by a particular repeating function corresponding to the rotation. Roughly, the sign of the twist pair (\(\pm 2\pi \)) gives the orientation of the rotation of the ribbon (to the ‘left’ or to the ‘right’).
Note that the charge of unbraided triplets is quantized, and so the charge is always an integer charge.
The combinatorial options allow for the production of the red, green and blue quarks of QCD. Each particle can be expressed as a triplet, each triplet corresponds to a distinct braid. See Bilson-Thompson (2005, p. 2) for the full list of particles and anti-particles that can be constructed, and their corresponding charge and colour. See Bilson-Thompson et al. (2007, 2012) for more extensive models in which a larger range of particles can be constructed.
A permutation matrix is a permutation of the identity matrix of order n, where each permutation is, itself, an identity matrix. So, for instance, the identity matrix of order 2, \(I_2\), is given as:
$$\begin{aligned} \begin{bmatrix} 1&\quad 0 \\ 0&\quad 1\\ \end{bmatrix} \end{aligned}$$Yielding two possible permutations:
$$\begin{aligned} \begin{bmatrix} 1&\quad 0 \\ 0&\quad 1\\ \end{bmatrix},\begin{bmatrix} 0&\quad 1 \\ 1&\quad 0\\ \end{bmatrix} \end{aligned}$$When we model a braid as a permutation matrix, we treat the ‘1’ in the identity matrix as representing a helon. Note that these are not ordinary permutation matrices, since standard permutation matrices have only a single possible type of ‘1’. Since there are three different types of helon, the permutation matrices that Bilson-Thompson appeals to use three different ‘types’ of ‘1’. The point, however, is that permutation matrices model the right combinatorial options, and so it is the combinations that are yielded by these matrices that Bilson-Thompson is using, and not the fact that the resulting permutations have the same basic properties as an identity matrix (i.e., playing the role of a multiplicative identity).
This is to be differentiated from versions of substantivalism that take material entities to be grounded in spacetime regions, rather than identical to them. See Lehmkuhl (2016) for a discussion of this ‘priority’ conception of super-substantivalism.
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Funding was provided by The Australian Research Council (Grant No. DE180100414), Australian Research Council (Grant No. DP180100105).
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Baron, S. Empirical incoherence and double functionalism. Synthese 199 (Suppl 2), 413–439 (2021). https://doi.org/10.1007/s11229-019-02462-9
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DOI: https://doi.org/10.1007/s11229-019-02462-9