Dualities in Physics: An Example from String Theory

Sebastian de Haro University of Cambridge and University of Amsterdam

Duality in contemporary mathematics — philosophical aspects

Bergische Universität Wuppertal, 4 September 2015

Based on:

• Dieks, D. van Dongen, J., de Haro, S. (2015), Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.07.007

• de Haro, S. (2015), Studies in History and Philosophy of Modern Physics, doi:10.1016/j.shpsb.2015.08.004

• de Haro, S., Teh, N., Butterfield, J. (2015), Studies in History and Philosophy of Modern Physics, submitted

Introduction

•Dualities have traditionally been of great interest in physics •The examples are numerous and diverse:

• Wave-particle duality, matrix-wave mechanics, electric-magnetic duality, harmonic oscillator duality, Kramers-Wannier duality (2d Ising models at high-low temperature),…

•The obvious suggestion to make is that two dual theories ‘say the same thing, in different words’• Indeed, for several notable dualities, this is the

consensus among physicists: the duality is a mere descriptive difference

2

Introduction

• There are two basic reasons why physicists are interested in dualities. Dualities in physics (more than symmetries) are normally:

(1) Striking: the two ‘sayings’ (i.e. the two formulations) are very disparate. In string theory: in T-duality, the two theories differ about the radius of a compact dimension of space: where one says it is 𝑟, the other says it is 1/𝑟; in S-duality, they differ about which particles are light and which are heavy; in gauge-gravity duality, they differ about the dimensionality of space(2) Useful: one main way the duality can be useful is if it relates a regime where problems are difficult to solve (say because couplings are strong) to one where they are easy to solve (say because couplings are weak)

3

Introduction

•Such a difference—of particles’ masses, or radius or dimension of space—seems so marked that you might well doubt that the formulations are ‘saying the same thing’. You might well say instead that they contradict each other; so that the example is best taken as a case of under-determination (of theory by all possible data), not as any kind of theoretical equivalence• This is a reasonable reaction: and it has been argued in

the literature

• experiments at RHIC (Brookhaven, NY)

4

Introduction

• Indeed I will argue that, while I endorse the string theory consensus, more is needed in order to go from duality to full theoretical equivalence: an interpretation of the theory is needed• This will confirm the consensus in string theory: that at least some of

these striking dualities do relate different formulations of a single theory• As the physics jargon has it: they ‘describe the same physics’; meaning of

course not just observational, but also theoretical, equivalence

• Some string theorists go further. They take the ongoing search for an M theory (a theory that unifies the 5 known string theories) to be the search for a formulation which will relate to the present formulations, on the two sides of such dualities, in much the way a gauge-invariant formulation of a gauge theory (a quantum field theory with as symmetry group a compact Lie group) relates different choices of gauge• In other words, they view duality as a symmetry of a larger theory (M

theory). The theories on either side of the duality are special cases of this larger theory (say, the symmetry is broken/fixed in one way or in another)

5

Introduction

• I will focus on one particular string-theoretic duality: gauge/gravity duality; and its main instantiation, the AdS/CFT correspondence

• Gauge/gravity duality relates a theory of gravity in 𝐷 dimensions to a quantum field theory (no gravity!) in 𝐷 − 1 dimensions: also called ‘holographic’

• There is independent motivation for studying gauge/gravity dualities: the gravitational theories are quantum:• So the duality should provide conceptual insights into the difficult

programme of quantising the gravitational force, by relating a theory of gravity to a gauge theory

• It has also already been used at the RHIC experiment in Brookhaven (NY)

6

Aim of this talk

• To expound on conceptual aspects of the best studied gauge/gravity duality (AdS/CFT), with an eye on foundational and philosophical issues that it bears on:• Provide a physicist’s definition of ‘duality’ and make two general

comments:

i. Duality between two theories as being ‘gauge related’ in the general philosophical sense of being physically equivalent: ‘saying the same thing, in different words’

ii. Work out a physical interpretation of the duality that takes us from duality to theoretical equivalence

7

AdS/CFT: crude description

• AdS/CFT is a duality between:

• A theory of gravity in anti-de Sitter space (AdS: a solution of Einstein’s equations with negative curvature); and a string theory in AdS. The low energy limit of string theory is Einstein’s theory of gravity

• A special kind of quantum field theory: a conformal field theory, i.e. a quantum field theory that is invariant under conformal transformations (coordinate transformations that give rise to a scale factor)

• Discovered by Maldacena in 1997

• Concrete realisation of the general phenomenon of ‘holography’ conjectured by ’t Hooft in 1993

8

Duality: a simple definition

• Regard a theory as a triple ℋ,𝒬,𝐷 : states, physical quantities, dynamics• ℋ = states (in Hilbert space)

• 𝒬 = specific set of operators (self-adjoint, renormalizable, invariant under symmetries)

• 𝐷 = dynamics (e.g. Lagrangian and integration measure)

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

• There exist 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

and 𝑑ℋ commutes with (is equivariant for) the two theories’ dynamics9

Duality as a unitary transformation

• Given the isomorphisms 𝑑ℋ:ℋ𝐴 → ℋ𝐵 and 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵,

𝑑ℋ is in fact a unitary operator:𝑑ℋ𝑑ℋ

† = 𝑑ℋ† 𝑑ℋ = 𝟙 ; and

𝑑𝒬 𝒪𝐴 = 𝑑ℋ 𝒪𝐴 𝑑ℋ†

• Thus, this is an ordinary unitary transformation on the Hilbert space, at the level of the states and the observables

• So the two theories are different representations of the same Hilbert space and set of quantities

• This is my first comment (i) on ‘duality’: this suggests that, on the simple definition adopted, the two theories are physically equivalent: they ‘say the same thing, in different words’ (as different representations). I say suggests because we still have to work out the interpretation (which will be my point (ii))

10

Duality (more refined version)

• For the theories of interest, we will need some more structure

• Add external parameters 𝒞 (e.g. couplings, sources)

• The theory is given as a quadruple ℋ,𝒬, 𝒞, 𝐷

• Duality is an isomorphism ℋ𝐴, 𝒬𝐴, 𝒞𝐴 ≃ ℋ𝐵, 𝒬𝐵, 𝒞𝐵 . There are three bijections: • 𝑑ℋ:ℋ𝐴 → ℋ𝐵• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵• 𝑑𝒞: 𝒞𝐴 → 𝒞𝐵

such that:

𝑂, 𝑠 𝑐 ,𝐷𝐴 = 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵 ∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴, 𝑐 ∈ 𝒞𝐴𝒪, 𝑠 𝑐 ,𝐷𝐴 = 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵(1)

11

Question

• Does gauge/gravity duality instantiate the simple definition of duality?

12

Comments

• I call the definition of duality ‘simple’ (even: ‘simplistic’) because a notion of duality that is applicable in some of the physically interesting examples may need a more general framework (e.g. a Hilbert space may be too restrictive for higher-dimensional QFTs)

• At present, no one knows how to rigorously define the theories involved in gauge/gravity dualities (except for lower-dimensional cases): not just the string theories, but also the conformal field theories involved

• But if one is willing to enter a non-rigorous physics discussion, then a good case can be made that:

(i) AdS/CFT can be cast in the language of states, quantities, and dynamics

(ii) When this is done, the AdS/CFT correspondence amounts to conjecturing a duality

13

The Gravity Side

• Hyperboloid in 𝐷 + 1 dimensions:

−𝑋02 − 𝑋𝐷

2 +

𝑖=1

𝐷−1

𝑋𝑖2 = −ℓ2

• Constraint can be solved introducing 𝐷 coordinates:𝑋0 = ℓ cosh 𝜌 cos 𝜏𝑋𝐷 = ℓ cosh 𝜌 sin 𝜏𝑋𝑖 = ℓ sinh 𝜌 Ω𝑖 𝑖 = 1,… , 𝑑 = 𝐷 − 1 , Ω𝑖 = unit vector

• Leading to: d𝑠2 = ℓ2 −cosh2 𝜌 d𝜏2 + d𝜌2 + sinh2 𝜌 dΩ𝐷−22

• Symmetry group SO 2, 𝑑 apparent from the construction

• Riemann tensor given in terms of the metric (negative curvature):

𝑅𝜇𝜈𝜆𝜎 = −1

ℓ2𝑔𝜇𝜆𝑔𝜈𝜎 − 𝑔𝜇𝜎𝑔𝜈𝜆

𝑋𝑖

𝑋0

𝑋𝐷

ℓ

14

Geometry of AdS

The Gravity Side (cont’d)

• Useful choice of local, Poincaré coordinates:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝜂𝑖𝑗 d𝑥

𝑖d𝑥𝑗 , 𝑖 = 1,… , 𝑑 = 𝐷 − 1

• 𝜂𝑖𝑗 = flat metric (Lorentzian or Euclidean signature)

• Fefferman and Graham (1985): for a space that satisfies Einstein's equations with a negative cosmological constant, and given a conformal metric at infinity, the line element can be written as:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥

𝑖d𝑥𝑗

𝑔𝑖𝑗 𝑟, 𝑥 = 𝑔 0 𝑖𝑗 𝑥 + 𝑟 𝑔 1 𝑖𝑗 𝑥 + 𝑟2𝑔 2 𝑖𝑗 𝑥 + ⋯

• Einstein’s equations now reduce to algebraic relations between 𝑔 𝑛 𝑥 𝑛 ≠ 0, 𝑑 and 𝑔 0 𝑥 , 𝑔 𝑑 𝑥 (de Haro et al. 2001)

• Such a space is called ALAdS (asymptotically locally AdS)

15

• This metric includes pure AdS, but also: AdS black holes (any solution with zero stress-energy tensor and negative cosmological constant). AdS/CFT is not restricted to the most symmetric case! Hence the name ‘gauge/gravity’

• So far we considered Einstein’s equations in vacuum. The above generalizes to the case of gravity coupled to matter

• Scalar field 𝜙 𝑟, 𝑥 , solve its equation of motion (Klein-Gordon equation) coupled to gravity:𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + 𝑟 𝜙 1 𝑥 +⋯+ 𝑟

𝑑𝜙 𝑑 𝑥 +⋯

• Again, 𝜙 0 𝑥 and 𝜙 𝑑 𝑥 are the boundary conditions and all other coefficients 𝜙 𝑛 𝑥 are given in terms of them (as well as the metric coefficients)

Adding Matter

16

The Gravity Side (cont’d)

AdS/CFT Dictionary

• 𝐷-dim AdS

• ALAdS

• d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥

𝑖d𝑥𝑗

• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯+ 𝑟𝑑𝑔 𝑑 𝑥

• Field 𝜙 𝑟, 𝑥 , mass 𝑚

• 𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + ⋯+ 𝑟𝑑𝜙 𝑑 𝑥

• CFT on ℝ𝐷−1

• QFT with a fixed point• Metric 𝑔 0 (𝑥)

• 𝑇𝑖𝑗 𝑥 =ℓ𝑑−1

16𝜋𝐺𝑁𝑔 𝑑 𝑥 + ⋯

• Operator 𝒪 𝑥 with scaling dimension Δ 𝑚• Coupling 𝜙 0 𝑥

• 𝒪 𝑥 = 𝜙 𝑑 𝑥

Fields Operators

Normalizable mode (sub-leading) Exp. val. (state)

Non-normalizable mode (leading) Coupling 17

Gravity (AdS) Gauge (CFT)

Example: AdS5 × 𝑆5 ≃ SU 𝑁 SYM

AdS5 × 𝑆5

• Type IIB string theory

• Limit of small curvature:supergravity (Einstein’s theory + specific matter fields)

• Symmetry of AdS: diffeo’s that preserve form of the metric generate conformal transformations on the bdy

• Symmetry of 𝑆5

SU 𝑁 SYM

• Supersymmetric, 4d Yang-Mills theory with gauge group SU(𝑁)

• Limit of weak coupling: ’t Hooft limit (planar diagrams)

• Classical conformal invariance of the theory

• Symmetry of the 6 scalar fields

• Limits are incompatible (weak/strong coupling duality: useful!)• Only gauge invariant quantities (operators) can be compared• Symmetry:

SO 2,4 × SO 6

18

AdS/CFT Duality

• AdS/CFT can be described in terms of the quadruple ℋ,𝒬, 𝒞, 𝐷 : • Normalizable modes correspond to exp. vals. of operators (choice of state)

• Fields correspond to operators

• Boundary conditions (non-normalizable modes) correspond to couplings

• Formulation otherwise different (off-shell Lagrangian, different dimensions!)

• Two salient points of :• Physical quantities, such as boundary conditions, that are not determined by

the dynamics, now also agree: they correspond to couplings in the CFT

• This is the case in any duality that involves parameters that are not expectation values of operators, e.g. T-duality (𝑅 ↔ 1/𝑅), electric-magnetic duality (𝑒 ↔ 1/𝑒)

• It is also more general: while ℋ,𝒬, 𝐷 are a priori fixed, 𝒞 can be varied at will (Katherine Brading: ‘modal equivalence’). We have a multidimensional space of theories

• Dualities of this type are not isomorphisms between two given theories (in the traditional sense) but between two sets of theories

ℋ

𝒬

𝒞

𝐷

(1)

19

AdS/CFT Duality (Continued)

• String theory in (AL)AdS space = QFT on boundary

• Formula 1 is generated by:

𝑍string 𝜙 0 = 𝜙 0,𝑥 =𝜙 0 𝑥

𝒟𝜙 𝑒−𝑆 𝜙 = exp d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

• The correlation functions of all operators match

• Physical equivalence, mathematical structure different

• Large distance ↔ high energy divergences

• As mentioned before, the AdS/CFT correspondence is still not proven (even by physics standards!). But on a heuristic approach, it is readily seen that it can be written in the language of states and operators; and it is a duality in the sense introduced earlier

(1′)

𝑐𝐵 = 𝑑𝒞 𝑐𝐴

20

Comments

• Earlier I suggested that dual theories are not necessarily theoretically equivalent; and that the step from duality to theoretical equivalence requires a physical interpretation of the theory

• I argued that, if AdS/CFT is indeed correct, then it is a duality (heuristically), and I exhibited the explicit dictionary

• If the duality is somehow broken (e.g. it only holds up to some order in an expansion in the couplings) then there is an exciting possibility: emergence of space-time: which I will not pursue here

• I will assume that the duality is exact and try to clarify when we have theoretical equivalence• To that end, let us recall Poincaré’s example about the Lobachevski plane

21

http://math.ucr.edu/home/baez/escher.gifhttp://mathworld.wolfram.com/images/eps-gif/HyperbolicTessellation_1000.gif

• (Outside view) Consider two-dimensional beings whose measuring rods expand/contract with temperature: highest at the centre, it goes to zero at the edge of the circle

• (Inside view) We can deceive these beings into thinking they live on the Lobachevski plane: with a non-Euclidean geometry

𝑇 = 0

𝑇 = 𝑅2

22

Poincaré’s example

Poincaré’s example

• Poincaré: there is no way to distinguish between the two situations. What is to count as the difference between a real non-Euclidean geometry and a Euclidean world with distorting fields?

• Nothing in the fact of the matter determines which hypothesis is ‘correct’ and the two theories are equivalent: there is theoretical equivalence

23

Poincaré’s example

• Poincaré’s analysis is helpful in that it suggests two ways in which theories can be interpreted physically (see next slide)

• Though his notion, as appearing from the example, might seem to be tied up with verificationism, this is in fact not necessary. The notion of duality introduced earlier certainly does not appeal to any notion of verification

• So consider internal and external viewpoints on the duality, as follows:

24

Interpretation

• External point of view: the meaning of the symbols (quantities) is externally fixed. Duality thus relates different physical quantities:• Example: in gauge/gravity duality, 𝑟 is fixed by the interpretation in the

gravity theory to mean ‘radial distance’. In the gauge theory, the corresponding symbol is fixed to mean ‘energy scale’. The two symbols clearly describe different physical quantities

• The two theories describe different physics hence they are notempirically equivalent

• The symmetry of the terms related by duality is broken by the different physical interpretation given to the symbols

• Only one of the two sides provides a correct interpretation of empirical reality

25

Interpretation

• Internal point of view: the meaning of the symbols (quantities) is not a priori fixed. It is only fixed ‘from within the theory’:• There is one single set of physical quantities that the two theories describe:

by definition of duality as isomorphism

• Because the meaning of the symbols is not a priori fixed, the two theories give equivalent descriptions of the same physics (same physical quantities)

• Even though (based on past conventions) the symbols in the two theories seem to have different interpretations (𝑟 as ‘radial distance’ or as ‘energy scale’), they really describe the same physics because they both describe the same observations equally well. There is no distinction between the two

• Matching with experiment is done by matching numbers: between a subset of the physical quantities and the observational outcomes (e.g. fitting of parameters in physical models!)

• No physical situation could tell one theory from the other. The two theories make exactly the same claims—not just about observational matters, but also about all theoretical matters: in physics jargon, they ‘describe the same physics’ 26

External vs. internal interpretations

• Is there a preferred interpretation? Do we always have an internal interpretation at our disposal?

• The difference between the external and internal viewpoints clearly depends on the epistemic and metaphysical commitments made:• The internal viewpoint seems incompatible with strong realism such as:

‘there is exactly one best description of the world’

• The internal viewpoint meshes well with anti-realism as well as weak forms of realism, e.g. structural realism (‘scientific theories are correct about the structure of reality’)

• But there is a second interpretational aspect, prompted by the wording ‘external’ vs. ‘internal’: ‘external’ suggests that there is a relevant environment to which the theory may be coupled (a context that fixes the meaning of the symbols). ‘Internal’ suggests that such context is not available

27

External vs. internal interpretations

• There are cases of two disjoint parts of reality that: match exactly, ‘are isomorphic’, in the taxonomy used by some theory (i.e. as regards the properties and relations the theory treats)—but are otherwise different, i.e. distinct and known to be distinct• For instance: (under the relevant idealisations) the small angular swing of the

clock pendulum in my neighbour’s living room is isomorphic with the linear motion of the bob at the end of the spring that my colleague uses for her class demonstrations in classical mechanics. The two systems are isomorphic yet clearly distinct. This is because location (my neighbour’s living room vs. my colleague’s classroom) are not part of the isomorphism

• Thus it is tempting to see the difference in the way meaning is assigned to symbols as matching a difference about the kind of theories considered:• (Parts): theories that only provide partial descriptions of empirical reality, i.e.

that describe parts of the world• (Whole): theories of the whole world, i.e. of the universe

28

External vs. internal interpretations

• In the case of (Whole) (theories of the universe) the idea of distinct but isomorphic existences indeed falls by the wayside: namely, when we aim to write down a cosmology. For such a theory, there will be, by hypothesis, only one token of its type of system, viz. the universe

• Thus, the internal viewpoint is certainly possible for (Whole)

• If, in addition, one takes some form of anti-realist or structural realist position, then the internal viewpoint is the only one available: for there are no criteria by which one theory should describe the world better than the other: they describe the world equally well

• But one should resist the temptation of seeing the difference between the external/internal viewpoints as matching the contrast (Parts)/(Whole)• 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 quantum mechanics: the two descriptions are completely equivalent descriptions of reality, yet we do not regard them as different theories: merely different descriptions of the same phenomena

29

External vs. internal interpretations

• In conclusion: given a duality, and given (Whole), the internal point of view is the correct one (unless very strong and, in this context, unnatural metaphysical assumptions are made)• It is also better suited to a ‘science first’ position on metaphysics

• The internal viewpoint also applies to cases of (Parts)

• Of course, this is not to deny that one formulation may be superior on pragmatic grounds (e.g. computational simplicity in a particular regime), as discussed for gauge-gravity duality

30

Conclusions

• Dualities in physics can be seen in terms of isomorphism between two theories

• But duality by itself does not lead to theoretical equivalence between physical theories: for which an interpretation is needed

• The basic interpretative fork is between the internal and external viewpoints, depending on how symbols are interpreted

• For theories of the whole universe (as is the case with string dualities), the internal point of view is the more natural one: duality thus amounting to full physical equivalence

31

Conclusions

• In string theory, dualities play the role of ‘guiding principles’ towards the construction of new theories• Gauge-gravity duality might be ‘resolved’ in a theory with a sufficiently large

symmetry that the string theory and the CFT appear as two ‘choices of gauge’

• Dualities in string theory are often seen as pointing to an underlying ‘unifying’ theory (M theory)

32

Thank you!

33