Holography lectures by Joan Simon
This is an introductory lecture series for PhD students on Holography given by Joan Simon
- GR as an effective gauge field theory
- Classical black hole thermodynamics
- Unruh effect stressing entanglement in QFT
- Quantum black holes leading to the holographic principle
- Motivating AdS/CFT (no string theory)
- Formulating AdS/CFT
Higgs Centre Seminar Room (JCMB)
We discussed GR as an effective gauge field theory. This allowed us to
introduce the general principles of EFT
apply them to GR
revisit GR is non-renormalisable
introduce the ADM formalism (hamiltonian formalism)
rederive the hamiltonian in general covariant theories vanishes on-shell, since it generates a gauge symmetry, unless its conjugate vector field does have support at the boundary of your manifold
use such boundary contributions to reproduce the mass and angular momentum conserved charges in GR
We continued our discussion on the gauge nature of GR :
stressing the difference between local and large diffeomorphisms
the latter giving rise to asymptotic symmetries
Having stressed the problem of finding a UV completion of GR remains opened, we listed two properties we believe such theory must satisfy :
there are no local gauge invariant observables
the graviton can not be a composite particle (Weinberg- Witten theorem)
This allowed us to stress the lack of locally conserved energy momentum stress tensor in general covariant theories.
To include further items in the above list, we said we would study black holes in these meetings. We started by
listing main features of these solutions
using the RN black hole as a toy model, viewing it as a 2-parameter space of solutions, we showed by explicit calculation that the variation of the mass and charge satisfy the first of thermodynamics if the area of the event horizon is interpreted as some entropy
in fact, we pointed out some formal analogy between the classical laws of black holes and the laws of thermodynamics satisfied by matter systems.
In the first part of the lecture, we noticed that requiring the euclidean continuation of the black hole metric to be smooth at the "event horizon" required euclidean time to be compact. Furthermore, requiring the periodicity at space like infinity to match the formal temperature of the black hole fixed the proportional between area and entropy to be 1/4, reproducing Hawking's discovery.
In the second part of the lecture, we introduced the covariant phase space applied to generally covariant theories. We pointed out connections with ADM and calculation of conserved charges at null infinity (BMS-like discussions). Even though the formalism is far more general, we derived the first law of black holes mechanics for pure GR by picking a Cauchy surface between the event horizon and space like infinity. This allowed us to identify the surface gravity as the scalar multiplying the area of the event horizon. We discussed the conditions under which such surface gravity is constant on the horizon hyper surface (0th law of black hole thermodynamics) and how the latter allows to identify the term in the first law.
We want to understand whether the temperature parameter appearing in our GR analysis has any physical meaning. We discussed two ways to describe thermal system in matter systems :
- as thermal density matrices
- as reduced density matrices in open systems
We noticed that in a quantum system involving two harmonic oscillators, there could be more than one "vacuum", using squeezed states. We want to explore whether a similar feature holds in black hole physics. We finished this part by making some remarks on the role of quantum entanglement in these discussions and how quantum correlations diverge when the insertion of these operators tends to be the same. We argued this QFT feature must be responsible for why entanglement entropy diverges in such continuum theories.
In the second part, we reviewed how transition amplitudes and ground state wave functions can be computed using euclidean path integrals in quantum mechanics. We then extended these methods to QFT.
Building on our previous lecture, we described how to compute density matrices and their matrix elements using euclidean path integrals, reminding ourselves how to compute the partition function at finite temperature.
In the second part of the lecture, we reviewed the geometry of constant proper accelerated observers in special relativity, the appearance of observer dependent horizons and how to choose coordinates describing the so called right and left Rindler wedges. We argued the Hilbert space for a single scalar field in Minkowski factorises into the Hilbert space of the right and left wedges, based on locality considerations.
Armed with path integrals and the discussion on the Rindler geometry, we computed the wave function for the Minkowski ground state and proved that it can be written as a thermo-field double state, so that when tracing out over one of the Rindler wedges, we obtain a thermal density matrix with a very precise temperature, i.e. the Unruh temperature.
We close the lecture by making some remarks on the similarities and the challenges we will face to extend this result for the black hole.
We extended Rindler's discussion to the two-sided maximally extended black hole.
We outlined the scattering problem for the black hole formed by gravitational collapse, mentioned the geometric optics approximation used by Hawking to analytically solve the gluing problem to compute the wave functions of the scalar field with the relevant boundary conditions and gave the final answer.
We ended up with a heuristic calculation estimating the time scale for the evaporation of a black hole and the kind of arguments leading Hawking to conclude gravity violates unitarity in quantum mechanics.
We asked whether black holes should be considered as quantum objects :
thermodynamic considerations of matter in the presence of black holes suggest the second law gets violated unless we include the Bekenstein-Hawking area term. Even if we do, the entropy of matter systems should be bounded by its energy and size (Bekentein's bound). This triggered the generalised second law.
revisiting the process of black hole formation, we reach the conclusion the entropy of matter systems in some bounded region of space should be smaller of equal than the entropy we assign to a black hole in the same region.
When combining these arguments with standard arguments in local QFT, we encounter some difficulty with unitarity. For those willing to save unitarity and include quantum physics in gravity, this triggers the holographic principle.
We finished the lecture giving evidence that such principle encodes dynamical information. Indeed, one can derive Einstein's equations from the holographic principle, the equivalence principle and the use of Clausius relation in thermodynamics, following Jacobson's argument.
The partition function plays a key role in QM and QFT to extract different observables. If gravity requires a quantum treatment, how do we compute a gravitational partition function ? More importantly, if black holes are thermal objects, can we compute a partition function that reproduces these results from first principles ?
Following Gibbons and Hawking, we consider a saddle point semiclassical approximation for flat space thermodynamics. After addition of countqerterms, we reproduce the relevant thermodynamics this way. We pointed out important issues such as existence of instabilities or the Hagedorn transition in string theory.
If these ideas are correct, they should apply more generically. We introduced de Sitter and checked the euclidean action reproduces the entropy of an static patch by explicit calculation, though we acknowledge we lack a UV understanding of this physical result.
We finished by outlining the two arguments that we shall follow to motivate the AdS/CFT : large N and decoupling limits.
We reviewed the large N limit in physical theories with matrix valued degrees of freedom. In the 't Hooft limit, we learn the 1/N expansion is controlled by the topology of the Feynman diagrams.
Correlation functions of single trace operators factorise, at leading order, suggesting these behave like classical point particles where 1/N can be interpreted as an effective Planck's constant.
In this lecture, we argue that the large N limit of matrix valued degrees of freedom can be interpreted as a strong theory. This is achieved by observing the inclusion of interactions in a quantum mechanical consistent way in string theory is also controlled by the topology of the closed 2d surfaces.
Mathematically, the perturbative string amplitudes with external string states have the same expansion as the connected correlators of single trace operators in the matrix model with the identification , external strings as single trace operators and summing over the genus-h of the string world-sheet is equivalent as summing over Feynman diagrams of genus-h.
This relation is fairly general, but is not constructive, i.e. it does not identify the relevant string theory capturing the expansion of a given gauge theory. We close with some remarks involving the holographic principle suggesting to consider string theory in extra dimensions as the dual to QCD in d=4. In particular, if the gauge theory is conformal, the on-shell gravity metric respecting the global symmetries of the gauge theory must be AdS.
We introduced the main logic and ideas behind the decoupling limits (low energy limits) used to argue the existence of AdS/CFT.
First, we explored these under arguably very general and reasonable conditions. Once more, as the physical system comes close to being extremal, i.e. scale invariant, some AdS geometry emerges in the gravitational description of the system.
Second, we discussed the conditions under which gravity is weakly coupled and which CFTs have a chance to have gravity duals and, if so, when they may be described by Einstein's like theories.
Finally, we went through the decoupling argument in a specific example borrowing some statements from string theory in order to have precise relations between microscopic parameters that may be useful in future meetings.
Before formulating the AdS/CFT, we reviewed the geometry of AdS, both in Poincare patch and global AdS, identified its timelike conformal boundary and its isometries, commenting on how these act geometrically.
We then moved to describe string theory in these backgrounds, stressing the difference between classical string theory and classical "gravity" theory, where the latter only keeps the massless closed string modes.
We reviewed some CFT basics, including the operator-state correspondence, and concluded reminding the matching of parameters between the CFT and the string theory on AdS.
As the first consequence of the duality, we stressed the UV/IR connection between bulk and boundary, both at zero and finite temperatures, giving some heuristic evidence for the extra dimension in AdS to provide some geometerisation of the energy scale in the boundary theory.
In the last part of the lecture, we ask for the dual of CFT operators. We guessed these should be bulk operators. To identify which ones, we deform the boundary CFT by one of such operators, and ask for the meaning of such operation in the bulk. Using as an example and its map to the string coupling, which is given by the expectation value of the dilation, we learn that the Lagrangian density in YM is dual to the dilation field, whose boundary value provides the change in in the boundary theory. We abstracted this lesson in general and gave two examples of such abstraction for currents and the boundary stress tensor.
We continued our operator-bulk field analysis, by computing the boundary stress tensor in 2d CFTs in terms of bulk geometry defined at the boundary of AdS. We checked our result be deriving the conformal anomaly and reproducing the Brown-Henneaux central charge formula.
We started the quantisation of a scalar field in AdS ...