Abstract
Spacetime as we know and love it is lost in most approaches to quantum gravity. For many of these approaches, as inchoate and incomplete as they may be, one of the main challenges is to relate what they take to be the fundamental non-spatiotemporal structure of the world back to the classical spacetime of general relativity (GR). The present essay investigates how spacetime is lost and how it may be regained in one major approach to quantum gravity, loop quantum gravity.
I thank audiences at the University of London’s Institute of Philosophy and at the spacetime workshop at the Bergisch University of Wuppertal. I am also grateful to my fellow Young Guns of General Relativity, Erik Curiel, John Manchak, Chris Smeenk and Jim Weatherall for valuable discussions and comments, and to Francesca Vidotto for correspondence. Finally, I owe thanks to two perceptive referees for their comments. Work on this project has been supported in part by a Collaborative Research Fellowship by the American Council of Learned Societies, by a UC President’s Fellowship in the Humanities, and by the Division for Arts and Humanities at the University of California, San Diego. Some of the material in this essay has its origin in [47].
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Notes
- 1.
Given this formidable success of the classical theory, one might wonder why we need a quantum theory of gravity at all. There are good reasons to think that we do, but they do not fully align with the standard lore one finds in the physics literature ([49] §1).
- 2.
For a very recent critical view, see, e.g. [25].
- 3.
- 4.
A useful introduction to the Lagrangian and the Hamiltonian formulation of GR is given in ([46], Appendix E). Wald’s textbook of 1984 only deals with the ADM version of Hamiltonian GR and, as time travel was not yet invented in 1984, does not treat Ashtekar’s version, pioneered in 1986.
- 5.
Of course, for most cases we care about, Hamiltonian theories afford a corresponding equivalent Lagrangian theory, and vice versa. Currently, a debate rages in philosophy of physics over which of the two, if any, is more fundamental or more perspicuous. Nothing I say here should be taken to entail a stance in that debate.
- 6.
There are, of course, purely internal degrees of freedom of particles, such as classical spin, which admit of a Hamiltonian treatment without the system necessarily being extended in space. Now, even a point particle with internal degrees of freedom is at least a physical system in space, and it certainly also evolves over external time.
- 7.
- 8.
For more details on how the constraints arise in some Hamiltonian systems, see ([21], Ch. 1). My exposition largely follows this reference.
- 9.
They are not referred to as tertiary, quaternary etc. constraints, but only collectively as ‘secondary’ constraints.
- 10.
Cf. ([21], §1.2.1).
- 11.
Cf. ([21] §1.2.2).
- 12.
Cf. ([8] §10.2.2).
- 13.
Second-class constraints can be regarded as resulting from fixing the gauge of a ‘larger’ system with an additional gauge invariance. They can be replaced by a corresponding set of first-class constraints which capture the additional gauge invariance. Second-class constraints are thus eliminable. In fact, in some cases, it may prove advantageous to thus ‘enlarge’ a system as this permits the circumvention of some technical obstacles ([21] §1.4.3), albeit at the price of introducing new ‘unphysical’ degrees of freedom. Without loss of generality, we can thus consider a Hamiltonian system whose constraints are all first-class.
- 14.
This manner of counting the physical degrees of freedom is well defined for any finite number of degrees of freedom, and perhaps for countably many too. For uncountably many degrees of freedom, new subtleties arise. Cf. ([21] §1.4.2).
- 15.
- 16.
For an explanation of the failures of determinism in this setting, cf. ([47] §4.1), on which the past few pages have been based. Also, and at the peril of burying an absolutely central point in a footnote, this severance of space and time threatens the general covariance so central to GR. How general covariance gets implemented in Hamiltonian GR and the subtleties that arise in doing so are discussed in ([47] §4.4). What follows explicates the gist of this implementation.
- 17.
For a somewhat rigorous execution in the case of the so-called ADM and Ashtekar-Barbero versions of Hamiltonian GR, cf. [47], §4.2.1 and §4.2.2, respectively.
- 18.
A spacetime diffeomorphism is a one-to-one and onto \(C^\infty \)-map from \(\mathscr {M}\) onto itself which has a \(C^\infty \)-inverse. Diffeomorphisms induce transformations in the fields defined on the manifolds. Intuitively, a map between manifolds is active if it ‘moves around’ the points without recourse to any coordinate system. Thus, an active transformation is not a change in coordinate systems, but a transformation pushing around the physical fields on the manifold. But this metaphorical picture should be enjoyed with the adequate mathematical caution.
- 19.
This is the received view, but it should be noted that there has been recent dissent, e.g. in ([14] §3).
- 20.
For a detailed analysis and justification, cf. ([47] §3, particularly §3.2).
- 21.
For a discussion of philosophical reactions to this situation, cf. ([23] §2.3).
- 22.
- 23.
For a more systematic explication of global hyperbolicity and neighbouring concepts, see ([41] 593).
- 24.
A more detailed analysis of dynamics in LQG can be found in ([47] §5.3).
- 25.
For the technical background of this basis and its interpretation, cf. ([35] §2.3).
- 26.
More precisely, they are represented by labelled graphs embedded in some background space. Thus, they are not invariant under spatial diffeomorphisms, i.e., when they are ‘pushed around’ on the embedding manifold. In order to fully solve the diffeomorphism constraints, then, we need equivalence classes of spin network states under three-dimensional diffeomorphisms on the background manifold. Sometimes, these equivalence classes, represented by abstract labelled graphs, are called ‘s-knot states’ in the literature. So I am being slightly sloppy by using the locution ‘spin network states’ ambiguously.
- 27.
It should be kept in mind, however, that these operators are not Dirac observables and should therefore be taken with a grain of salt. They are partial observables in the sense of [33].
- 28.
Cf. e.g. [28].
- 29.
Cf. Figure 1 in [22].
- 30.
- 31.
This does not entail that the fundamental non-localities could not have observable consequences, such as those proposed by [32].
- 32.
For an up-to-date review on emergent properties, cf. [31].
- 33.
At least at the level of ordinary quantum mechanics; in relativistic quantum theories, matters become more subtle. Cf. ([38] §2.2).
- 34.
Strictly speaking, LQG basic variables are the holonomies and fluxes introduced in §2.1, which are not identical to the connection and the canonically conjugate electric field of the connection representation but are constructed from them.
- 35.
The remainder of this section draws on ([47] §9).
- 36.
Consider the n-body problem: while the phase space of states of an n-particle system in a physical space of m dimensions is topologically \(\mathbb {R}^{2mn}\) and therefore finite-dimensional in classical mechanics, the corresponding quantum space of states is the infinite-dimensional Hilbert space \(L^2(\mathbb {R}^{mn})\), the space of square-integrable functions on \(\mathbb {R}^{mn}\).
- 37.
No attempt shall be made to substantially consider the wider literature on the topic. Cf. [43] for an analysis of various proposals for reduction as an inter-theoretic relation, with a particular eye on the physical sciences.
- 38.
The clause “appropriately related quantities in the theory to be approximated” in Definition 6 above occludes substantive work that must be completed to achieve such “appropriate relation”. I am grateful to Erik Curiel for pushing me on this point—I most certainly deserve the pushing here.
- 39.
I wish to thank Jeremy Butterfield for suggesting this relaxation.
- 40.
Thanks to Erik Curiel for holding me to task here.
- 41.
For a more thorough discussion of Landsman’s argument, cf. ([47] §9.2.1).
- 42.
- 43.
- 44.
As ([44] §11) points out, there are deep connections between the various semi-classical programmes.
- 45.
For an intuitive introduction, see ([34] §6.7.1). The picture is that of the gravitational field like a (quantum cloud of) fabric(s) of weaves which appears to be smooth if seen from far but displays a discrete structure if examined more closely. Hence weave states.
- 46.
The ‘upper case’ spin network states \(|S\rangle \) live in \(\mathscr {K}^\star \), the pre-kinematical Hilbert space, i.e. the Hilbert space containing all spin network states which solve the Gauss constraints, but not necessarily the spatial diffeomorphism constraints. Thus, the spin network states in \(\mathscr {K}^\star \) are not represented by abstract graphs, as are those in the full kinematical Hilbert space \(\mathscr {H}_K\), but as embedded graphs on a background manifold. This choice is just conveniently following the established standard in the literature on weave states; we will see below in Footnote 47 that this poses no problem as everything can be directly carried over to the spatially diffeomorphically invariant level.
- 47.
The weave states as introduced above have merely been defined at the pre-kinematic level, i.e. they are not formulated in terms invariant under spatial diffeomorphisms (cf. also Footnote 46). The reason for this choice lies mostly in that this is the canonical choice in the literature, but also because in this way, the weave states can be directly related to three-metrics, rather than equivalence classes of three-metrics. This, however, does not constitute a problem whatsoever, as the characterization of weave states carries over into the context of diffeomorphically invariant spin network states in \(\mathscr {H}_K\), as follows. If we introduce a map \(P_{\text{ diff }}: \mathscr {K}^\star \rightarrow \mathscr {H}_K\) which projects states in \(\mathscr {K}^\star \) related by a spatial diffeomorphism unto the same element of \(\mathscr {H}_K\), then the state \(\mathscr {H}_K\ni |s\rangle = P_{\text{ diff }} |S\rangle \) is a weave state of the classical three-geometry \([q_{ab}]\), i.e., the equivalence class of three-metrics \(q_{ab}\) under spatial diffeomorphisms, just in case \(|S\rangle \) is a weave state of the classical three-metric \(q_{ab}\) as defined above.
- 48.
For details, cf. ([47] 181).
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Wüthrich, C. (2017). Raiders of the Lost Spacetime. In: Lehmkuhl, D., Schiemann, G., Scholz, E. (eds) Towards a Theory of Spacetime Theories. Einstein Studies, vol 13. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3210-8_11
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