The disappearance and re-emergence of space and …...Empirical incoherence and the emergence of spacetime The disappearance and re-emergence of space and time in quantum theories

The disappearance of space and time in QGEmpirical incoherence and the emergence of spacetime

The disappearance and re-emergence ofspace 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


Organization of lecture

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

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



The central problem... particularly if one insists on a fundamental role for spacetime

ThesisMany approaches to quantum gravity (QG) suggest or imply thatspace and time do not exist at the most fundamental ontological level.Thus deprived of their former status as part of the fundamentalfurniture 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 andchairs.



Time in physics

Time in physics does not have manyof the features we would usually thinkof as essential for time:

It is isomorphic to (an openinterval of) R, or imposes atotal order on events.

It contains aphenomenologically specialmoment—the ‘now’.

It has a dynamical quality—a‘flow’.

It is asymmetric, i.e., it has a‘direction’.

ThesisPhysics has denied all of these.



Physical spaceHenri Poincaré, Science and Hypothesis

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



Space and time in canonical QGSpace and time in causal set theory

Why we need a quantum theory of gravity

Source: http://blog.michaelkcooke.com/

General relativity (GR) can’t be thelast word: quantum effects must betaken into consideration in earlyuniverse and evolution of blackholes

⇒ must find theory that combinesquantum effects and (strong)gravitational fields

⇒ need a quantum theory of gravity

various approaches, string theoryand loop quantum gravity (LQG)most visible, causal set theory


http://blog.michaelkcooke.com/



Mapping QGSource: http://universe-review.ca/F15-particle.htm


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



A road to unification for someSource: http://sciencereview.berkeley.edu/articleex.php?issue=18&article=features_04_quantum


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



The two main rivals: (1) String theory

Brian Giberson, “String Theory”

starts out from the standard modelof particle physics, a quantumtheory

sense in which gravity can be saidto be ‘naturally’ included; details farfrom clear

originally developed assuming aspacetime background (as inertcontainer), but questions havebeen raised about it (dualities:physically equivalent stringtheories use different spacetimes)

outstanding problems: ‘landscapeproblem’, repeated falsepredictions of ‘supersymmetricparticles’




The two main rivals: (2) Loop quantum gravity (LQG)

Source: http://www.cpt.univ-mrs.fr/∼rovelli/

starts out from GR,incorporates its core lessons(dynamical spacetime)

tries to incorporate quantumeffects by ‘quantizing’ GR

outstanding problems: (1)problem of time/ill-understooddynamics, (2) classicallimit/emergence of spacetimefrom basic quantum structure


http://www.cpt.univ-mrs.fr/~rovelli/



The dissolution of spacetime in QG

In string theory as well as in loop quantum gravity (LQG) (andother approaches to QG), indications are coalescing that spaceand time are no longer fundamental entities as classicallyconceived (substantivally or relationally), but merely ‘emergent’phenomena that arise from the basic physics.

In particular (as in LQG): no continuous spacetime, discretestructure

In the language of physics: spacetime theories such as GR as‘effective’ and spacetime itself ‘emergent’, much likethermodynamics is an effective theory and temperature isemergent, as it is built up from the collective behaviour of gasmolecules.

Unlike the fact that temperature is emergent, the idea that theuniverse is not in space and time shocks our very idea ofphysical existence as profoundly as any scientific revolution.




The project‘Emergent Spacetime in Quantum Theories of Gravity’

philosophers have only just started tostudy the implications of QG and thedisappearance of spacetime

Funded project with Nick Huggett(University of Illinois, Chicago):‘Emergent Spacetime in QuantumTheories of Gravity’

joint framework, Huggett studiesstring theory, Wüthrich does LQG,causal set theory

The remainder of this talk focuses onmy contribution to this project.




On the conceptual foundations of LQG

LQG attempts to transpose the central lesson of GR into aquantum theory

central innovation of GR: spacetime isn’t fixed ‘background’,which determined the inertial forces, but a dynamical structurewhich 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 in3-dim ‘spatial’ hypersurfaces, parametrized by 1-dim ‘time’

3-dim ‘space’ considered as dynamical system which evolvesover ‘time’; 3-dim hypersurfaces represent (instantaneous)states of dynamical theory




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


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



But hasn’t relativity taught us that no one shall put asunder whatMinkowski has joined together, i.e. that no dissection ofspacetime into space and time can be privileged?

⇒ True, but in order to make up for this, the Hamiltonian systems issubject to ‘constraints’, which mathematically ‘undo’ thedissection.

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)




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 weare forced into accepting that, fundamentally, there cannot be time, orat least that there cannot be genuine change.

genuine physical magnitudes: Dirac observables

have to commute with Hamiltonian⇒ must be constants ofmotion

⇒ There cannot be change even in the very minimal sense that thephysical quantities can have different values at different times.

⇒ last remnant of a dynamical aspect of time is lost




The problem of time in canonical quantum gravity

Problem arises from treating Hamiltonian GR treating as westandardly treat gauge theories, i.e., from treating Diff(M) as agauge symmetry just as gauge groups in gauge theories

Technically: space of initial data is a presymplectic space withgauge symmetries, dynamical trajectories lie in gauge orbits

⇒ change is pure gauge, which means, if gauge symmetries areunderstood to relate mathematically distinct representations ofthe same physical possibility, that there is no change in any ofthe genuinely physical properties of the system

⇒ no change, perhaps no time




Quantized space‘spin network states’: quantum states of the gravitational field;physical space is supposed to be quantum superposition of spinnetwork states

These spin network states can be represented by labelledgraphs embedded in some space:

2i

1i

1j

2j

3j




Quantized space: spin network states

(To be precise: we should really be looking at abstract graphsand equivalence classes of spin network states, as how the spinnetworks are embedded in manifold is physically irrelevant.)

‘spin’-representations on vertices (represented by nodes ingraph) correspond to ‘size’ of the ‘space atoms’, those on edgescorrespond 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 adiscrete quantum structure.

⇒ Space, as it figures in our conceptions of the world, is emergentphenomenon.




Causal set theory and its motivations

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

Theorems by Hawking et al. (J Math Phys 17 (1976): 174-181) andMalament (J Math Phys 18 (1977): 1399-1404): given causal structureand volume information, one finds metric gab of the manifoldM;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 acausal set

discreteness of fundamental structure has technical andconceptual utility




Causal set theory: the axioms

Axiom (Causal set theory)

The fundamental structure is a causal set, i.e. an ordered pair 〈C,≺〉consisting of a set C of elementary events and a relation, denoted bythe 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




Representing causal sets: (almost) a Hasse diagram




Two major challenges

1 Inverse problem (sometimes ‘entropy crisis’) present in discreterelational approaches to QG (causets, graphs, etc): vast majorityof basic structures are not approximated by a relativisticspacetime (in fact, the larger the causets the bigger the problem)

2 find dynamics in some principled and physically motivated waysuch that the dynamics will ‘drive’ the causal sets to those whichdo approximate a low-dimensional spacetime

so far, only classical dynamics (no quantum interferenceeffects taken into account)known dynamics do not tend to result in causal set whichgive rise to manifold-like spacetimes




Time in causal set theory

fundamental relation is causal, not temporal; temporal relationsare supposed to emerge

⇒ Strictly speaking, there is no time in causal set theory.

Otherwise, ≺ looks somewhat like time (it doesn’t pick a Now ora Flow):

1 Its antisymmetry and irreflexivity conjointly implyasymmetry (∀a,b ∈ C,a ≺ b → ¬(b ≺ a)).

2 discrete, not continuous (not isomorphic to open interval ofR)

3 It orders events only partially, not totally.

⇒ looks like a discrete B-theoretic version of (special-)relativistictime without metric relations (durations)




Space in causal set theory









set of all ‘spacelike-related’ events to ‘here-now’ h: events thatdo not causally precede or are causally preceded by ‘here-now’

Problem: in the green set, ∃ pairs of events which are causallyrelated

⇒ ‘space’ should be subset of these, such that no such pairsremain

Definition (Causal-set space)

Space as judged from the ‘here-now’ is the maximal set of events in Ccontaining h such that no two elements are causally related.

FactThe resulting subset of C is structureless—it is just a set of eventswith no relations defined on it. Technically, it is an ‘antichain’.




Recovering geometry: topology

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 itscardinality.

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. Defineopen sets of A as sets of events to the past (future) of events inAn \ A⇒ (coarse) topology




Recovering geometry: metric

recovering metric (spacelike distances):

)y(+J\) x(+J

)y({J\) x({J

z

w

yx

problem: non-uniqueness




The problem of space in causal set theory

space by itself has no dimensions: usual dimension estimatorsfor embedding causal sets into d-dimensional Minkowskispacetime start out from length of causal chains, i.e., there is noteven a proxy for dimension

it has no affine or differentiable structure: causal sets arediscrete, so even the whole of a causal set has no such structure(except, perhaps, in trivial way in that structureless points can beconsidered a zero-dimensional affine space)

it also has no metric structure: there simply isn’t anything whichsupport ‘distances’ in this structureless set—metric emerges(together with the rest of the essential features) in the limit ofrelativistic spacetimes

non-locality: discreteness of the causal set and the Lorentzinvariance demanded of the emerging spacetime conspire torender the physics non-local in a way unfamiliar to generalrelativity.




Discreteness + Lorentz symmetry = Non-locality

p

r

q




Conclusion⇒We have a ‘problem of space’!

i.e., at the emergent, low-energy level, we seem to have a richlystructured space, for which, however, there is no analogue at thefundamental, high-energy level (at least not a structuredanalogue).

Recall that this is analogous to the problem of time/change(although, to repeat, I don’t claim that the problem of space isjust 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.



The threat of empirical incoherenceThe 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 theoryundermines our empirical justification for accepting it as true. (cf.Barrett 1999, §4.5.2)

A theory denying the fundamental existence of spacetime is thusalleged to be empirically incoherent because

the empirical justification of a theory derives only from ourobservation of the effects of a localized something situatedin spacetime;yet if such a theory were true, there could be no suchlocalized something.

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




Saving empirical coherence

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

Avoiding the charge of empirical incoherence requires that atheory must be shown to admit emergent spacetime,

i.e., it must be shown how general-relativistic spacetimeemerges as an approximation in quantum theories of gravity

This task is also important in the ‘context of justification’, as itsdischarge provides an account of why the classical spacetimetheory (GR) was as successful as it was.

This is no mean feat, and quite generally the pièce de résistancefor formulating such a theory!




The emergence of spacetime in causal sets

?

iabg ;hMi¹, hC




The emergence of spacetime in causal sets

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

Definition

A causal set 〈C,≺〉 is said to approximate a classical spacetime〈M,gab〉 iff there is an injective function φ : C →M s.t.

1 the causal relations are preserved, i.e. ∀a,b ∈ C,a ≺ b iffφ(a) ∈ J−(φ(b));

2 on average, φ maps one element of C onto each Planck-sizedvolume of 〈M,gab〉.




The inverse problem for causal sets (Smolin 2006)

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

i¹, hCiabg ;hM




How sprinkling randomly saves Lorentz symmetryFrom David Rideout, http://www.phys.syr.edu/~rideout


http://www.phys.syr.edu/~rideout



The inverse problem for causal sets (Smolin 2006)

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

iabg ;hMi¹, hC




Classical sequential growth dynamics

David Rideout and Rafael Sorkin. Classical sequential growth dynamics for causal sets. Physical ReviewD61 (1999): 024002.

sequential growth dynamics, to produce a probability measure on set ofhistories

growth by stochastic Markov process obeying the following dynamicalconditions:

1 internal temporality: in auxiliary external order of their birth, thenthe 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 thattransition probability between two causal sets related dynamicallyas along a directed path is path-independent

3 Bell causality: roughly, what transpires at spacelike separationfrom the node to which the newborn event is attached does not toinfluence the corresponding transition amplitude




Classical sequential growth dynamics




The hope is...

...that this framework permits a principled way of assigningtransition amplitudes such that ‘manifold-like’ causal sets willend up with a high probability, at the expense ofnon-manifold-like causets.

Some timid success so far, but much remains to be done.

Major problem: all of this classical; really, the dynamics wouldhave to be quantum, e.g., using quantum approach to sums overhistories

⇒ This will further complicate what space and time are in quantumcausal set theory...




Conclusions

Space and time (or spacetime) seem to vanish in manyapproaches to quantum gravity.

I introduced examples of LQG and causal set theory, but theproblem is generic.

This threatens the empirical coherence of these theories.

Arguably, threat can only be averted by showing how relativisticspacetime emerges from fundamental structure.

This is necessary anyway, to explain why GR was as successfulas it was.


Documents

The disappearance and re-emergence of space and …...Empirical incoherence and the emergence of spacetime The disappearance and re-emergence of space and time in quantum theories