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Loop Quantum Gravity and Non-Separability

May 3, 2014

[This post is taken from part of an early draft of Christopher G. Weaver’s paper entitled “Against Causal Reductionism”]

     David Lewis’s Humean supervenience thesis (HST) says that the world’s fundamental structure consists of the arrangement of qualitative, intrinsic, categorical, and natural properties of space-time points (or perhaps some other suitable replacement[1]) and the spatio-temporal relations of such points. All derivative structure globally supervenes on such fundamental structure. Furthermore, “Humean supervenience” writes Lewis, “is named in honor of the greater denier of necessary connections. It is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another.”[2] The HST therefore entails that the fundamental physical state of the world is separable.[3] However, if fundamental physics delivers to us an end-game fundamental physical theory that is non-separable, then the HST is false. Say that a fundamental physical theory is non-separable when,

… given two regions A and B, a complete specification of the states of A and B separately fails to fix the state of the combined system A + B. That is, there are additional facts—nonlocal facts, if we take A and B to be spatially separated—about the combined system, in addition to the facts about the two individual systems.[4]

     Many theoreticians have pointed out how the HST is untenable by reason of quantum physics.[5] The existence of entangled quantum states is an implication of every interpretation of quantum mechanics.[6] Entangled quantum states do not globally supervene on local matters of particular fact, “[t]hat is, the local properties of each particle separately do not determine the full quantum state and, specifically, do not determine how the evolutions of the particles are linked.”[7] In fact, the non-separability of quantum mechanics was one reason why Einstein believed the theory to be incomplete.[8] There are responses to this objection from quantum mechanics, but I will not pursue that specific debate here. What I will argue is that one of the leading theories of quantum gravity is non-separable (and not because of quantum entanglement).

Canonical Quantization of GTR

     The leading canonical quantum gravity model (CQG) is loop quantum gravity (LQG). Following Rickles (2008), I note that according to the CQG approaches, GTR provides the details about how the metric (which characterizes the evolution of space-time) evolves.[9] Normally on CQGs, space and time come apart, where the former evolves against the background of the latter. Such separation is obtained by the introduction of an approximate equivalence and a foliation:

(1) ℳ ≅ ℝ x s

where s is a 3D hypersurface that is compact, and where the foliation is:

(2) 𝔍t: s → (∑t  ⊂ ℳ)

     Every hypersurface ∑t amounts to a temporal instant, and the manifold then is an agglomeration of such instants understood as a one-parameter family. In the context of CQGs, there are a number of avenues from such an agglomeration to a bona fide manifold. The fact that there are such avenues is generated by the diffeomorphism gauge symmetry of GTR. The diffeomorphism constraint that is a vector field, the Hamiltonian constraint that is a scalar field, plus various gauge functions on the spatial manifold generates diffeomorphism gauge transformations[10] Moreover, these constraints and functions evolve space forward one space-like hypersurface at a time.[11] The entire theory remains generally covariant and so the laws hold for coordinate systems related by coordinate transformations that are both arbitrary and smooth.[12] CQGs, therefore, understand both the geometry of the manifold and the gravitational field in terms of the evolutions of various fields, which are defined over space-like hypersurfaces ∑i on an assumed foliation.

Loop Quantum Gravity

     Again, the leading and most popular CQG is loop quantum gravity.[13] Proponents of this approach maintain that GTR can be simplified, and that one can understand the theory in terms of gauge fields.[14] Quantum gauge fields can be understood in terms of loops. By analogy with electrodynamics, we can say that space-time geometry is encoded in electric fields of gravitational gauge fields. The loops appropriately related to such electric fields weave the very tapestry of space itself.[15] According to LQG then, the fundamental objects are networks of various interacting discrete loops.[16] Many proponents of LQG maintain that these fundamental networks are arrangements of spin networks.[17]

     Spin networks do an amazing amount of work for LQG. They not only provide one with the means to solve the Wheeler-de Witt equation (see Jacobson and Smolin (1988)), but arrangements of such networks give rise to both the geometry of space-time (Markopoulou (2004, p. 552)), and a fundamental orthonormal basis for the Hilbert space in LQG’s theory of gravity.[18] Furthermore, the role of spin networks in LQG recommends that LQG is non-separable. The causal structure of space-time is not determined by the categorical and local qualitative properties of space-time points and their spatio-temporal relations, nor by individual loops and spatial relations in which such loops stand. Let me explain.

     On one interpretation of LQG, spin networks are types of causal sets, and so LQG in the quantum cosmology context has some similarities with quantum causal history (QCH) approaches.[19] Thus, LQG implies that the causal structure of the cosmos is determined by partially ordered and locally finite (in terms of both volume and entropy) sets. Such sets are regarded as events which one associates with Hilbert spaces with finitely many dimensions. These pluralities identified as events are “regarded as representing the fundamental building blocks of the universe at the Planck scale.”[20] Notice that these building blocks are pluralities of loops.[21] Individual loops themselves do nothing to determine causal structure. Moreover, some loops are joined in such a way that they are not susceptible to separation even though they are in no way linked (e.g., Borromean rings).[22] The spatio-temporal relations of such loops do nothing to determine that self-same structure, for (again) spin networks of loops weave together space-time geometry itself.[23] What is more, even on non-causal set approaches to LQG the very dynamics and evolution of quantum gravitational systems on LQG involve shifts from spin networks to spin networks. On orthodox LQG (without causal sets) quantum states are sets “of ‘chunks’, or quanta of space, which are represented by the nodes of the spin network, connected by surfaces, which are represented by the links of the spin networks.”[24] Causal structure is therefore determined by interrelated systems of loops, not individual loops and their spatial-temporal relations.

     I conclude that on varying approaches to LQG, the fundamental entities of the theory determine spatio-temporal relations while failing to bear such relations. Such entities also constitute systems, the fundamental parts of which, are non-separable. Thus, if LQG is approximately correct with respect to what it has said so far about physics at the Planck scale, the universe is non-separable. LQG suggests that the HST is false.


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* Thanks to Dean Rickles and Aron C. Wall for comments on an earlier draft of this post. Any mistakes that remain are mine.

[1] The deliverances of physics determine whether or not the replacement is suitable.

[2] Lewis (1986a, p. ix) emphasis mine. John Hawthorne summarized the HST by stating that derivative facts supervene “on the global distribution of freely recombinable fundamental properties.” Hawthorne (2006, p. 245). Hawthorne does not endorse the HST.

[3] Lewis (1986b, p. 14); cf. the discussion in Maudlin (2007, p. 120) who characterized the separability of the view as follows, “[t]he complete physical state of the world is determined by (supervenes on) the intrinsic physical state of each space-time point (or each pointlike object) and the spatio-temporal relations between those points.” Maudlin (2007, p. 51).

[4] Wallace (2012, p. 293).

[5] See the discussions in Lewis (1986a, p. xi); Loewer (1996, pp. 103-105); and Maudlin (2007, pp. 61-64).

[6] Schrödinger (1935, p. 555) said that entanglement is the “…the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought”.

[7] Loewer (1996, p. 104).

[8] See Einstein (1948); cf. Brown who remarked, “…[Einstein’s] opposition to quantum theory was based on the fact that, if considered complete, the theory violates a principle of separability for spatially separated systems.” Brown (2005, p. 187)

[9] I am following Rickles’s discussion of CQGs. See Rickles (2008, pp. 323-327).

[10] Rickles (2008, p. 324); Sahlmann (2012, p. 189).

[11] In the present context, the Hamiltonian is a sum of the aforementioned constraints.

[12] I should add that if one says of space that it is 3+1 dimensional the theory breaks general covariance.

[13] See Rovelli (2011a), (2011b); Smolin (2001, pp. 125-145), (2002), (2004, pp. 501-509).

[14] The insight is Ashtekar’s (1986) who leaned on Sen (1981); cf. Smolin (2004, p. 501).

[15] “…the loops of the quantized electric field do not live anywhere in space. Instead, their configuration defines space.” Smolin (2004, p. 503).

[16] Some of these loops are knotted, meaning that “…it is impossible, by smooth motions within ordinary Euclidean 3-space, to deform the loop into an ordinary circle, where it is not permitted to pass stretches of the loop through each other…” Penrose (2005, p. 944).

[17] A spin network is a “graph, whose edges are labeled by integers, that tell us how many elementary quanta of electric flux are running through the edge. This translates into quanta of areas, when the edge pierces a surface” Smolin (2004, p. 504). Such networks constitute eigenstates of operators representing volume and area (Rickles (2005, p. 704)). The idea comes to us from Penrose (1971). Before the use of spin-networks theorists used multi-loop states. See Rovelli (2008, p. 28).

[18] There are theorems which establish each result. See Smolin (2004).

[19] Markopoulou and Smolin (1997) join “the loop representation formulation of the canonical theory [of gravity] to the causal set formulation of the path integral.” (ibid., p. 409). See also Hawkins, Markopoulou, and Sahlmann (2003, p. 3840); Rovelli (2008, p. 35); and Smolin (2005, p. 19).

[20] Hawkins, Markopoulou, and Sahlmann (2003, p. 3840).

[21] In fact, one should understand a spin network state in terms of “a sum of loop states.” Spin network states are quantum states (they are the very eigenstates of observables which help us get at volumes and areas via measurement) understood as pluralities of loop states. Quotations in this note come from Smolin (2005, p. 13).

[22] See Penrose (2005, p. 944).

[23] As Rovelli stated,

“[a] spin network state does not have position…a spin network state is not in space: it is space. It is not localized with respect to something else: something else (matter, particles, other fields) might be localized with respect to it. To ask ‘where is a spin network’ is like asking ‘where is a solution of the Einstein equations.’” (2004, pp. 20-21 emphasis in the original)

[24] Rovelli (2008, p. 38). Even if one wanted to rid LQG of the redundancy in spin networks (due to symmetry) by switching to spin-knots or s-knots (diffeomorphism equivalence classes of spin-networks), my case for the non-separability of LQG would only be strengthened (see Rovelli (2004, pp. 263-264); cf. Rickles (2005, p. 710)).

One Comment leave one →
  1. May 3, 2014 10:03 pm

    I’ve just learned that Graham Oppy’s “Humean Supervenience?” paper in Philosophical Studies (2000) 101: 77-105, raises some related concerns about the HST (see especially pages 91-94). Thanks to Graham Oppy for bringing the article to my attention. What is more, soon after reading Oppy’s paper, I realized that some emergent space-time cosmological models may serve as potential defeaters for the HST. If there really was a bona fide quantum era from which space-time emerged, or if space-time itself is in some sense non-fundamental, it would be difficult to maintain that there are fundamental spatio-temporal relations in the Humean subvenient base. Likewise, it would be difficult to maintain that space-time points are fundamental and a part of the relevant subvenient base.

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