Holographic relations between theories have become an important theme in quantum gravity research. These relations entail that a theory without gravity is equivalent to a gravitational theory with an extra spatial dimension. The idea of holography was first proposed in 1993 by ‘t Hooft on the basis of his studies of evaporating black holes. Soon afterwards the holographic AdS/CFT duality was introduced, which since has been intensively studied in the string theory community and beyond. Recently, Verlinde has proposed that Newton’s law of gravitation can be related holographically to the ‘thermodynamics of information’ on screens. I discuss the last two scenarios, with special attention to the status of the holographic relation in them and to the question of whether they make gravity and spacetime emergent. I conclude that only Verlinde’s scheme instantiates emergence in a clear and uncontroversial way. I suggest that a reinterpretation of AdS/CFT may create room for the emergence of spacetime and gravity there as well.

1. 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 Gravity
AEI Potsdam, 18 December 2014
Partly based on PhilSci 10606 with D. Dieks, J. van Dongen

2. 2
• Duality and emergence of space-time have
been a strong focus in quantum gravity and
string theory research in recent years

3. 3

4. 4
• 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.

5. • 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!

6. ’t Hooft’s Holographic Hypothesis
• The total number of degrees of freedom, 𝑛, in a region of spacetime
containing 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”.

7. ’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.

8. ’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.

9. 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?
9

10. Plan
•Duality: AdS/CFT
• Introduction
• Duality
• Renormalization group
• Diffeomorphism invariance and background
independence
• Interpretation
•Emergence: Verlinde’s scenario and AdS/CFT
10

11. AdS/CFT
• 𝐷-dim. anti-de Sitter space
• Can be extended to (AL)AdS
• In local coordinates:
d𝑠2
=
ℓ2
𝑟2
d𝑟2
− d𝑡2
+ d𝐱2
• Fields 𝜙 𝑟, 𝑥
• Mass 𝑚
• CFT on ℝ 𝐷−1
• QFT with a fixed point,
other backgrounds
• Operators 𝒪 𝑥
• Dimension Δ
11

12. 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 −𝑑 𝜙 𝑟, 𝑥
𝑟=0
= 𝜙 0 𝑥 = 𝑒 d 𝑑 𝑥 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)
12

13. Renormalization Group
• Radial integration: • Wilsonian renormalization:
Λ𝑏Λ0
𝑘
integrate out
New cutoff 𝑏Λ
rescale 𝑏Λ → Λ until 𝑏 → 0
AdS 𝑟
𝜕AdS 𝑟 𝜕AdS 𝜖
new boundary condition
integrate out
IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT
13

14. 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.
14

15. 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

16. 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?
16
𝑍string 𝑟Δ −𝑑
𝜙 𝑟, 𝑥
𝑟=0
= 𝜙 0 𝑥 = 𝑒 d 𝑑 𝑥 𝜙 0 𝑥 𝒪 𝑥
CFT
(1)

17. 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).
17

18. 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
18

19. 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

20. 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
20

21. 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.
21

22. 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.
22

23. •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! 23

24. • 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)
24
Emergence

25. 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?
25

26. 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!
26

27. 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
27

28. 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
28

29. 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
29

30. 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
30

31. The Appearance of Gravity
1) Holography: number of bits in the reservoir
𝑁 =
𝐴 𝑐3
𝐺ℏ
=
4𝜋𝑅2 𝑐3
𝐺ℏ
2) Equipartition: 𝐸 = 𝑀𝑐2 ~ 𝑁 𝑇
3) Bekenstein: Δ𝑆 ~ 𝑚 ∆𝑥
4) Second law of thermodynamics: 𝐹 = 𝑇
∆𝑆
∆𝑥
j 31

32. The Appearance of Gravity
1) Holography: number of bits in the reservoir
𝑁 =
𝐴 𝑐3
𝐺ℏ
=
4𝜋𝑅2
𝑐3
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
𝑅2
32

33. 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
𝐺ℏ
is 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
33

34. Analysing the process
34

35. • 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?
35

36. At which level does this require holography?
36
• The emergence of gravity only requires approximate holography.
• In Verlinde’s scheme, the microscopic bulk theory can be
dispensed with.

37. 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.
37

38. 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.
38

39. 39
Thank you!