Talk given at the 17th International Interdisciplinary Seminar, Netherhall House, London. 3 January 2015.

1. A Holographic Universe?
Sebastian de Haro
17th International Interdisciplinary Seminar
Netherhall House, London, 3 January 2015
Amsterdam University College, University of Amsterdam
Department of History and Philosophy of Science, University of Cambridge

2. ’t Hooft’s Holographic Hypothesis (1993)
• The maximum number of degrees of freedom, 𝑛, in a region of
space-time of area 𝐴, is related to the entropy of a black hole whose
horizon fits that area:
𝑛 × log 2 = 𝑆BH =
𝑘B 𝑐3 𝐴
4𝐺ℏ
• Normally in quantum theory, the number of degrees of freedom
grows like the volume rather than the area
• Thus, black holes seem to reduce the number of possible
configurations in theories that include gravity
• “We can represent all that happens inside [a volume] by degrees of
freedom on the surface”
• “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”.

3. (‘bulk’)
Could it be that gravity is “just the bulk
description” of a world without gravity?
Picture: http://scienceblogs.com/startswithabang/files/2010/03/star-wars-hologram.jpg
• So that: “the world is a hologram”?
• Would gravity and space-time somehow ‘emerge’?
• And what would this exactly mean?
Picture: http://www.zeropoint.ca/holoapple%20(3).jpg

4. ’t Hooft’s Holographic Hypothesis
• The observables “can best be described as if“ they were Boolean
variables on a lattice [i.e., 0s or 1s], 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 space-time's boundary (the horizon)
• He argued that quantum gravity theories that are formulated in a
four dimensional space-time, and that one would normally expect
to have a number of degrees of freedom that scales with the
volume, must be “infinitely correlated” at the Planck scale: see EPR
• 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

5. ’t Hooft’s Holographic Hypothesis
• ’t Hooft’s paper wavers between boundary and
bulk as fundamental ontologies for theories of
quantum gravity
• There is an interpretative tension here, that
resurfaces in other contexts where there are
holographic descriptions; also called:
• Holographic dualities: one-to-one maps between all
of the physics in the bulk and on the boundary.
• By ‘physics’ I mean: states and observables.
• The map preserves the relevant structures on both sides.

6. 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?
• What do dualities tell us about theories of
quantum gravity?

7. Plan
• The holographic principle: AdS/CFT correspondence
• Philosophical questions
(with Dieks and Dongen, PhilSci (2014) 10606)
• Emergence of space-time and gravity
• Whither quantum gravity?

8. The Holographic Principle
• ’t Hooft’s speculative idea became a piece
theoretical physics after Maldacena’s discovery of
the ‘AdS/CFT correspondence’ (1997) in string
theory
• Consider a world with a negative cosmological
constant (more on the positive case, in which we
live, later)
• This is called ‘anti-de Sitter’ space (‘AdS’), as opposed to
‘de Sitter’ space

9. • It looks like a horse saddle:
• The angles of a triangle on its surface add up to less than 180°
• Since it has a boundary (at infinity), one must specify
boundary conditions for particles/fields
• Similar to Newtonian mechanics, where one imposes initial
conditions for the positions and momenta of particles
Anti-de Sitter Space

10. The AdS/CFT Correspondence
• AdS/CFT relates the theory of gravity in the bulk of
AdS (with its boundary conditions) to a quantum
field theory (‘CFT’, stands for ‘conformal field
theory’) defined on the boundary (‘at infinity’)
• The notion of ‘spatial infinity’ can be made
mathematically precise (using Penrose’s ‘conformal
treatment of infinity’), but this is irrelevant for this talk
• In particular, there is nothing mysterious about this kind
of ‘infinity’ (though crucial for AdS/CFT!)

11. AdS/CFT
• For each physical field in the bulk of AdS there is a
corresponding field (operator) on the boundary
• The correspondence states that the values of all physical
quantities in the bulk are the same as those on the boundary.
Examples:
• One of the interesting aspects of the correspondence is that it
maps large distances (low energies: IR) to short distances
(high energies: UV)
geometry (4d)
fluctuations of the geometry
energy excitation (scalar field)
black hole temperature
small perturbations of the black hole
geometry (3d)
energy, momentum, pressure, shear stress (density)
gluon condensate
temperature of a metal
electrical and thermal properties (conductivity)
Bulk Boundary
For how to ‘reconstruct’ the bulk from the boundary, see: de Haro et al. (2001)

12. AdS/CFT
• Strictly speaking, the AdS/CFT correspondence has
the status of a ‘conjecture’, though there is massive
evidence for it: very many non-trivial checks have
been performed (and it is usually called a
‘correspondence’: compare e.g. Fermat’s last
‘theorem’ before it was proven!)
• AdS/CFT has been used to explain heavy ion collisions
at RHIC (Brookhaven, NY)
• Another application: quantum properties of materials
(graphene)

13. Philosophical Questions
• Is one side of the duality more fundamental?
• If the boundary side is more fundamental, space-time
could be ‘emergent’
• What is the physical interpretation of the duality?

14. 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 ‘energy 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

15. 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

16. Interpretation
• The internal point of view seems more natural for
theories of the whole physical world
• It implies that properties such as the ‘dimensionality’ of
space-time are not fundamental, but a matter of useful
description
• 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 more natural. Compare:
• Position/momentum duality in QM
• Equivalence of frames in special relativity
(no absolute rest frame)
• Here, the dimensionality of space is not absolute but
depends on a choice of ‘frame’

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

18. • 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 variables 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 clear 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!

19. • 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 is room for
emergence of space-time
(analogy: thermodynamics vs. statistical mechanics)
Emergence

20. • If the holographic relation exists at the fundamental level
(bottom), 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?
BulkBoundary
Fundamental
Derived

21. • Holographic models of gravity apply to an expanding
universe as well (de Sitter space):
Maldacena-Hartle-Hawking proposal
• In this case, the holographic surface is at future infinity
• Time, not space, is the emergent direction!
• The details are involved: ‘analytic continuation’
(de Haro and Petkou, JHEP 1411 (2014) 126)
• Much more work needed
• Holography seems to apply to spaces with no
cosmological constant as well: Verlinde’s scheme
• Newton’s force of gravity as an entropic force
Extensions of holographic duality

22. What do we know and what do we not know
about quantum gravity?
Experimentally confirmed
(theory + experiment)
Not experimentally confirmed
(theory + indirect tests)

23. What do we know and what do we not know
about quantum gravity?
• Consistent theories of quantum gravity are very difficult,
but not impossible, to come by:
• String theory and loop quantum gravity are currently the only
candidates by any reasonable criteria of what a ‘theory’ is
• String theory is able to reproduce general relativity and quantum
field theory in suitable limits (plus holography, and much more)
• Observational/experimental evidence (mainly from astronomy
and cosmology) key to confirm the theory
• Currently no experimental confirmation, but successful
experimental tests of aspects of the theory: RHIC, Hawking
radiation in analogue models
• String theory, and in particular holographic reformulations
of gravity, are mature and robust enough that a number of
lessons can (and should) be drawn

24. Whither quantum gravity?
• In the presence of holographic dualities: space,
time, and gravity are not the fundamental concepts
• Concepts such as the dimensionality of space are useful,
but not fundamental
• Space, time, and gravity may indeed be emergent
(under the specified conditions)
• Hence as we learn about quantum gravity, a
paradigm shift is taking place:
• Gravity is ‘special’, indeed fundamentally different from
the other forces
• From ‘unification’ to… ‘reformulation’ or ‘holographic
reconstruction’

25. Thank you!