Berlin Slides Dualities and Emergence of Space-Time and Gravity

Dualities and Emergent Gravity: AdS/CFT and Verlinde’s Scheme

Sebastian de Haro University of Amsterdam and University of Cambridge

Emergent Time and Emergent Space in Quantum GravityAEI Potsdam, 18 December 2014

Partly based on PhilSci 10606 with D. Dieks, J. van Dongen

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• Duality and emergence of space-time have been a strong focus in quantum gravity and string theory research in recent years

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• The notion of ‘emergence’ of space-time and/or gravity is often attached to the existence of a ‘duality’.

• An argument along the following lines is often made:

a) Theory F ('fundamental') and theory G ('gravity') are dual to one another.

b) Theory F does not contain gravity (and/or space-time) whereas theory G does.

c) Therefore space-time (and/or gravity) emerges in theory G. Theory F is to be regarded as more fundamental.

• But this argument is problematic: it replaces ‘duality’ by ‘emergence’.• Duality is a symmetric relation, whereas emergence is not

symmetric• We need to explain what breaks the symmetry • Emergence of space-time requires more than simply ‘the

space-time being dual to something that is not spatio-temporal’.

• It might lead to bad heuristics for constructing new theories, in particular when we are told that we should not pursue theory G but just work on theory F.

• I will discuss the notions of duality and emergence in holographic scenarios:• Duality: AdS/CFT• Emergence: Verlinde’s holographic scenario and AdS/CFT

• I will only discuss the possible emergence of gravity together with one, spatial dimension.• This is a non-trivial task: for obtaining the right classical

dynamics for the metric is hard!

’t Hooft’s Holographic Hypothesis

• The total number of degrees of freedom, 𝑛, in a region of spacetimecontaining a black hole, is:

𝑛 =𝑆

log 2=

𝐴

4𝐺log 2

• Hence, “we can represent all that happens inside [a volume] by degrees of freedom on the surface”

• “This suggests that quantum gravity should be described entirely by a topological quantum field theory, in which all degrees of freedom can be projected on to the boundary”

• “We suspect that there simply are no more degrees of freedom to talk about than the ones one can draw on a surface [in bit/Planck length2]. The situation can be compared with a hologram of a three dimensional image on a two dimensional surface”.

’t Hooft’s Holographic Hypothesis

• The observables “can best be described as if“ they were Boolean variables on a lattice, which suggests that the description on the surface only serves as one possible representation.

• Nevertheless, 't Hooft's account more often assumes that the fundamental ontology is the one of the degrees of freedom that scale with the spacetime's boundary. He argued that quantum gravity theories that are formulated in a four dimensional spacetime, and that one would normally expect to have a number of degrees of freedom that scales with the volume, must be “infitely correlated" at the Planck scale.

• The explanatory arrow here clearly goes from surface to bulk, with the plausible implication that the surface theory should be taken as more basic than the theory of the enclosed volume.

• There is no indication that a notion of emergence is relevant here.

’t Hooft’s Holographic Hypothesis

• ’t Hooft’s paper wavers between boundary and bulk as fundamental ontologies.

• There is an interpretative tension here, that resurfaces in other contexts where there are dualities.

Philosophical concerns regarding holographic dualities:

•Can one decide which side of the duality is more fundamental?

•Is one facing emergence of space, time, and/or gravity?

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Plan

•Duality: AdS/CFT• Introduction•Duality•Renormalization group•Diffeomorphism invariance and background

independence• Interpretation

•Emergence: Verlinde’s scenario and AdS/CFT

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AdS/CFT

•𝐷-dim. anti-de Sitter space • Can be extended to (AL)AdS

• In local coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 − d𝑡2 + d𝐱2

• Fields 𝜙 𝑟, 𝑥• Mass 𝑚

• CFT on ℝ𝐷−1

• QFT with a fixed point, other backgrounds

•Operators 𝒪 𝑥• Dimension Δ

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Duality Statement•One-to-one map of states and quantities (observables) between distinct theories preserving certain structures.•String theory in (AL)AdS space = QFT on boundary•Fields 𝜙 𝑟, 𝑥 ↔ Operators 𝒪 𝑥•Partition function 𝑑 = 𝐷 − 1 :

𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

•Physical equivalence, mathematical structure different•Large distance ↔ high energy divergences•Strictly speaking, the AdS/CFT correspondence has the status of a ‘conjecture’, though there is massive evidence for it (and it is usually called a ‘correspondence’: compare e.g. Fermat’s last ‘theorem’ before it was proven!)

(1)

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Renormalization Group

• Radial integration: • Wilsonian renormalization:

Λ𝑏Λ0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

AdS𝑟

𝜕AdS𝑟 𝜕AdS𝜖

new boundary condition

integrate out

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT

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Conditions for AdS/CFT Duality

• What could lead to the failure of AdS/CFT as a duality?

• Two conditions must be met for this bijection to exist. The observable structures of these theories should be:

i. Complete (sub-) structures of observables, i.e. no other observables can be written down than (1): this structure of observables contains what the theories regard to be ‘physical’ independently on each side of the duality.

ii. Identical, i.e. the (sub-) structures of observables are identical to each other.

If ii. is not met, we can have a weaker form of the conjecture: a relation that is non-exact. For instance, if the duality holds only in some particular regime of the coupling constants.

• There are no good reasons to believe that i. fails.

• Whether ii. is met is still open, but all available evidence indicates that it is satisfied, including some non-perturbative tests.

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Remarks on Background Independence

• Theories of gravity are usually required to be ‘background independent’. In Einstein’s theory of relativity, the metric is a dynamical quantity, determined from the equations of motion rather than being fixed from the outset.

• The concept of ‘background independence’ does not have a fixed meaning, see Belot (2011).

• Here I will adopt a ‘minimalist approach’: a theory is background independent if it is generally covariant and its formulation does not make reference to a background/fixed metric. In particular, the metric is determined dynamically from the equations of motion.

• In this minimalist sense, classical gravity in AdS is fully background independent: Einstein’s equations with negative cosmological constant.• Quantum corrections do not change this conclusion: they appear perturbatively as

covariant higher-order corrections to Einstein’s theory.

• Could background independence be broken by a choice of particular solutions of Einstein’s equations?• The equations of motion do not determine the boundary conditions, which need to

be specified additionally (de Haro et al. 2001). • But this is not a restriction on the class of solutions considered; as in classical

mechanics, the equations of motion simply do not contain the informtion about the boundary/initial conditions.

• This does not seem a case of lack of background independence of the theory. At most, it may lead to spontaneous breaking of the symmetry by a choice of a particular solution. 15

Diffeomorphism Invariance of (1)

• I have discussed background independence of the equations of motion. What about the observables?

• Partition function (1):• It depends on the boundary conditions on the metric (as do the classical

solutions).• It is diffeomorphism invariant, for those diffeomorphisms that preserve the

asymptotic form of the metric.

• Other observables obtained by taking derivatives of (1): they transform as tensors under these diffeomorphisms. These observables are covariant, for odd d (=boundary dimension):

• For odd 𝑑: • Invariance/covariance holds.

• For even 𝑑:• Bulk diffeomorphisms that yield conformal transformations of the boundary

metric are broken due to IR divergences (holographic Weyl anomaly). Is this bad?

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𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1)

Diffeomorphism Invariance (even 𝑑)

• The breaking of diffeomorphism invariance exactly mirrors the breaking of conformal invariance by quantum effects in the CFT.

• The partition function now depends on the representative of the conformal structure picked for regularization.

• The observables (1) such as the stress-tensor no longer transform covariantly, but pick up an anomalous term.

• Anomalies are usually quantum effects, proportional to ℏ. Here, the anomaly is (inversely) proportional to Newton’s constant 𝐺.

• The anomaly is robust: it is fully non-linear and it does not rely on classical approximations.

• This anomaly does not lead to any inconsistencies because the metric is not dynamical in the CFT (see Huggett’s talk).

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Philosophical Questions

• Is one side of the duality more fundamental?• If QFT more fundamental, space-time could be ‘emergent’• If the duality is only approximate: room for emergence

(e.g. thermodynamics vs. atomic theory)

• If duality holds good: one-to-one relation between the values of physical quantities. In this case we have to give the duality a physical interpretation

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Interpretation

•External view: meaning of observables is externally fixed. Duality relates different physical quantities• No empirical equivalence, numbers correspond to

different physical quantities• The symmetry of the terms related by duality is broken by

the different physical interpretation given to the symbols• Example: 𝑟 fixed by the interpretation to mean ‘radial

distance’ in the bulk theory. In the boundary theory, the corresponding symbol is fixed to mean ‘renormalization group scale’. The two symbols clearly describe different physical quantities. More generally, the two theories describe different physics hence are not empirically equivalent

• Only one of the two sides provides a correct interpretation of empirical reality 19

Interpretation

• Internal point of view: • The meaning of the symbols is not fixed beforehand• There is only one set of observables that is described by

the two theories. The two descriptions are equivalent. No devisable experiment could tell one from the other (each observation can be reinterpreted in the ‘dual’ variables)

• Cannot decide which description is superior. One formulation may be superior on practical grounds (e.g. computational simplicity in a particular regime)

• On this formulation we would normally say that we have two formulations of one theory, not two different theories

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Interpretation

•The internal point of view seems more natural for theories of the whole world

•Even if one views a theory as a partial description of empirical reality, in so far as one takes it seriously in a particular domain of applicability, the internal view seems the more natural description.• Compare: position/momentum duality in QM. Equivalence of

frames in special relativity.

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Interpretation

•The internal point of view seems more natural for theories of the whole world

•Even if one views a theory as a partial description of empirical reality, in so far as one takes it seriously in a particular domain of applicability, the internal view seems the more natural description.• Compare: position/momentum duality in QM. Equivalence of

frames in special relativity.

• We should worry about the measurement problem, but it is not necessarily part of what is here meant by ‘theories of the whole world’, because the statement is still true in the classical limit, where we get Einstein gravity.

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•Butterfields’s puzzling scenario about truth (2014): Does reality admit two or more complete descriptions which• (Different): are not notational variants of each other; and yet• (Success): are equally and wholly successful by all epistemic

criteria one should impose?

•On the external view, the two theories are not equally successful because they describe different physical quantities: only one of them may describe this world.•On the internal view, the two descriptions are equivalent

hence equally successful. • If they turn out to be notational variants of each other (e.g.

different choices of gauge in a bigger theory) then the philosophical conclusion is less exciting, but new physics is to be expected. This is what often happens when there is a duality. Currently there is no indication that the two theories are notational variants of each other.

• If the two theories are not notational variants of each other, then we do face the puzzling scenario!

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•On the external view, the two theories describe different physics• The dual theory is only a tool that might be useful, but does

not describe the physics of our world• Here, the idea of ‘emergence’ does not suggest itself

because whichever side describes our world, it does not emerge from something else.

•On the internal view there is a one-to-one relation between the values of physical quantities• Again emergence does not suggest itself: the two

descriptions are equivalent• If the duality is only approximate then there may be room

for emergence of space-time (analogy: thermodynamics vs. statistical mechanics)

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Emergence

Example: Verlinde’s Scheme

• Working out the idea of an approximate duality for the specific case of Newtonian gravity

• Gravity is special: it is universal. It applies to all matter and energy, regardless of specific interactions; it seems to relate to space itself

• This universality reminds one of the universal character of thermodynamical behavior, which is independent of microscopic details

• Gravity distinguishes itself from other forces because it is difficult to quantize; is it fundamentally different?

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Guiding idea about force as a thermodynamic phenomenon

•Entropic processes: as a result of random motion of its microscopic constituents a physical system will end up in a state of greater entropy, i.e. higher probability: the system seems to be directed

•Although there are no forces on the microscopic level, on the thermodynamic level the system appears driven, and this can be described by a “macroscopic force”

• Like a stretched polymer. Spring constant not a fundamental constant but depends on 𝑇!

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Applying this to Gravity

•Start with a theory without gravity on a two-dimensional screen, e.g. the surface of a sphere•Holography: this theory codifies information about

matter in an additional spatial dimension (“in the bulk”)•The microscopic details of this gravitation-free

theory remain unspecified: it is a theory of holographic degrees of freedom (Verlinde calls them “bits”)•Make gravity appear as a macroscopic

thermodynamic phenomenon

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Working this out

• Imagine a sphere, whose area is divided into small cells with each one degree of freedom (“bit”). Call this the ‘system’.

•On the sphere an entropic process takes place: this system is coupled to a reservoir at fixed temperature (the ‘environment’), and the distribution of dof of the system tends to equilibrium.

•This process will correspond to gravitational motion inside the sphere

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Appearance of Space

• In the surface theory, there are no spatial dimensions other than those within the surface itself

•Consider several spheres, namely different surface theories that relate to each other via ‘renormalization’ (‘coarse-graining’ steps)

•Coarse-graining:• Removing some dof reduces the area of the sphere• ‘Coarse-grained’ theories describe less dof, i.e. less space

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Appearance of Space

• Thus, a spatial dimension 𝑥 appears as a bookkeeping device that records the level of coarse graining on the sphere

• Entropy grows when a particle is thrown in (Bekenstein):Δ𝑆 ~ 𝑚 ∆𝑥

Picture: http://media02.hongkiat.com/black-white-photo-water/black-and-white-drops.jpg

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The Appearance of Gravity

1) Holography: number of bits in the reservoir

𝑁 =𝐴 𝑐3

𝐺ℏ=

4𝜋𝑅2𝑐3

𝐺ℏ

2) Equipartition: 𝐸 = 𝑀𝑐2 ~ 𝑁 𝑇

3) Bekenstein: Δ𝑆 ~ 𝑚 ∆𝑥

4) Second law of thermodynamics: 𝐹 = 𝑇∆𝑆

∆𝑥

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The Appearance of Gravity

1) Holography: number of bits in the reservoir

𝑁 =𝐴 𝑐3

𝐺ℏ=

4𝜋𝑅2𝑐3

𝐺ℏ

2) Equipartition: 𝐸 = 𝑀𝑐2 ~ 𝑁 𝑇

3) Bekenstein: Δ𝑆 ~ 𝑚 ∆𝑥

4) Second law of thermodynamics: 𝐹 = 𝑇∆𝑆

∆𝑥

From which we get Newton’s law: 𝐹 ~𝑀𝑚

𝑅232

Some Distinctions in Verlinde’s Scheme

•We should not regard the process of ‘throwing a particle in’ as increasing the number of bits in the theory on the sphere• Remember duality: the bits on the screen are dual to the

particles near the screen• Throwing a particle in thus decreases the number of bits in the

system. More precisely: it increases the number of bits in the reservoir and decreases the number of bits in the system.

• So the boundary theory is not a theory about what is inside the screen (the reservoir) but about the bits that are within one Compton wavelength of the screen (the system)

• The relation 𝑁 =𝐴𝑐3

𝐺ℏis the definition of the relation

between the bulk and the boundary. It is not a statement about entropy. We can rescale the area and rescale ℏ at the same time without changing anything

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Analysing the process

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• The holographic relation may well be a bijective map. • There is no reason in this case to think that one side is

more fundamental than the other (left-right).• But the thermodynamic limit introduces the emergence

of gravity in an uncontroversial sense (top-bottom).

Does Gravity Emerge?

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At which level does this require holography?

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• The emergence of gravity only requires approximate holography.• In Verlinde’s scheme, the microscopic bulk theory can be

dispensed with.

Emergence of Space and Gravity

• Gravity could thus emerge in the same way (via coarse graining) in other situations where gauge/gravity duality does not hold exactly (e.g. cosmological scenarios: dS/CFT).

• But this idea can be applied more generally to AdS/CFT, where the renormalization group flow introduces coarse graining over high-energy degrees of freedom.

• In this case, Einstein gravity may emerge from the fundamental bulk theory, whether the latter contains gravity or not.

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Conclusions

• In holographic scenarios with an exact duality, the microscopic surface theory is not necessarily more fundamental than the microscopic bulk theory.• The bulk does not emerge from the boundary in such

cases.

•However, the appearance of gravity in the thermodynamic limit makes it a clear case of emergence, connected with robustness and novelty of behavior. This robustness explains the universality of gravity.•That gravity is emergent could give rise to new

predictions: the law of gravity is not exact but subject to fluctuations.

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Thank you!