A Holographic Universe?Sebastian de Haro

17th International Interdisciplinary SeminarNetherhall House, London, 3 January 2015

Amsterdam University College, University of AmsterdamDepartment of History and Philosophy of Science, University of Cambridge

’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 =𝑘B𝑐

3𝐴

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”.

(‘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

’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

’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.

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?

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?

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

• 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

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!)

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 condensatetemperature 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)

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)

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?

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

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

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’

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

• 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!

• 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

• 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

• 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

What do we know and what do we not know about quantum gravity?

Experimentally confirmed(theory + experiment)

Not experimentally confirmed(theory + indirect tests)

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

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’

Thank you!