Duality and Emergence

Sebastian De Haro University of Cambridge and University of Amsterdam

Oxford, 9 June 2016

Plan of the Talk

A. Review of previous work on emergence in gauge/gravity duality1. Emergence and duality

2. Two kinds of emergence

3. Example: emergence of a classical spacetime

B. Report on the relation between duality and emergence:1. Diffeomorphism symmetry

2. Emergence of diffeomorphisms

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Part APrevious work on emergence in

gauge/gravity duality

1. Emergence and Duality

• Emergence and duality: two notions that seem to be closely connected but are clearly distinct—perhaps even mutually exclusive!• Duality: theoretical/formal equivalence between two theories.

• Emergence: focus on novelty, hence lack of equivalence between two theories or phenomena.

• In string theory and in quantum field theory, duality and emergence often seem to coexist. • Dualities are important theoretical tools.

• Dualities seem to be related to emergence: e.g. the dual theory contains new particles or phenomena.

• How can these contrasting features be combined?• If two theories or phenomena are equivalent (the equivalence relation is

invertible), then it is hard to see how there can be novelty.

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Emergence

• In the rest of the talk, I will adopt Butterfield’s (2011) conception of emergence:

“Properties or behaviour of a system which are novel and robustrelative to some appropriate comparison class. Here ‘novel’ meanssomething like: ‘not definable from the comparison class’, and maybe‘showing features (maybe striking ones) absent from the comparisonclass’. And ‘robust’ means something like: ‘the same for variouschoices of, or assumptions about, the comparison class’.”

• Butterfield argues that his notion of emergence is compatible with reduction/deduction. In reduction there can be novelty (e.g. taking a limit).

• A similar question, of the compatibility of emergence with duality, arises in quantum field theory (QFT) and string theory.

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Emergence and Duality

• Starting from Butterfield’s conception of emergence, these features can be found also in the characterisation of dualities in physics:• One description contains ‘novel’ aspects compared to its dual.

• The novel aspects are robust: they do not depend on details of the dual.

• The emerging/emerged theories are defined at different scales (coupling or other parameter).

• The ‘emergent’ feature is obtained by the use of limits.

• But there is an important difference: • Duality is symmetric (the dual descriptions are mutually ‘novel’ with respect

to each other); while:

• Emergence is apparently asymmetric.

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Duality

• A duality is a formal relation, a special case of theoretical equivalence: an isomorphism between two theories.

• In certain cases, two dual theories may also be taken to be physically equivalent, i.e. two descriptions of a single theory for the same physical reality (De Haro (2016b)).• For two theories to be equivalent, it is sufficient that each theory be

unextendable, i.e. roughly, be a complete toy-cosmology.

• Gauge/gravity duality is one of the cases in which the two duals qualify for physical equivalence: they can be taken to describe identical physics.

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Duality

• Regard a bare theory as a triple ℋ,𝒬,𝐷 : states, physical quantities, dynamics • ℋ = states: in the cases we will consider: a Hilbert space• 𝒬 = physical quantities: a specific set of operators: self-adjoint,

renormalizable, invariant under symmetries• 𝐷 = dynamics: a choice of Hamiltonian, alternately a Lagrangian

• A duality is an isomorphism between two bare theories ℋ𝐴, 𝒬𝐴, 𝐷𝐴and ℋ𝐵 , 𝒬𝐵, 𝐷𝐵 , as follows:

• There exist structure-preserving bijections: • 𝑑ℋ:ℋ𝐴 → ℋ𝐵,

• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵

and pairings (expectation values) 𝒪, 𝑠 𝐴 such that:𝒪, 𝑠 𝐴 = 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠

𝐵∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴

as well as triples 𝒪; 𝑠1, 𝑠2 𝐴 .

• The duality map commutes with (is equivariant for) the two theories’ dynamics.

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Duality, symmetry, models, and interpretation

• A bare theory is a triple ℋ,𝒬,𝐷 up to isomorphism: in the quantum case, unitary equivalence.

• Thus we regard two dual ‘theories’ as models of the same (bare) theory.• I will still sometimes refer to them as two ‘theories’.

• But the distinction will become important when we come to symmetries.

• Duality must map the symmetries of the bare theory but it need not map the symmetries of the models.

• These notions of bare theory and of duality are formal: I have not mentioned interpretation.

• Interpretations are construed as surjective maps from the bare theory to the physical quantities, satisfying suitable conditions.

• The maps could endow the models with the same or with different interpretations. • If the interpretations are the same, there is physical equivalence.

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Coarse-graining

• In the examples we will discuss, coarse-graining will be the mechanism which allows us to study emergence.• Coarse-graining: a systematic approximation scheme in which degrees of

freedom which are less important to a particular situation are removed.

• Measured by a parameter 𝑟.

• In the limit 𝑟 → 0, novel behaviour appears. But also before the limit: in a weak, but vivid sense.

• Robustness: relative independence from the details of the degrees of freedom which are removed (autonomy).

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Emergence and Duality

• There can be emergence in the context of a duality. We can accommodate an asymmetric relation of emergence by weakening/modifying the duality.

• The notion of duality as an isomorphism between theories suggests how the duality can be modified:

(BrokenMap) Take a duality that is not exact, e.g. the map fails to be a bijection for sufficiently small lengths and high energies.

• The duality map goes in one direction: from theory F (fine-grained) to theory G (coarse-grained).

• Then G can emerge from F, after appropriate approximations (coarse-graining) and redefinitions.

(Approx) Each of the theories has a coarse-graining or approximation (especially, a spectrum of energy/length scales). So there are two families of theories, each theory given by a setting of its scale/parameter.

• The duality map relates the theories pairwise at each level of coarse-graining.• Novelty can emerge on both sides of the duality.

• The two kinds of emergence can be combined.

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Two kinds of emergence

𝑑′: 𝐺′ 𝐹′

𝑑: 𝐺 𝐹

𝐺′′ 𝐹′′

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𝑑′′

(BrokenMap)

(Approx)

e.g. add new terms to the classical action

Our Example: Gauge/Gravity Duality

• Gauge/gravity duality: a theory of gravity in (𝑑+1) dimensions is equivalent to a quantum field theory (no gravity!) in 𝑑 dimensions.• Also called ‘holographic’.

• The theories differ in the number of dimensions that they assign to space (and, of course, in much else!).

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Our Example: Gauge/Gravity Duality

• Gauge/gravity duality has been an important focus in quantum gravity research in recent years.• Not just nice theoretical models: one of its versions (AdS/CFT)

successfully applied: RHIC experiment in Brookhaven (NY).

• Rather general phenomenon, not restricted to string theory: e.g. also in loop quantum gravity! (Smolin, Dittrich,…)

• It is often claimed that spacetime ‘disappears/dissolves’ at high energies; and ‘emerges’ in a suitable semi-classical limit.

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Gauge/Gravity Duality

• (𝑑 + 1)-dim AdS: the maximally symmetric solution of Einstein’s equations with a negative cosmological constant.

• Generalizing: arbitrary asymptotic boundary at spatial infinity. • The boundary data are a conformal class

of metrics.

• Fields must be provided with boundary conditions.• Scalar field 𝜙 𝑟, 𝑥 , mass 𝑚

• Long-distance (IR) divergences.

• CFT (conformal field theory) on 𝑆𝑑

• The QFT has a UV fixed point.• The metric (conformal class of metrics)

is arbitrary.

• Operators are classified by their scaling dimensions and spin.• Operator 𝒪 𝑥 , scaling dimension Δ 𝑚

• High-energy (UV) divergences.

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Gravity (AdS) Gauge (CFT)

Maldacena (1997)

Witten (1998)

De Haro et al. (2001)

Emergence in gauge/gravity duality

• The full string theory is approximated by semi-classical gravity:• The semi-classical approximation becomes better near the boundary.

• The approximation can thus be parametrised by the radial distance 𝑟, which corresponds to the energy scale in the boundary theory.

• Radial flow in the bulk geometry towards the UV (𝑟 → ∞) can be interpreted as the renormalization group flow of the boundary theory towards the IR.

• Wilsonian renormalization group methods can be used.

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Holographic Renormalization Group

• Radial flow between two radial cut-offs 𝜖, 𝜖′

• Wilsonian renormalization: integrate out degrees of freedom between two cut-offs Λ, 𝑏Λ (𝑏 < 1)

Λ𝑏Λ0

𝑘

integrate out

New cutoff 𝑏Λ

rescale 𝑏Λ → Λ until 𝑏 → 0

IR cutoff 𝜖 in AdS ↔ UV cutoff Λ in QFT

AdS𝜖′

𝜕AdS𝜖′ 𝜕AdS𝜖

new boundary condition

integrate out

cut-off surface

Holographic Renormalization Group

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domain wall

Freedman et al. (1999)

AdS1 AdS2

Part BDuality and symmetry

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Gauge Symmetry

• The formulation of physical theories often requires the introduction of variables (degrees of freedom) that are not physical.

• A symmetry often ensures that the physical quantities are independent of these (unphysical) degrees of freedom.

• Such symmetries are called gauge symmetries.

• Natural question for duality: do dualities relate the gauge symmetries of the two theories?

• Usual expectation about gauge symmetry: since the physical quantities in a duality are gauge invariant, gauge symmetries need not (do not) map through the duality.

• A stronger claim is often made: gauge symmetries on one side shouldbe ‘invisible’ to the other side.

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Symmetry and Duality

• But given the distinction between the theory (the triple of states, quantities, and dynamics) and its models (the gravity side and the QFT), we should not expect all symmetries to be invisible.

• The theory may have some symmetries which are shared by its models. So we should distinguish:• Symmetries of the theory: the symmetries which are mapped by the duality.

• Proper symmetries of the models: which are not mapped by the duality.

• A verdict on a symmetry being ‘physical’ or ‘redundant’ depends on interpretation (De Haro et al. (2016)).

• The symmerties of the theory are potentially physical. For instance, the physical quantities fall in representations of this symmetry.• They correspond to Caulton’s (2015) ‘synthetic symmetries’.

• The proper symmetries of the models are representational redundancies.• Caulton’s ‘analytic symmetries’. 21

Greaves and Wallace (2014)

Diffeomorphism Symmetry

• Diffeomorphisms give, roughly, different ways to coordinatise the points of a manifold.

• Einstein’s general relativity is invariant under diffeomorphism symmetry. That is, it shouldn’t matter how we parametrise the manifold.

• QFTs are not invariant under arbitrary diffeomorphisms.

• Natural question for gauge/gravity duality: do diffeomorphisms map across the duality? Are diffeomorphisms symmetries of the theory or proper symmetries of the models?

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Aim: assess emergence from QFT

• Horowitz and Polchinski (2006) introduce the notion of ‘invisibility’: “the gauge variables of AdS/CFT are trivially invariant under the bulk diffeomorphisms, which are entirely invisible in the gauge theory.”

• Their claim:(1) Diffeomorphisms are present in the gravity theory but not in the QFT.

This gives a verdict: diffeomorphisms as proper symmetries of the model.

(1) They conclude that diffeomorphisms emerge from the QFT.

• My aim: to assess the claim of emergence from the QFT. Worries:(1) Not all diffeomorphisms are invisible.

(2) Therefore, it is not clear that all diffeomorphisms emerge from the QFT.

• There is one well-known class of diffeomorphisms which is visible and which can acquire a physical meaning: namely inducing conformal transformations. These act on the QFT quantities: they are visible: they are shared symmetries of the two models.

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Invisible Diffeomorphisms

• Let us analyse these diffeomorphisms in more detail.

• Call a diffeomorphism that preserves appropriate structures of model X: X-invisible.

• Gravity-invisible diffeomorphisms: those diffeomorphisms that preserve some structures (to be specified) of the gravity theory.

• This class turns out larger than expected.

• QFT-invisible diffeomorphisms: those diffeomorphisms (of the gravity theory) that leave all the quantities of the QFT (states and operators) unaffected.

• This class turns out smaller than expected.

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Defining the class of metrics

• A Poincaré metric ො𝑔 on 𝑀 is, roughly speaking, a metric of signature (𝑝 + 1, 𝑞), such that:

(1) ො𝑔 has conformal infinity 𝑀,𝑔 : a conformal manifold 𝑀 of signature 𝑝, 𝑞 , and 𝑔 a representative of the conformal class.

(2) ො𝑔 satisfies Einstein’s equations in vacuum.

• Theorem (Fefferman and Graham (1985, 2012)): Let ො𝑔 be a Poincaré metric on 𝑀. Then there exists an open neighbourhood 𝑈 near the boundary of 𝑀 on which there is a unique diffeomorphism 𝜙:𝑈 → 𝑀such that 𝜙ȁboundary is the identity map, and 𝜙∗ ො𝑔 is in Poincaré normal form, with 𝑀 given by 𝑟 = 0:

𝜙∗ ො𝑔 =1

𝑟2d𝑟2 + 𝑔 𝑥, 𝑟

• In other words, when working with Poincaré metrics, we may, without loss of generality, consider those that are in normal form.

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Visible and Invisible Diffeomorphisms

• A diffeomorphism 𝜙:𝑈 → 𝑈 (𝑈 open in 𝑀) is said to be invisible relative to ො𝑔,𝑀, 𝑔 , if it satisfies:

(i: invisible relative to ො𝑔): 𝜙∗ ො𝑔 is in Poincaré normal form.

(ii: invisible relative to 𝑀): 𝜙ȁboundary = idboundary .

(iii: invisible relative to 𝑔): 𝜙 is an isometry of the metric on 𝑀 (the representative of the conformal class).

• These three definitions prompt three notions of gravity-invisibility:

(a) Strongly gravity-invisible: fix all of (i)-(iii).

(b) Weakly gravity-invisible: fix (i)&(ii) or (i)&(iii).

(c) Gravity-invisible: fix (ii)&(iii).

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Visible and Invisible Diffeomorphisms

(a) Theorem. If 𝜙 is strongly gravity-invisible (fixes (i)-(iii)) then 𝜙 is the identity, i.e. the class is trivial.

(b) Proposition. If 𝜙 is weakly gravity-invisible (fixes (i)&(ii) or (i)&(iii))then 𝜙boundary is a conformal transformation on 𝑀.

(c) Claim. There exist non-trivial gravity-invisible diffeomorphisms.

• Thus:

(1) The weakly gravity-invisible diffeomorphisms give the conformal transformations of the QFT: QFT-visible diffeomorphisms can be identified with weakly gravity-invisible diffeomorphisms. • These are symmetries of the theory.

(2) One can also show that the gravity-invisible diffeomorphisms are indeed QFT-invisible. This class is non-trvial. • These are symmetries of the model.• Hence the idea of a ‘bulk argument’: an analogue of Einstein’s hole

argument.

• No claim here that these three classes exhaust all the interesting diffeomorphisms. 27

Back to Emergence

• The analysis of emergence is different for QFT-visible and QFT-invisible diffeomorphisms because of their different properties.

• QFT-visible diffeomorphisms are present on both sides of the duality: there is no good argument for the emergence of one from the other.

• QFT-invisible diffeomorphisms are present in the gravity theory but not in the QFT. No emergence in the sense discussed earlier.• It should be understood epistemically: as novelty in the descriptive apparatus

of one theory with respect to the other.

• It is the emergence of symmetries of the model, which are representational redundancies.

• In both cases, we can still have (Approx) emergence, i.e. emergence on the gravity side from an underlying string theory.

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Summary and conclusions

• Emergence can take place when duality is broken by coarse-graining.

• Two kinds of emergence, depending on which duality condition is weakened: (BrokenMap) vs. (Approx).• (BrokenMap): there is no exact duality.• (Approx): coarse graining replaces the original duality by a series of dualities

between approximate theories.

• Gauge/gravity duality was discussed as a case of (Approx). • The mechanism for emergence is the holographic renormalization group. • Classical features of spacetime emerge after coarse-graining IR degrees of

freedom.

• We have distinguished the diffeomorphisms which are symmetries of the theory (visible) and proper symmetries of a model (invisible).

• The class of QFT-invisible diffeomorphisms was identified with the class of gravity-invisible diffeomorphisms.• QFT-visible diffeomorphisms: no clear emergence from the QFT.• QFT-invisible: emergence only in an epistemic sense.

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References

Review of gauge/gravity: De Haro, Mayerson, Butterfield (2016). “Conceptual Aspects of Gauge/Gravity Duality”. Foundations of Physics,forthcoming.Part A (review):1. De Haro (2016). “Dualities and Emergent Gravity: Gauge/Gravity

Duality”. Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.08.004, forthcoming.

2. De Haro, Teh, Butterfield (2016). “Comparing Dualities and Gauge Symmetries” Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2016.03.002, forthcoming.

3. Dieks, Dongen, Haro (2015). “Emergence in Holographic Scenarios for Gravity”. Studies in History and Philosophy of Modern Physics, 52(B), pp. 203–216.

Part B (new work):1. De Haro (2016b). “Spacetime and Physical Equivalence”. Space and

Time after Quantum Gravity, N. Huggett and C. Wüthrich (Eds.), forthcoming.

2. De Haro (2016a). “Invisibility of Diffeomorphisms”, forthcoming.30