The Disappearance and Re-Emergence of Space and Time in Quantum Theories of Gravity

The disappearance of space and time in QG Empirical incoherence and the emergence of spacetime

The disappearance and re-emergence of space and time in quantum theories of gravity

Christian Wüthrich (based on joint work with Nick Huggett)

University of California, San Diego

Black Forest Summer School: Physics and Philosophy of Time 26 July 2013

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Empirical incoherence and the emergence of spacetime Organization of lecture

1 The disappearance of space and time in QG Space and time in canonical QG Space and time in causal set theory

2 Empirical incoherence and the emergence of spacetime The threat of empirical incoherence The emergence of spacetime in causal sets

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Empirical incoherence and the emergence of spacetime The central problem ... particularly if one insists on a fundamental role for spacetime

Thesis Many approaches to quantum gravity (QG) suggest or imply that space and time do not exist at the most fundamental ontological level. Thus deprived of their former status as part of the fundamental furniture of the world, together, perhaps, with quarks and leptons, they merely ‘emerge’ from the deeper physics that does not rely on, or even permit, their (fundamental) existence, rather like tables and chairs.

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Empirical incoherence and the emergence of spacetime Time in physics

Time in physics does not have many of the features we would usually think of as essential for time: It is isomorphic to (an open interval of) R, or imposes a total order on events. It contains a phenomenologically special moment—the ‘now’. It has a dynamical quality—a ‘ﬂow’. It is asymmetric, i.e., it has a ‘direction’.

Thesis Physics has denied all of these.

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Empirical incoherence and the emergence of spacetime Physical space Henri Poincaré, Science and Hypothesis

In the ﬁrst place, what are the properties of space properly so called?... The following are some of the more essential:— 1st it is continuous; 2nd, it is inﬁnite; 3rd, it is of three dimensions; 4th, it is homogeneous—that is to say, all its points are identical one with another; 5th, it is isotropic. (1905, 52)

General relativity (GR) can’t be the last word: quantum effects must be taken into consideration in early universe and evolution of black holes ⇒ must ﬁnd theory that combines quantum effects and (strong) gravitational ﬁelds ⇒ need a quantum theory of gravity Source: http://blog.michaelkcooke.com/ various approaches, string theory and loop quantum gravity (LQG) most visible, causal set theory

Source: http://universe-review.ca/F15-particle.htm

Source: http://sciencereview.berkeley.edu/articleex.php?issue=18&article=features_04_quantum

starts out from the standard model of particle physics, a quantum theory sense in which gravity can be said to be ‘naturally’ included; details far from clear originally developed assuming a spacetime background (as inert container), but questions have been raised about it (dualities: physically equivalent string theories use different spacetimes) outstanding problems: ‘landscape problem’, repeated false Brian Giberson, “String Theory” predictions of ‘supersymmetric particles’

starts out fromGR, incorporates its core lessons (dynamical spacetime) tries to incorporate quantum effects by ‘quantizing’GR outstanding problems: (1) problem of time/ill-understood dynamics, (2) classical limit/emergence of spacetime

Source: http://www.cpt.univ-mrs.fr/∼rovelli/ from basic quantum structure

In string theory as well as in loop quantum gravity (LQG) (and other approaches to QG), indications are coalescing that space and time are no longer fundamental entities as classically conceived (substantivally or relationally), but merely ‘emergent’ phenomena that arise from the basic physics. In particular (as in LQG): no continuous spacetime, discrete structure In the language of physics: spacetime theories such as GR as ‘effective’ and spacetime itself ‘emergent’, much like thermodynamics is an effective theory and temperature is emergent, as it is built up from the collective behaviour of gas molecules. Unlike the fact that temperature is emergent, the idea that the universe is not in space and time shocks our very idea of physical existence as profoundly as any scientiﬁc revolution.

philosophers have only just started to study the implications of QG and the disappearance of spacetime Funded project with Nick Huggett (University of Illinois, Chicago): ‘Emergent Spacetime in Quantum Theories of Gravity’ joint framework, Huggett studies string theory, Wüthrich does LQG, causal set theory The remainder of this talk focuses on my contribution to this project.

LQG attempts to transpose the central lesson of GR into a quantum theory central innovation of GR: spacetime isn’t ﬁxed ‘background’, which determined the inertial forces, but a dynamical structure which interacts with matter LQG is based on a reformulation of GR as a so-called ‘Hamiltonian system’, which re-interprets spacetimes as (3 + 1)-dimensional (instead of 4-dim), with constraints ⇒ forces a ‘foliation’ of spacetime by an equivalence relation in 3-dim ‘spatial’ hypersurfaces, parametrized by 1-dim ‘time’ 3-dim ‘space’ considered as dynamical system which evolves over ‘time’; 3-dim hypersurfaces represent (instantaneous) states of dynamical theory

Source: http://sciencereview.berkeley.edu/articleex.php?issue=18&article=features_04_quantum

But hasn’t relativity taught us that no one shall put asunder what Minkowski has joined together, i.e. that no dissection of spacetime into space and time can be privileged? ⇒ True, but in order to make up for this, the Hamiltonian systems is subject to ‘constraints’, which mathematically ‘undo’ the dissection. Limitations of approach: not all models of GR are considered (only globally hyperbolic vacuum models). (Dynamical equations play somewhat different role, as we’ll see; matters of interpretation and observation will arise)

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Space and time in canonical QG Empirical incoherence and the emergence of spacetime Space and time in causal set theory The problem of time in canonical quantum gravity (or at least one version of the problem)

Thesis (The problem of time) If we only accept a few seemingly innocuous assumptions, then we are forced into accepting that, fundamentally, there cannot be time, or at least that there cannot be genuine change.

genuine physical magnitudes: Dirac observables have to commute with Hamiltonian ⇒ must be constants of motion ⇒ There cannot be change even in the very minimal sense that the physical quantities can have different values at different times. ⇒ last remnant of a dynamical aspect of time is lost

Problem arises from treating Hamiltonian GR treating as we standardly treat gauge theories, i.e., from treating Diff(M) as a gauge symmetry just as gauge groups in gauge theories Technically: space of initial data is a presymplectic space with gauge symmetries, dynamical trajectories lie in gauge orbits ⇒ change is pure gauge, which means, if gauge symmetries are understood to relate mathematically distinct representations of the same physical possibility, that there is no change in any of the genuinely physical properties of the system ⇒ no change, perhaps no time

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG Space and time in canonical QG Empirical incoherence and the emergence of spacetime Space and time in causal set theory Quantized space ‘spin network states’: quantum states of the gravitational ﬁeld; physical space is supposed to be quantum superposition of spin network states These spin network states can be represented by labelled graphs embedded in some space:

j i2 2

j3 j1

(To be precise: we should really be looking at abstract graphs and equivalence classes of spin network states, as how the spin networks are embedded in manifold is physically irrelevant.) ‘spin’-representations on vertices (represented by nodes in graph) correspond to ‘size’ of the ‘space atoms’, those on edges correspond to ‘size’ of the surface connecting adjacent ‘chunks’ of space ⇒ space is ‘granular’ at Planck scale ⇒ The smooth space of the classical theory is supplanted by a discrete quantum structure. ⇒ Space, as it ﬁgures in our conceptions of the world, is emergent phenomenon.

In a nutshell Causal set theory is based on the assumption that the fundamental (spatio-temporal, gravitational?) structure is a discrete set of basal events partially ordered by causality.

Theorems by Hawking et al. (J Math Phys 17 (1976): 174-181) and Malament (J Math Phys 18 (1977): 1399-1404): given causal structure and volume information, one ﬁnds metric gab of the manifold M; same is true for dimension, topology, and differential structure IOW: causal structure determines geometry (but not ‘size’) ⇒ motivates causal set approach: the fundamental structure is a causal set discreteness of fundamental structure has technical and conceptual utility

Axiom (Causal set theory) The fundamental structure is a causal set, i.e. an ordered pair hC, ≺i consisting of a set C of elementary events and a relation, denoted by the inﬁx ≺, deﬁned on C satisfying the following conditions:

1 ≺ induces a partial order on C: irreﬂexivity: ∀a ∈ C, ¬(a ≺ a) antisymmetry: ∀a, b ∈ C, ¬(a ≺ b and b ≺ a) transitivity: ∀a, b, c ∈ C, if a ≺ b and b ≺ c, then a ≺ c

2 local ﬁnitarity: ∀a, c ∈ C, card({b ∈ C|a ≺ b ≺ c}) < ∞

Remarks: antisymmetry ⇒ no closed timelike curves local ﬁnitarity ⇒ structure must be discrete

1 Inverse problem (sometimes ‘entropy crisis’) present in discrete relational approaches to QG (causets, graphs, etc): vast majority of basic structures are not approximated by a relativistic spacetime (in fact, the larger the causets the bigger the problem)

2 ﬁnd dynamics in some principled and physically motivated way such that the dynamics will ‘drive’ the causal sets to those which do approximate a low-dimensional spacetime so far, only classical dynamics (no quantum interference effects taken into account) known dynamics do not tend to result in causal set which give rise to manifold-like spacetimes

fundamental relation is causal, not temporal; temporal relations are supposed to emerge ⇒ Strictly speaking, there is no time in causal set theory. Otherwise, ≺ looks somewhat like time (it doesn’t pick a Now or a Flow): 1 Its antisymmetry and irreﬂexivity conjointly imply asymmetry( ∀a, b ∈ C, a ≺ b → ¬(b ≺ a)). 2 discrete, not continuous (not isomorphic to open interval of R) 3 It orders events only partially, not totally. ⇒ looks like a discrete B-theoretic version of (special-)relativistic time without metric relations (durations)

set of all ‘spacelike-related’ events to ‘here-now’ h: events that do not causally precede or are causally preceded by ‘here-now’ Problem: in the green set, ∃ pairs of events which are causally related ⇒ ‘space’ should be subset of these, such that no such pairs remain

Deﬁnition (Causal-set space) Space as judged from the ‘here-now’ is the maximal set of events in C containing h such that no two elements are causally related.

Fact The resulting subset of C is structureless—it is just a set of events with no relations deﬁned on it. Technically, it is an ‘antichain’.

S A Major, D Rideout, S Surya. Classical and Quantum Gravity 23 (2006): 4743.

D Rideout, P Wallden. Classical and Quantum Gravity 26 (2009): 155013.

⇒ A maximal antichain is completely characterized by its cardinality. recovering topology: (future-)thicken antichain A to

An := {x ∈ Fut(A) ∪ A : |Past(x) \ Past(A)| ≤ n},

for some n ∈ N, similarly for (past-)thickened antichain. Deﬁne open sets of A as sets of events to the past (future) of events in An \ A ⇒ (coarse) topology

recovering metric (spacelike distances):

J+(x) J+(y) \ z

y x

J{(x) J{(y) \

problem: non-uniqueness

space by itself has no dimensions: usual dimension estimators for embedding causal sets into d-dimensional Minkowski spacetime start out from length of causal chains, i.e., there is not even a proxy for dimension it has no afﬁne or differentiable structure: causal sets are discrete, so even the whole of a causal set has no such structure (except, perhaps, in trivial way in that structureless points can be considered a zero-dimensional afﬁne space) it also has no metric structure: there simply isn’t anything which support ‘distances’ in this structureless set—metric emerges (together with the rest of the essential features) in the limit of relativistic spacetimes non-locality: discreteness of the causal set and the Lorentz invariance demanded of the emerging spacetime conspire to render the physics non-local in a way unfamiliar to general relativity.

q r

Conclusion ⇒ We have a ‘problem of space’!

i.e., at the emergent, low-energy level, we seem to have a richly structured space, for which, however, there is no analogue at the fundamental, high-energy level (at least not a structured analogue). Recall that this is analogous to the problem of time/change (although, to repeat, I don’t claim that the problem of space is just as puzzling as the problem of time). Admittedly, structure (e.g. topological) is induced on space, i.e., on the antichain, by causally ‘thick’ space.

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG The threat of empirical incoherence Empirical incoherence and the emergence of spacetime The emergence of spacetime in causal sets Are spacetime-less theories empirically incoherent?

Deﬁnition (Empirical incoherence) A theory is empirically incoherent just in case the truth of the theory undermines our empirical justiﬁcation for accepting it as true. (cf. Barrett 1999, §4.5.2)

A theory denying the fundamental existence of spacetime is thus alleged to be empirically incoherent because the empirical justiﬁcation of a theory derives only from our observation of the effects of a localized something situated in spacetime; yet if such a theory were true, there could be no such localized something.

Barrett, Jeffrey A., The Quantum Mechanics of Minds and Worlds, Oxford University Press (1999).

Nick Huggett und CW, „Emergent spacetime and empirical (in)coherence”, Studies in the History and Philosophy of Science (in press).

Avoiding the charge of empirical incoherence requires that a theory must be shown to admit emergent spacetime, i.e., it must be shown how general-relativistic spacetime emerges as an approximation in quantum theories of gravity This task is also important in the ‘context of justiﬁcation’, as its discharge provides an account of why the classical spacetime theory (GR) was as successful as it was. This is no mean feat, and quite generally the pièce de résistance for formulating such a theory!

; gab , hM i hC ¹ i

Lee Smolin. The case for background independence. In Dean Rickles et al. (eds.), The Structural Foundations of Quantum Gravity. Oxford: Oxford University Press (2006), 196-239.

Deﬁnition A causal set hC, ≺i is said to approximate a classical spacetime hM, gabi iff there is an injective function φ : C → M s.t. 1 the causal relations are preserved, i.e. ∀a, b ∈ C, a ≺ b iff φ(a) ∈ J−(φ(b));

2 on average, φ maps one element of C onto each Planck-sized volume of hM, gabi.

If the mapping φ is a random ‘Poisson sprinkling’ of events onto the spacetime manifold, then it is easy to ﬁnd a causal set that approximates a given globally hyperbolic general-relativistic spacetime with bounded curvature (i.e. below the ‘discreteness scale’ of the causal set, the manifold is approximately ﬂat).

; gab hM i , hC ¹ i

Christian Wüthrich The disappearance and re-emergence of space and time The disappearance of space and time in QG The threat of empirical incoherence Empirical incoherence and the emergence of spacetime The emergence of spacetime in causal sets How sprinkling randomly saves Lorentz symmetry From David Rideout, http://www.phys.syr.edu/~rideout

But almost no causal set approximates a low-dimensional spacetime. Theory does neither have the conceptual resources to select (in principled way), nor a dynamical principle to generate, those causal sets that approximate low-dimensional spacetime.

; gab hM i , hC ¹ i

David Rideout and Rafael Sorkin. Classical sequential growth dynamics for causal sets. Physical Review D61 (1999): 024002.

sequential growth dynamics, to produce a probability measure on set of histories growth by stochastic Markov process obeying the following dynamical conditions: 1 internal temporality: in auxiliary external order of their birth, then the events’ labels l ∈ N0 must obey the condition that if x ≺ y, then l(x) < l(y) 2 discrete general covariance: labeling is unphysical, in that transition probability between two causal sets related dynamically as along a directed path is path-independent 3 Bell causality: roughly, what transpires at spacelike separation from the node to which the newborn event is attached does not to inﬂuence the corresponding transition amplitude

...that this framework permits a principled way of assigning transition amplitudes such that ‘manifold-like’ causal sets will end up with a high probability, at the expense of non-manifold-like causets. Some timid success so far, but much remains to be done. Major problem: all of this classical; really, the dynamics would have to be quantum, e.g., using quantum approach to sums over histories ⇒ This will further complicate what space and time are in quantum causal set theory...

Space and time (or spacetime) seem to vanish in many approaches to quantum gravity. I introduced examples of LQG and causal set theory, but the problem is generic. This threatens the empirical coherence of these theories. Arguably, threat can only be averted by showing how relativistic spacetime emerges from fundamental structure. This is necessary anyway, to explain why GR was as successful as it was.

Christian Wüthrich The disappearance and re-emergence of space and time