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Colloquium2013

This document discusses the nature of gravity and its relationship to other forces and fields. It provides evidence that gravity is an emergent phenomenon that arises from an underlying non-gravitational theory. Specifically: 1) Gravity behaves differently than other forces in that it curves spacetime itself rather than being mediated by particle exchanges. However, quantum gravity theories propose gravitons as force-carrying particles. 2) Holographic duality theories from the 1990s demonstrated that gravitational theories in higher dimensions are equivalent to non-gravitational theories in lower dimensions. 3) Modern developments like string theory and the AdS/CFT correspondence provide concrete examples of holography and establish gravity as an emer

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Sumit R. Das University of Kentucky
Is Gravity different from other forces ? • The description of gravitational forces due to Einstein is rather different from that of the other known forces. • Forces like electromagnetism, weak interactions or strong interactions are due to exchange of particles – described astoundingly well by Quantum Mechanics of fields • All this happens in a given, fixed, space-time.
• On the other hand, gravity hardly appears to be a force of this sort. • Rather, it is the reaction of any object with energy to the curvature of space-time – in General Relativity space time itself is dynamical.
• Nevertheless, in some sense, gravitational forces can be thought of arising from exchange of particles – these are called “gravitons”. Instead of spin 1 they have spin 2. • Beginning in the 1930’s, physicists have tried to make sense of a theory of “quantum gravity” by applying the standard rules of quantum mechanics to gravitons. • The result has been rather discouraging.
• Now-a-days we do much better – we think of gravity coming from exchange of closed strings rather than particles. • These are rather tiny – so usual experiments should perceive them as point-like objects • The lowest (quadrupole ) mode of oscillation is the graviton. (Yoneya; Scherk and Schwarz, 1974) • The presence of an infinite number of higher modes, however, smooth out the high energy problems faced in quantum gravity based on spin-2 objects alone. – (Green and Schwarz, 1984)
• In recent years, this has led to a rather different picture of gravity. • In this picture, gravitational force is -in a sense -not fundamental. • Rather, it is an effective description of a theory pretty much like the theories of strong and electro-weak interactions – a theory which lives in a fixed space-time. GRAVITY IS AN EMERGENT PHENOMENON.
There are two key features of this connection (1) This non-gravitational theory lives in a lower number of dimensions – as if providing a hologram of what happens in the “bulk”. In some cases the hologram is on the boundary of the space-time. (2) When the hologram is highly quantum, its equivalent description in terms of gravity is classical – just Einstein’s equations
GAUGE THEORIES • For reasons which are not completely clear, nature prefers theories which are characterized by gauge invariance. • The simplest example is electrodynamics. The observables here are the electric field and the magnetic field - and at the classical level that is all you need. You can describe the motion of charged particles and electromagnetic waves. • This is not quite true at the quantum level.
Aharanov Bohm Effect • Charged particles which go around an infinitely thin solenoid along different paths acquire different phases – even though the particles move in a field free region. • Somehow one has to introduce potentials ,
• But this is a redundant description • Potentials which are related by gauge transformations represent the same physics – and lead to the same and • This looks strange – why not use gauge invariant variables ? • We saw that the obvious ones don’t suffice – it turns out that we need to consider non-local variables called Wilson loops • For the Aharanov-Bohm problem the particle is sensitive to this quantity – which is non-zero due to Stokes’ Theorem
Quarks, Mesons and Baryons • The need to understand gauge invariant variables became pronounced in the theory of strong interactions. • Hadrons are made of quarks which interact with each other by exchanging gluons. • Quarks have charge – they also have color. • However they are permanently confined – the p physical objects we see are neutrons, protons a and mesons – these are colorless, or color I singlets. Can we express physics in terms of mesons, baryons and glueballs ?
Strings • Before the advent of QCD – the theory of quarks and gluons it was realized that hadrons behave like strings – this was the origin of string theory (Nambu, Nielsen, Susskind). • After QCD it was soon realized that strings are flux tubes. • gauge strings be • diems t  p 0 Ties up nicely with the idea that the gauge invariant observables are in fact Wilson loop like objects - these must the strings of QCD But there was one puzzle – what is the dimensionless number which serves as the string coupling ?
In quantum electrodynamics, there is a cloud of virtual charges around an electron, which makes it a rather fuzzy extended object . Yet we can treat it as point-like. • This is because there is a small coupling constant in QED • What could be the small number for strings ? QCD with massless quarks does not have a dimensionless parameter !
Large N limit • ‘t Hooft came up with a surprising answer to this question. • Quarks have 3 colors – technically this means that the gauge group of QCD is SU(3). • Consider instead a theory with colors, and the special limit • ‘t Hooft showed that in this large N expansion the coupling which characterizes interaction of strings is given by • In fact the limit is a classical limit
The Isotropic Oscillator • To see why a large-N limit is a classical limit consider a familiar problem in elementary quantum mechanics – the isotropic harmonic oscillator, in N dimensions. • We want to look at the singlet sector of the theory – i.e zero angular momentum • So we use spherical polar coordinates in N dimensions and perform a rescaling of the wave-function
• The Schrodinger equation now becomes • When is large this is like a one dimensional problem with an effective potential • Note that the Planck constant of this theory is in fact • Therefore the large limit is like a classical limit. In fact the full quantum problem is very well approximated by the solution of the classical equations of motion.
But why higher dimensions ? • This aspect is still a bit mysterious. • However the essential physics can be understood from another quantum mechanics problem – that of a hermitian matrix • The singlet sector of this model is described in terms of the eigenvalues of the matrix • These eigenvalues can be thought of as the coordinates of non-relativistic fermions moving along a one dimensional line
• When is large there are lots of lots of fermions – it is then useful to think of the problem in terms of a density of fermions . • But this is a field in 1+1 dimensions. • We started out with a theory in 0+1 dimensions with degrees of freedom. • Now we find that for large it may be expressed as a theory in 1+1 dimensions. • This fact was known for a very long time – at least since 1980
• When the problem was revisited in early 1990’s there were several surprises (1) The theory is secretly relativistic – and the interactions between blobs of fermions are local in space and time. (S.R.D. and A. Jevicki, 1990) (2) This theory is in fact a String Theory (Gross & Milkovic; Brezin & Kazakov; Gross and Klebanov) – rewriting in terms of densities explains why it is so. (3) The interaction between blobs in fact include gravitational forces (Polchinski and Naatsumme). A GRAVITATIONAL THEORY IS DESCRIBED BY A NONGRAVITATIONAL THEORY IN LOWER DIMENSIONS HOLOGRAPHY
• Of course gravity in 1+1 dimensions is rather boring – though not trivial. • Are there higher dimensional examples of this ? • The answer came from a rather different line of thinking.
Black Holes • In 1970’s the work of Bekenstein and Hawking showed that black holes in fact radiate, and may be considered as a thermodynamic object with a characteristic temperature and an entropy. • The entropy formula is rather intriguing : Black holes do not appear to be extensive in the usual sense.
Bekenstein Bound • Bekenstein pushed this a bit more. • He argued that once we take into account of gravity and black holes, the maximum possible entropy of anything inside a large region is proportional not to the volume, but to the area of the boundary
The Holographic Principle • To “explain” this, ‘t Hooft and Susskind came up with a surprising interpretation of this bound. • They proposed the “Holographic Principle” Gravitational physics in a d+1 dimensional world is completely equivalent to non-gravitational physics in (d-1)+1 dimensions. • The latter may be thought to live on the “boundary” “Ordinary” non-gravitational Physics on boundary Gravity in “Bulk”
From Black Holes to Holography • A concrete realization of this principle came from thinking about black holes in string theory. • String Theory provides the microscopic description of a large class of black holes. They look like objects extended in some internal dimensions – D branes. • The low energy excitations of D-branes are described by a non-abelian gauge theory which lives on the brane
From Black Holes to Holography • A concrete realization of this principle came from thinking about black holes in string theory. • String Theory provides the microscopic description of a large class of black holes. They look like objects extended in some internal dimensions – D branes. • The low energy excitations of D-branes are described by a non-abelian gauge theory which lives on the brane
• Using this microscopic picture, thermodynamic properties of these black holes – as well properties of Hawking radiation could be reproduced (Strominger & Vafa; Callan & Maldacena, S.R.D. and S.D. Mathur; Maldacena and Strominger; Dhar, Mandal, Wadia). • In this picture, when a particle falls into a black hole, it can be converted into these gauge field quanta – this appears as absorption by the black hole. • Hawking radiation is the opposite process
• Maldacena provided a key insight into this absorption process • He viewed absorption in the gravity picture as a conversion of closed strings living far away from the brane into closed strings which are localized near the brane
• Maldacena provided a key insight into this absorption process • He viewed absorption in the gravity picture as a conversion of closed strings living far away from the brane into closed strings which are localized near the brane
• He then argued that the success of absorption calculations imply that in an appropriate limit the closed string modes near the brane must be completely equivalent to the gauge theory quanta moving purely along the brane.
• A gauge field theory in some number of dimensions is therefore equivalent to a theory containing gravity in a higher number of dimensions. • We do not have the explicit constructions like the lower dimensional examples. Nevertheless this does constitute a CONCRETE REALIZATION OF HOLOGRAPHY – and perhaps the most useful one yet.
AdS/CFT • The holographic correspondence is understood well when the theory of gravity lives in a space-time whose non-compact part is asymptotically anti-de-Sitter (AdS). AdS space has a scale symmetry - In fact conformal symmetry (Streching different parts of space time by different amounts)
• The field theory lives on the boundary of AdS . (Gubser,Klebanov & Polyakov; Witten) • This field theory should, therefore, be also conformally invariant time – a Conformal Field Theory Closed Strings In bulk r Gauge Theory On boundary In fact, the simplest situation involves the space-time - the bulk theory is something called IIB superstring theory – and the field theory on the boundary is a highly supersymmetric version of Yang-Mills called N=4 Super-Yang-Mills theory.
The parameters of these two theories are related as follows = = = = length scale of the AdS space string coupling constant Yang-Mills coupling constant (square) string length
• The relationship theory of strings, • The relationship tells us that the classical limit of the is the limit tells us that in this limit, we can still get something non-trival if • This is precisely ‘t Hooft’s large N limit ! • The two lines of thinking which led to holography have now converged nicely. And is indeed the string coupling constant. • The radial direction of AdS, as well as the 5 angles on the have emerged from the large-N gauge theory.
• This would not have been very useful – if it were not for another feature of the relationship Suppose we take the large-N limit as well as the limit of strong ‘t Hooft coupling, • We will have a very weak curvature of the AdS space • In this circumstance, the low energy approximation of closed string theory – General Relativity – should be reliable. • Now there is a direct duality between gauge theory and classical gravity.
The Dictionary • The vacuum of the Yang-Mills theory corresponds to pure AdS spacetime with no excitations – and no deformations. • For each field in the bulk, there is a dual operator in the field theory on the boundary Scalar U(1) Gauge graviton Scalar Conserved Current Energy-Momentum tensor
• All these bulk fields vanish in pure AdS • Deformations are obtained by solving the bulk equations of motion. For example a deformation of a massive scalar would have the following behavior near the boundary • The dual description will be a conformal field theory which is deformed by a source • While the function determines the expectation value of the operator • There are of course similar formulae for other bulk fields.
Excited States One way to have a nonzero with a vanishing source is to have the dual field theory in an excited state. horizon An equilibrium thermal state of the boundary theory with temperature T is dual to a black hole in the bulk – with a Hawking temperature = T boundary
• How about a state with a nonzero charge density for a global charge ? • Recall a conserved current on the boundary corresponds to a gauge field in the bulk – so a charged black hole in the bulk will describe a boundary theory Nonzero temperature Nonzero chemical potential • Furthermore, if we have an extremal black hole in the bulk – a black hole with vanishing Hawking temperature, we describe a zero temperature state with nonzero chemical potential.
GRAVITY FIELD THEORY • The AdS/CFT correspondence is useful in both directions. • It threw valuable light on the information loss problem associated with Hawking radiation by mapping it to a problem in field theory. • However, the field theory is always strongly coupled – so unless there is supersymmetry (or slightly broken supersymmetry) it is difficult to get quantitative results to illuminate issues in gravity. • We will, however, talk about its use for understanding issues in cosmology later – but for now let us explore the other direction – using gravity to understand properties of strongly coupled field theory.
Critical Phenomena • One class of important phenomena in physics which involve conformal symmetry and conformal field theory is Critical Phase transitions. • Critical Phenomena are interesting because they are universal – many different materials behave similarly near the critical point. • Typically this involves another interesting way to obtain a nonzero in the absence of a source - spontaneous symmetry breaking. • For example, holographic superconductors – or more accurately holographic superfluids.
• The setup is a charged black brane in four dimensional AdS space, and a charged scalar field which couples to the gravity and the gauge field. The boundary theory has a global U(1) symmetry. There is a dual operator which is the order parameter for this. horizon boundary • There is always a trivial solution charged black brane. This clearly has • This is the disordered phase. & : this is the
• At low enough temperatures the trivial solution is unstable. • The stable solution has a nonzero localized near the horizon – hairy black hole. (Gubser; Hartnoll, Herzog, Horowitz) This leads to a in the boundary field theory. U(1) is broken horizon boundary The phase transition is critical and mean field.
Exotic Criticality • There have been studies of quantum critical phenomena in both the “top-down” and “bottom-up” approaches. This has led to some novel behaviors. • For example – there are quantum critical points in 2+1 dimensions with Berezinski-Kosterlitz-Thouless behavior. (Jensen, Karch and Son; Iqbal, Liu, Mezei and Si) • And there are indications of fermi surfaces and strange metal behavior. (Liu, McGreevy and Vegh;……)
Dynamics Perhaps the main usefulness of the holographic approach is that studying dynamics is not conceptually very different from statics. In fact, some of the most interesting results in this subject pertain to transport coefficients. Suppose we turn on a source in the boundary theory at some time. This is simply a time-dep boundary condition for the dual bulk field. Solving for the bulk solution then allows a calculation of the response. Disturbance falls into the black hole – this is perceived as dissipation in the boundary theory
The universal viscosity/entropy • At the level of linear response, this calculation is that of the absorption cross-section of the appropriate wave by the black hole. • For example, turning on a source for the energy momentum tensor leads to shear viscosity – the corresponding bulk disturbance is a graviton with polarizations along the boundary. • In usual Einstein-Hilbert action, this graviton behaves like a massless minimally coupled scalar.
• The absorption cross-section for such a mode is universal - it does not depend on the details of the metric (S.R.D., G. Gibbons and S.D. Mathur) • Thus, • If one works out the numerical factors, one gets • This is a remarkable prediction for any field theory which has a gravity dual – regardless of its details (Kovtun, Son and Starinets). • This number is much smaller than any known liquid. ??
• Intriguingly, the quark-gluon liquid produced at RHIC has a ratio which is also quite small – and there is no theoretical framework in QCD which explains this. • N = 4 Yang-Mills at zero temperature is very different from QCD at zero temperature. However at high temperatures – beyond the de-confinement transition – these two theories are qualitatively not so different. • This has led many people to believe that N = 4 may not be a bad zeroth order approximation to QCD in this regime. In fact several properties of the quark-gluon liquid seem to find qualitative explanations in a gravity dual.
• The description of hydrodynamics by gravity is not restricted to the linear regime. In fact there is a beautiful connection between Einstein’s equations and the equations of non-linear hydrodynamics which by itself is worth exploring. (Bhattacharyya, Hubeny, Rangamani and Minwalla) Such a connection first came up in the “membrane paradigm” – T. Damour Perhaps even more interestingly the same methods can be applied for situations very far from equilibrium – situations in which conventional theoretical tools are rare.
Holographic Quantum Quench • One such problem is quantum quench. • This is the behavior of a quantum system with a coupling or an external parameter which varies with time • Starting with some equilibrium state, e.g. the ground state at zero temperature, what is the nature of the final state – and how does this approach the final state – a standard question in many areas of physics – e.g. cosmology
• Recent years have witnessed vigorous activity because this kind of question has now become experimentally accessible in cold atom systems. • Among other things this involves two important questions (1) Does the final state resemble a thermal state – of so in what sense ? (2) If this quench happens across a critical point, does the subsequent behavior of the system have universal features ? Unfortunately such far-from-equilibrium behavior of strongly coupled quantum systems is out of reach of conventional theoretical methods – though there are some exceptions e.g. in 1+1 dimensions (Calabrese and Cardy) CAN HOLOGRAPHY HELP ?
• In the holographic context a time dependent coupling is simply a time dependent boundary condition for the dual bulk field – this sends out a disturbance in the interior
• Indeed this leads to thermalization . • In the holographic context, this manifests itself as formation of a horizon. The most common situation involves black hole formation in the bulk . (Chesler & Yaffe; Bhattacharya & Minwalla) • In other situations thermalization is signaled by apparent horizons. (S.R.D., T. Nishioka and T. Takayanagi)
• This problem becomes particularly interesting when the coupling passes through a critical point. Then, even with an initially slow coupling, adiabaticity breaks down in a universal fashion. • For example, consider a magnet exactly at the critical temperature in the presence of a time dependent external magnetic field. • When the field crosses zero, the system is thrown out of adiabatic evolution. How does the system relax to the new ground state ? H t T adiabatic
• For such slow quenches, Kibble and Zurek conjectured a set of universal scaling properties, e.g. the order parameter in the region close to the critical point behaves as • Here the external parameter (e.g. magnetic field ) crosses the critical point in a linear fashion • The behavior is universal – determined by the correlation length critical exponent and the dynamical critical exponent . • Unlike equilibrium critical phenomena there is no conceptual framework like Renormalization Group which explains this kind of universality.
• We studied this issue in several models with holographic critical points both at zero and non-zero temperatures. • It turns out this provides an analytic understanding of KibbleZurek scaling. • The scaling behavior arises because (1) The bulk scalar has a zero mode at the critical point. (2) In the critical region there is a novel small expansion I in fractional powers of . (3) To leading order the dynamics is dominated by the e zero mode. • This leads to a simple equation for the dynamics of the order parameter. The equation has scaling solutions [P. Basu and S.R.D. (2011); P. Basu, D. Das, S.R.D. & T. Nishioka(2012); P. Basu, D. Das, S.R.D. and K. Sengupta (2013)]
Dynamics of order parameter Scaling as a function of the quench rate
• There are some new results for fast holographic quench which exhibit scaling behavior. (Buchel, Lehner, Myers & Niekerk) • Suppose we perturb a CFT by a relevant operator with dimension • Where is e.g. of the form • Then the one point function scales as • Once again the result is universal and holds for arbitrary protocols so long as the behavior is linear near t=0.
Big Bang / Big Crunch • Remarkably the same setup also allows us to investigate a rather different problem – the problem of cosmological singularities. • These are space-like regions of very high curvatures. Einstein equations cannot be used to evolve the system in time across such regions – as at the Big Bang. • Space-like singularities are puzzling – they are not things. They cannot be resolved by trying to find objects which replace them. • They just happen to you. t x
Big Bang / Big Crunch • Can gauge-gravity duality help ? • This problem has been studied by various groups in various ways • (Hertog & Horowitz; A.Awad, S.R.D., A. Ghosh, J. Michelson, K. Narayan, J.H. Oh & S. Trivedi; Craps, Hertog & Turok)
• Suppose we are in global AdS , and the ‘t Hooft coupling of the dual gauge theory is time dependent . • At early times, the ‘t Hooft coupling is large – so there is a nice gravity description. • At intermediate times, the coupling becomes small – the bulk curvatures become large – and this is physically like a space-like singularity. • Can we use the dual gauge theory to ask if there is a smooth time evolution ?
• Suppose we are in global AdS , and the ‘t Hooft coupling of the dual N = 4 gauge theory is time dependent . • At early times, the ‘t Hooft coupling is large – so there is a nice gravity description. • At intermediate times, the coupling becomes small – the bulk curvatures become large – and this is physically like a spacelike singularity. • Can we use the dual gauge theory to ask if there is a smooth time evolution ? ?? High Curvature AdS
• It turns out that in some situations, it is possible to argue that the gauge theory indeed allows a smooth time evolution through this “singularity”. • We have not yet been, however, able to figure out the precise nature of the state at late times, though we can argue that big black holes are not formed. • It is also unclear whether the present knowledge of the AdS/CFT dictionary is sufficient to calculate physically interesting quantities like the fluctuation spectrum at late times.
• The relationship between gauge theory and gravity has thrown valuable light on a major mystery in gravity – the problem of information loss in black holes. • If we are successful, we will get the first true insight into another major mystery in gravity – the problem of space-like singularities.
THANK YOU

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Colloquium2013

  • 1.
  • 2.
    Is Gravity differentfrom other forces ? • The description of gravitational forces due to Einstein is rather different from that of the other known forces. • Forces like electromagnetism, weak interactions or strong interactions are due to exchange of particles – described astoundingly well by Quantum Mechanics of fields • All this happens in a given, fixed, space-time.
  • 3.
    • On theother hand, gravity hardly appears to be a force of this sort. • Rather, it is the reaction of any object with energy to the curvature of space-time – in General Relativity space time itself is dynamical.
  • 4.
    • Nevertheless, insome sense, gravitational forces can be thought of arising from exchange of particles – these are called “gravitons”. Instead of spin 1 they have spin 2. • Beginning in the 1930’s, physicists have tried to make sense of a theory of “quantum gravity” by applying the standard rules of quantum mechanics to gravitons. • The result has been rather discouraging.
  • 5.
    • Now-a-days wedo much better – we think of gravity coming from exchange of closed strings rather than particles. • These are rather tiny – so usual experiments should perceive them as point-like objects • The lowest (quadrupole ) mode of oscillation is the graviton. (Yoneya; Scherk and Schwarz, 1974) • The presence of an infinite number of higher modes, however, smooth out the high energy problems faced in quantum gravity based on spin-2 objects alone. – (Green and Schwarz, 1984)
  • 6.
    • In recentyears, this has led to a rather different picture of gravity. • In this picture, gravitational force is -in a sense -not fundamental. • Rather, it is an effective description of a theory pretty much like the theories of strong and electro-weak interactions – a theory which lives in a fixed space-time. GRAVITY IS AN EMERGENT PHENOMENON.
  • 7.
    There are twokey features of this connection (1) This non-gravitational theory lives in a lower number of dimensions – as if providing a hologram of what happens in the “bulk”. In some cases the hologram is on the boundary of the space-time. (2) When the hologram is highly quantum, its equivalent description in terms of gravity is classical – just Einstein’s equations
  • 8.
    GAUGE THEORIES • Forreasons which are not completely clear, nature prefers theories which are characterized by gauge invariance. • The simplest example is electrodynamics. The observables here are the electric field and the magnetic field - and at the classical level that is all you need. You can describe the motion of charged particles and electromagnetic waves. • This is not quite true at the quantum level.
  • 9.
    Aharanov Bohm Effect •Charged particles which go around an infinitely thin solenoid along different paths acquire different phases – even though the particles move in a field free region. • Somehow one has to introduce potentials ,
  • 10.
    • But thisis a redundant description • Potentials which are related by gauge transformations represent the same physics – and lead to the same and • This looks strange – why not use gauge invariant variables ? • We saw that the obvious ones don’t suffice – it turns out that we need to consider non-local variables called Wilson loops • For the Aharanov-Bohm problem the particle is sensitive to this quantity – which is non-zero due to Stokes’ Theorem
  • 11.
    Quarks, Mesons andBaryons • The need to understand gauge invariant variables became pronounced in the theory of strong interactions. • Hadrons are made of quarks which interact with each other by exchanging gluons. • Quarks have charge – they also have color. • However they are permanently confined – the p physical objects we see are neutrons, protons a and mesons – these are colorless, or color I singlets. Can we express physics in terms of mesons, baryons and glueballs ?
  • 12.
    Strings • Before theadvent of QCD – the theory of quarks and gluons it was realized that hadrons behave like strings – this was the origin of string theory (Nambu, Nielsen, Susskind). • After QCD it was soon realized that strings are flux tubes. • gauge strings be • diems t  p 0 Ties up nicely with the idea that the gauge invariant observables are in fact Wilson loop like objects - these must the strings of QCD But there was one puzzle – what is the dimensionless number which serves as the string coupling ?
  • 13.
    In quantum electrodynamics,there is a cloud of virtual charges around an electron, which makes it a rather fuzzy extended object . Yet we can treat it as point-like. • This is because there is a small coupling constant in QED • What could be the small number for strings ? QCD with massless quarks does not have a dimensionless parameter !
  • 14.
    Large N limit •‘t Hooft came up with a surprising answer to this question. • Quarks have 3 colors – technically this means that the gauge group of QCD is SU(3). • Consider instead a theory with colors, and the special limit • ‘t Hooft showed that in this large N expansion the coupling which characterizes interaction of strings is given by • In fact the limit is a classical limit
  • 15.
    The Isotropic Oscillator •To see why a large-N limit is a classical limit consider a familiar problem in elementary quantum mechanics – the isotropic harmonic oscillator, in N dimensions. • We want to look at the singlet sector of the theory – i.e zero angular momentum • So we use spherical polar coordinates in N dimensions and perform a rescaling of the wave-function
  • 16.
    • The Schrodingerequation now becomes • When is large this is like a one dimensional problem with an effective potential • Note that the Planck constant of this theory is in fact • Therefore the large limit is like a classical limit. In fact the full quantum problem is very well approximated by the solution of the classical equations of motion.
  • 17.
    But why higherdimensions ? • This aspect is still a bit mysterious. • However the essential physics can be understood from another quantum mechanics problem – that of a hermitian matrix • The singlet sector of this model is described in terms of the eigenvalues of the matrix • These eigenvalues can be thought of as the coordinates of non-relativistic fermions moving along a one dimensional line
  • 18.
    • When is largethere are lots of lots of fermions – it is then useful to think of the problem in terms of a density of fermions . • But this is a field in 1+1 dimensions. • We started out with a theory in 0+1 dimensions with degrees of freedom. • Now we find that for large it may be expressed as a theory in 1+1 dimensions. • This fact was known for a very long time – at least since 1980
  • 19.
    • When theproblem was revisited in early 1990’s there were several surprises (1) The theory is secretly relativistic – and the interactions between blobs of fermions are local in space and time. (S.R.D. and A. Jevicki, 1990) (2) This theory is in fact a String Theory (Gross & Milkovic; Brezin & Kazakov; Gross and Klebanov) – rewriting in terms of densities explains why it is so. (3) The interaction between blobs in fact include gravitational forces (Polchinski and Naatsumme). A GRAVITATIONAL THEORY IS DESCRIBED BY A NONGRAVITATIONAL THEORY IN LOWER DIMENSIONS HOLOGRAPHY
  • 20.
    • Of coursegravity in 1+1 dimensions is rather boring – though not trivial. • Are there higher dimensional examples of this ? • The answer came from a rather different line of thinking.
  • 21.
    Black Holes • In1970’s the work of Bekenstein and Hawking showed that black holes in fact radiate, and may be considered as a thermodynamic object with a characteristic temperature and an entropy. • The entropy formula is rather intriguing : Black holes do not appear to be extensive in the usual sense.
  • 22.
    Bekenstein Bound • Bekensteinpushed this a bit more. • He argued that once we take into account of gravity and black holes, the maximum possible entropy of anything inside a large region is proportional not to the volume, but to the area of the boundary
  • 23.
    The Holographic Principle •To “explain” this, ‘t Hooft and Susskind came up with a surprising interpretation of this bound. • They proposed the “Holographic Principle” Gravitational physics in a d+1 dimensional world is completely equivalent to non-gravitational physics in (d-1)+1 dimensions. • The latter may be thought to live on the “boundary” “Ordinary” non-gravitational Physics on boundary Gravity in “Bulk”
  • 24.
    From Black Holesto Holography • A concrete realization of this principle came from thinking about black holes in string theory. • String Theory provides the microscopic description of a large class of black holes. They look like objects extended in some internal dimensions – D branes. • The low energy excitations of D-branes are described by a non-abelian gauge theory which lives on the brane
  • 25.
    From Black Holesto Holography • A concrete realization of this principle came from thinking about black holes in string theory. • String Theory provides the microscopic description of a large class of black holes. They look like objects extended in some internal dimensions – D branes. • The low energy excitations of D-branes are described by a non-abelian gauge theory which lives on the brane
  • 26.
    • Using thismicroscopic picture, thermodynamic properties of these black holes – as well properties of Hawking radiation could be reproduced (Strominger & Vafa; Callan & Maldacena, S.R.D. and S.D. Mathur; Maldacena and Strominger; Dhar, Mandal, Wadia). • In this picture, when a particle falls into a black hole, it can be converted into these gauge field quanta – this appears as absorption by the black hole. • Hawking radiation is the opposite process
  • 27.
    • Maldacena provideda key insight into this absorption process • He viewed absorption in the gravity picture as a conversion of closed strings living far away from the brane into closed strings which are localized near the brane
  • 28.
    • Maldacena provideda key insight into this absorption process • He viewed absorption in the gravity picture as a conversion of closed strings living far away from the brane into closed strings which are localized near the brane
  • 29.
    • He thenargued that the success of absorption calculations imply that in an appropriate limit the closed string modes near the brane must be completely equivalent to the gauge theory quanta moving purely along the brane.
  • 30.
    • A gaugefield theory in some number of dimensions is therefore equivalent to a theory containing gravity in a higher number of dimensions. • We do not have the explicit constructions like the lower dimensional examples. Nevertheless this does constitute a CONCRETE REALIZATION OF HOLOGRAPHY – and perhaps the most useful one yet.
  • 31.
    AdS/CFT • The holographiccorrespondence is understood well when the theory of gravity lives in a space-time whose non-compact part is asymptotically anti-de-Sitter (AdS). AdS space has a scale symmetry - In fact conformal symmetry (Streching different parts of space time by different amounts)
  • 32.
    • The fieldtheory lives on the boundary of AdS . (Gubser,Klebanov & Polyakov; Witten) • This field theory should, therefore, be also conformally invariant time – a Conformal Field Theory Closed Strings In bulk r Gauge Theory On boundary In fact, the simplest situation involves the space-time - the bulk theory is something called IIB superstring theory – and the field theory on the boundary is a highly supersymmetric version of Yang-Mills called N=4 Super-Yang-Mills theory.
  • 33.
    The parameters ofthese two theories are related as follows = = = = length scale of the AdS space string coupling constant Yang-Mills coupling constant (square) string length
  • 34.
    • The relationship theoryof strings, • The relationship tells us that the classical limit of the is the limit tells us that in this limit, we can still get something non-trival if • This is precisely ‘t Hooft’s large N limit ! • The two lines of thinking which led to holography have now converged nicely. And is indeed the string coupling constant. • The radial direction of AdS, as well as the 5 angles on the have emerged from the large-N gauge theory.
  • 35.
    • This wouldnot have been very useful – if it were not for another feature of the relationship Suppose we take the large-N limit as well as the limit of strong ‘t Hooft coupling, • We will have a very weak curvature of the AdS space • In this circumstance, the low energy approximation of closed string theory – General Relativity – should be reliable. • Now there is a direct duality between gauge theory and classical gravity.
  • 36.
    The Dictionary • Thevacuum of the Yang-Mills theory corresponds to pure AdS spacetime with no excitations – and no deformations. • For each field in the bulk, there is a dual operator in the field theory on the boundary Scalar U(1) Gauge graviton Scalar Conserved Current Energy-Momentum tensor
  • 37.
    • All thesebulk fields vanish in pure AdS • Deformations are obtained by solving the bulk equations of motion. For example a deformation of a massive scalar would have the following behavior near the boundary • The dual description will be a conformal field theory which is deformed by a source • While the function determines the expectation value of the operator • There are of course similar formulae for other bulk fields.
  • 38.
    Excited States One wayto have a nonzero with a vanishing source is to have the dual field theory in an excited state. horizon An equilibrium thermal state of the boundary theory with temperature T is dual to a black hole in the bulk – with a Hawking temperature = T boundary
  • 39.
    • How abouta state with a nonzero charge density for a global charge ? • Recall a conserved current on the boundary corresponds to a gauge field in the bulk – so a charged black hole in the bulk will describe a boundary theory Nonzero temperature Nonzero chemical potential • Furthermore, if we have an extremal black hole in the bulk – a black hole with vanishing Hawking temperature, we describe a zero temperature state with nonzero chemical potential.
  • 40.
    GRAVITY FIELD THEORY • TheAdS/CFT correspondence is useful in both directions. • It threw valuable light on the information loss problem associated with Hawking radiation by mapping it to a problem in field theory. • However, the field theory is always strongly coupled – so unless there is supersymmetry (or slightly broken supersymmetry) it is difficult to get quantitative results to illuminate issues in gravity. • We will, however, talk about its use for understanding issues in cosmology later – but for now let us explore the other direction – using gravity to understand properties of strongly coupled field theory.
  • 41.
    Critical Phenomena • Oneclass of important phenomena in physics which involve conformal symmetry and conformal field theory is Critical Phase transitions. • Critical Phenomena are interesting because they are universal – many different materials behave similarly near the critical point. • Typically this involves another interesting way to obtain a nonzero in the absence of a source - spontaneous symmetry breaking. • For example, holographic superconductors – or more accurately holographic superfluids.
  • 42.
    • The setupis a charged black brane in four dimensional AdS space, and a charged scalar field which couples to the gravity and the gauge field. The boundary theory has a global U(1) symmetry. There is a dual operator which is the order parameter for this. horizon boundary • There is always a trivial solution charged black brane. This clearly has • This is the disordered phase. & : this is the
  • 43.
    • At lowenough temperatures the trivial solution is unstable. • The stable solution has a nonzero localized near the horizon – hairy black hole. (Gubser; Hartnoll, Herzog, Horowitz) This leads to a in the boundary field theory. U(1) is broken horizon boundary The phase transition is critical and mean field.
  • 44.
    Exotic Criticality • Therehave been studies of quantum critical phenomena in both the “top-down” and “bottom-up” approaches. This has led to some novel behaviors. • For example – there are quantum critical points in 2+1 dimensions with Berezinski-Kosterlitz-Thouless behavior. (Jensen, Karch and Son; Iqbal, Liu, Mezei and Si) • And there are indications of fermi surfaces and strange metal behavior. (Liu, McGreevy and Vegh;……)
  • 45.
    Dynamics Perhaps the mainusefulness of the holographic approach is that studying dynamics is not conceptually very different from statics. In fact, some of the most interesting results in this subject pertain to transport coefficients. Suppose we turn on a source in the boundary theory at some time. This is simply a time-dep boundary condition for the dual bulk field. Solving for the bulk solution then allows a calculation of the response. Disturbance falls into the black hole – this is perceived as dissipation in the boundary theory
  • 46.
    The universal viscosity/entropy •At the level of linear response, this calculation is that of the absorption cross-section of the appropriate wave by the black hole. • For example, turning on a source for the energy momentum tensor leads to shear viscosity – the corresponding bulk disturbance is a graviton with polarizations along the boundary. • In usual Einstein-Hilbert action, this graviton behaves like a massless minimally coupled scalar.
  • 47.
    • The absorptioncross-section for such a mode is universal - it does not depend on the details of the metric (S.R.D., G. Gibbons and S.D. Mathur) • Thus, • If one works out the numerical factors, one gets • This is a remarkable prediction for any field theory which has a gravity dual – regardless of its details (Kovtun, Son and Starinets). • This number is much smaller than any known liquid. ??
  • 48.
    • Intriguingly, thequark-gluon liquid produced at RHIC has a ratio which is also quite small – and there is no theoretical framework in QCD which explains this. • N = 4 Yang-Mills at zero temperature is very different from QCD at zero temperature. However at high temperatures – beyond the de-confinement transition – these two theories are qualitatively not so different. • This has led many people to believe that N = 4 may not be a bad zeroth order approximation to QCD in this regime. In fact several properties of the quark-gluon liquid seem to find qualitative explanations in a gravity dual.
  • 49.
    • The descriptionof hydrodynamics by gravity is not restricted to the linear regime. In fact there is a beautiful connection between Einstein’s equations and the equations of non-linear hydrodynamics which by itself is worth exploring. (Bhattacharyya, Hubeny, Rangamani and Minwalla) Such a connection first came up in the “membrane paradigm” – T. Damour Perhaps even more interestingly the same methods can be applied for situations very far from equilibrium – situations in which conventional theoretical tools are rare.
  • 50.
    Holographic Quantum Quench •One such problem is quantum quench. • This is the behavior of a quantum system with a coupling or an external parameter which varies with time • Starting with some equilibrium state, e.g. the ground state at zero temperature, what is the nature of the final state – and how does this approach the final state – a standard question in many areas of physics – e.g. cosmology
  • 51.
    • Recent yearshave witnessed vigorous activity because this kind of question has now become experimentally accessible in cold atom systems. • Among other things this involves two important questions (1) Does the final state resemble a thermal state – of so in what sense ? (2) If this quench happens across a critical point, does the subsequent behavior of the system have universal features ? Unfortunately such far-from-equilibrium behavior of strongly coupled quantum systems is out of reach of conventional theoretical methods – though there are some exceptions e.g. in 1+1 dimensions (Calabrese and Cardy) CAN HOLOGRAPHY HELP ?
  • 52.
    • In theholographic context a time dependent coupling is simply a time dependent boundary condition for the dual bulk field – this sends out a disturbance in the interior
  • 53.
    • Indeed thisleads to thermalization . • In the holographic context, this manifests itself as formation of a horizon. The most common situation involves black hole formation in the bulk . (Chesler & Yaffe; Bhattacharya & Minwalla) • In other situations thermalization is signaled by apparent horizons. (S.R.D., T. Nishioka and T. Takayanagi)
  • 54.
    • This problembecomes particularly interesting when the coupling passes through a critical point. Then, even with an initially slow coupling, adiabaticity breaks down in a universal fashion. • For example, consider a magnet exactly at the critical temperature in the presence of a time dependent external magnetic field. • When the field crosses zero, the system is thrown out of adiabatic evolution. How does the system relax to the new ground state ? H t T adiabatic
  • 55.
    • For suchslow quenches, Kibble and Zurek conjectured a set of universal scaling properties, e.g. the order parameter in the region close to the critical point behaves as • Here the external parameter (e.g. magnetic field ) crosses the critical point in a linear fashion • The behavior is universal – determined by the correlation length critical exponent and the dynamical critical exponent . • Unlike equilibrium critical phenomena there is no conceptual framework like Renormalization Group which explains this kind of universality.
  • 56.
    • We studiedthis issue in several models with holographic critical points both at zero and non-zero temperatures. • It turns out this provides an analytic understanding of KibbleZurek scaling. • The scaling behavior arises because (1) The bulk scalar has a zero mode at the critical point. (2) In the critical region there is a novel small expansion I in fractional powers of . (3) To leading order the dynamics is dominated by the e zero mode. • This leads to a simple equation for the dynamics of the order parameter. The equation has scaling solutions [P. Basu and S.R.D. (2011); P. Basu, D. Das, S.R.D. & T. Nishioka(2012); P. Basu, D. Das, S.R.D. and K. Sengupta (2013)]
  • 57.
    Dynamics of orderparameter Scaling as a function of the quench rate
  • 58.
    • There aresome new results for fast holographic quench which exhibit scaling behavior. (Buchel, Lehner, Myers & Niekerk) • Suppose we perturb a CFT by a relevant operator with dimension • Where is e.g. of the form • Then the one point function scales as • Once again the result is universal and holds for arbitrary protocols so long as the behavior is linear near t=0.
  • 59.
    Big Bang /Big Crunch • Remarkably the same setup also allows us to investigate a rather different problem – the problem of cosmological singularities. • These are space-like regions of very high curvatures. Einstein equations cannot be used to evolve the system in time across such regions – as at the Big Bang. • Space-like singularities are puzzling – they are not things. They cannot be resolved by trying to find objects which replace them. • They just happen to you. t x
  • 60.
    Big Bang /Big Crunch • Can gauge-gravity duality help ? • This problem has been studied by various groups in various ways • (Hertog & Horowitz; A.Awad, S.R.D., A. Ghosh, J. Michelson, K. Narayan, J.H. Oh & S. Trivedi; Craps, Hertog & Turok)
  • 61.
    • Suppose weare in global AdS , and the ‘t Hooft coupling of the dual gauge theory is time dependent . • At early times, the ‘t Hooft coupling is large – so there is a nice gravity description. • At intermediate times, the coupling becomes small – the bulk curvatures become large – and this is physically like a space-like singularity. • Can we use the dual gauge theory to ask if there is a smooth time evolution ?
  • 62.
    • Suppose weare in global AdS , and the ‘t Hooft coupling of the dual N = 4 gauge theory is time dependent . • At early times, the ‘t Hooft coupling is large – so there is a nice gravity description. • At intermediate times, the coupling becomes small – the bulk curvatures become large – and this is physically like a spacelike singularity. • Can we use the dual gauge theory to ask if there is a smooth time evolution ? ?? High Curvature AdS
  • 63.
    • It turnsout that in some situations, it is possible to argue that the gauge theory indeed allows a smooth time evolution through this “singularity”. • We have not yet been, however, able to figure out the precise nature of the state at late times, though we can argue that big black holes are not formed. • It is also unclear whether the present knowledge of the AdS/CFT dictionary is sufficient to calculate physically interesting quantities like the fluctuation spectrum at late times.
  • 64.
    • The relationshipbetween gauge theory and gravity has thrown valuable light on a major mystery in gravity – the problem of information loss in black holes. • If we are successful, we will get the first true insight into another major mystery in gravity – the problem of space-like singularities.
  • 65.