Reduction and Emergence in Holographic Scenarios for

GravitySebastian de Haro

(University of Amsterdam)

Foundations of DualitiesFlorence, 15-16 September 2014

Based on PhilSci 10606 with: D. Dieks, J. van Dongen

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“Starting from first principles and general assumptions, we present a heuristic argument that shows that Newton’s law of gravitation naturally arises in a theory in which space emerges through a holographic scenario. Gravity is identified with an entropic force caused by changes in the information associated with the position of material bodies.”

Erik Verlinde, 2010

(Related work: Jacobson 1995)

“Starting from first principles and general assumptions, we present a heuristic argument that shows that Newton’s law of gravitation naturally arises in a theory in which space emerges through a holographic scenario. Gravity is identified with an entropic force caused by changes in the information associated with the position of material bodies.”

Erik Verlinde, 2010

A new spin on holography?

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?•Conceptual analysis can be useful for physics. Consider e.g. the following argument:

(a) theory A and theory B are dual to one another; (b) an exact formulation for theory A is known, but such a formulation for B is not known; therefore (c) theory A is more fundamental than theory B.

• This argument is fallacious because it appeals to the pragmatic situation that a formulation of B is not known (b). By duality (a), theoretical superiority of A cannot be claimed. An exact formulation of B might be found tomorrow!

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Plan

•AdS/CFT duality• Introduction•Duality•Diffeomorphism invariance and background

independence•Renormalization group• Interpretation

•Verlinde’s holographic scenario• The scenario• Emergence

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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

• CFT on other backgrounds

•Operators 𝒪 𝑥• Dimension Δ

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Duality Statement

•String theory in (AL)AdS space = CFT 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

•One-to-one map of observables (and states) preserving certain structures.•Physical equivalence, mathematical structure

different• Large distance ↔ high energy divergences

(1)

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Conditions for AdS/CFT 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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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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Remarks on Background Independence (odd 𝑑)

• Two possible meanings of background independence, e.g. Belot (2011): A weak one: the theory can be defined for every background (metric, fields,…). A strong one: the theory is independent of the background metric chosen.

• In the first sense, (1) is background independent. In fact, the equations of motion are fully background independent. Partial background dependence comes in the observables through the dependence on a boundary conditionfor the asymptotic metric. More precisely, they depend on a choice of conformal class for the asymptotic metric. The equations of motion are fully background independent, they can include

matter. The equations of motion do not determine the boundary conditions. These have

to be imposed additionally. The boundary conditions make the observables to partly depend on background,

but observables exist (and match) for every background, even not locally Minkowski. The observables (1) themselves are scalars.

• We should not be surprised that there is a dependence on the boundary metric, since Einstein’s equations do not require more. These are part of the space of solutions.

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𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1)

Diffeomorphism Invariance of (1)

• Normally the diffeomorphism invariance of the equations of motion is discussed. The theory is here by construction diffeomorphism invariant. We will therefore consider diffeomorphism invariance of the observables (1).

• Regarding the metric in the bulk: the partition function is diffeomorphism invariant, for those diffeomorphisms that preserve the asymptotic form of the metric.

• Observables obtained by taking derivatives transform as tensors under these diffeomorphisms. These observables are covariant.

• 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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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.

• There seems to be no way of getting rid of this anomaly: it is fully non-linear and it does not rely on classical approximations. Perhaps this is the best one can do in spaces with a negative cosmological constant.

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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

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Remarks

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

different physical quantities

• Internal point of view: • The two descriptions are equivalent. No devisable

experiment could tell one from the other (each observation can be reinterpreted in the ‘dual’ language)

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

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• Holography and emergence of gravity?

• Erik Verlinde’s proposal

Emergence

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Motivating Thoughts

• Hints from string theory, the holographic conjecture/principle: there are solid indications from quantum gravity research that gravitational theories within a volume correspond to a theory without gravity on the boundary of the volume

• AdS/CFT duality gives a concrete and detailed example of the idea. Renormalization steps in the CFT on the surface correspond to different sizes of the bulk

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Could it be that gravity is “just the bulk-description” of a world without gravity?

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Further Motivating Thoughts

• 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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Program of Research

•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-less theory remain unspecified: it is a mere information processing device, a theory of holographic “bits”

•Make gravity appear as a macroscopic thermodynamic phenomenon

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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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Working this out

• Imagine a sphere, whose area is divided into small cells with each one “bit”. This information suffices to describe the inside (holography)

•On the sphere an entropic process takes place: the distribution of 0-s and 1-s tends to an 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’ (or ‘coarse-graining’)

•Coarse-graining:•Removing some bits reduces the area of the sphere• ‘Coarse-grained’ theories encode less information, i.e.

describe 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 on sphere

𝑁 ~ 𝐴 = 4𝜋𝑅2

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 on sphere

𝑁 ~ 𝐴 = 4𝜋𝑅2

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

𝑅225

Thermodynamic limit(coarse graining)

micro-macro

Macroscopic theoryon sphere: no gravity

Microscopic theoryon sphere

Microscopic theoryin bulk

Holography

Macroscopic theoryin bulk: gravity

Thermodynamic limit(coarse graining)

micro-macro

Holography

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).

Thermodynamic limit(coarse graining)

micro-macro

Holography

Thermodynamic limit(coarse graining)

micro-macro

Holography

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.• Gravity could thus emerge in situations where gauge/gravity

duality does not hold exactly (e.g. cosmological scenarios)

Conclusions

• In the holographic scenario, the microscopic surface theory is not necessarily more fundamental than the microscopic bulk theory.

•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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