#### David Wakeham

Doctor of Philosophy in Physics (PhD)

**Research Topic**

Connections between quantum gravity and information

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- Mark Van Raamsdonk

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Aspects of quantum information in quantum field theory and quantum gravity (2019)

In this thesis we discuss applications of quantum information theoretic concepts toquantum gravity and the low-energy regime of quantum field theories.The first part of this thesis is concerned with how quantum information spreadsin four-dimensional scattering experiments for theories coupled to quantum electrodynamicsor perturbative quantum gravity. In these cases, every scattering processis accompanied by the emission of an infinite number of soft photons or gravitons,which cause infrared divergences in the calculation of scattering probabilities.There are two methods to deal with IR divergences: the inclusive and dressedformalisms. We demonstrate that in the late-time limit, independent of the method,the hard outgoing particles are entangled with soft particles in such a way that thereduced density matrix of the hard particles is essentially completely decohered.Furthermore, we show that the inclusive formalism is ill-suited to describe scatteringof wavepackets, requiring the use of the dressed formalism. We construct theHilbert space for QED in the dressed formalism as a representation of the canonicalcommutation relations of the photon creation/annihilation algebra, and argue that itsplits into superselection sectors which correspond to eigenspaces of the generatorsof large gauge transformations.In the second part of this thesis, we turn to applications of quantum informationtheoretic concepts in the AdS/CFT correspondence. In pure AdS, we find anexplicit formula for the Ryu-Takayanagi (RT) surface for special subregions in thedual conformal field theory, whose entangling surface lie on a light cone. Theexplicit form of the RT surface is used to give a holographic proof of Markovicityof the CFT vacuum on a light cone. Relative entropy of a state on such specialsubregions is dual to a novel measure of energy associated with a timelike vector flow between the causal and entanglement wedge. Positivity and monotonicity ofrelative entropy imply positivity and monotonicity of this energy, which yields aconsistency conditions for solutions to quantum gravity.

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Baryons, Branes, and (Striped) Black Holes : Applications of the Gauge / Gravity Duality to Quantum Chromodynamics and Condensed Matter Physics (2014)

The gauge / gravity duality, or holographic correspondence, is a theoretical tool that allows the description of strongly coupled field theory through a dual classical gravity theory. In this thesis, we advance the use of numerical methods in applications of the holographic correspondence to the study of strongly coupled field theories in three situations.Firstly, we study the relationship between chemical potential and charge density across myriad examples of Lorentz invariant 3+1 dimensional holographic field theories with the minimal structure of a conserved charge. Solving for the classical gravitational configurations dual to the field theories and extracting the charge density and chemical potential, we enumerate the relationships that can exist in a wide range of holographic theories.Secondly, we study the spontaneous formation of inhomogeneous (striped) order, a phenomenon that has been observed in the cuprates, in a 2+1 dimensional strongly coupled field theory. By numerically solving the equations of motion using finite difference techniques, we construct the full nonlinear striped black brane solutions that provide the gravity dual to this field theory. We evaluate the thermodynamics and show that the system undergoes a second order phase transition to the striped phase as the temperature is lowered.Finally, we apply the holographic correspondence to study particular aspects of quantum chromodynamics (QCD). First, we develop a phenomenological holographic model to describe the colour superconductivity phase of QCD, which is believed to exist at large quark density. We construct the phase diagram for our model, which includes confined, deconfined, and superconducting phases. In a separate project, we revisit the construction of the baryon in the Sakai-Sugimoto model of holographic QCD. In this model, gauge field configurations on the probe D8 flavour branes with non-trivial topological charge (instantons) correspond to baryons in the dual field theory. In order to extend previous studies, we relax an assumption of spherical symmetry and, utilizing pseudospectral methods, numerically construct the deformed instanton in the bulk. Compared to previous studies, we find significantly more realistic values for the mass and size of the baryon.

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Momentum-space entanglement and the gravity of entanglement in AdS/CFT (2014)

In the first part of this thesis we explore the entanglement structure of relativistic field theories in momentum space. We discuss a Wilsonian path integral formulation and a perturbative approach. Using perturbation theory we obtain results for specific quantum field theories. These are understood through scaling and decoupling properties of field theories. Convergence of the perturbation theory taking loop diagrams into account is also discussed. We then discuss the entanglement structure in systems where Lorentz invariance is broken by a Fermi surface. The Fermi surface helps the convergence of perturbation theory and entanglement of modes near the Fermi surface is shown to be amplified, even in the presence of a large momentum cutoff. In the second part of this thesis we explore the connection between entanglement and gravity in the context of the AdS/CFT correspondence. We show that there are certain thermodynamic-like relations common to all conformal field theories, which when mapped via the AdS/CFT correspondence to the bulk are tantamount to Einstein's equations, to lowest order in the metric.

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The AdS/CFT Correspondence: Bulk to Boundary Map and Applications (2014)

The holographic principle connects theories with gravity to lower dimensional theories without gravity. Notably, the AdS/CFT correspondence — the first concrete realization of the holographic principle — provides a one to one map between string theory in Anti de Sitter space, and a strongly coupled, large N, SU(N) super Yang-Mills gauge theory in one less dimension.In this thesis, within the context of holographic field theories, I improve on the current understanding of the map between gravity (bulk) and gauge theory (boundary) degrees of freedom. Furthermore, I explore some of the applications of the AdS/CFT correspondence to the study of strongly coupled field theories.I study the map between bulk and boundary degrees of freedom mainly by trying to determine what is the gravity dual of a subset of the boundary field theory. In the process of doing so I show how extremal surfaces, entanglement entropy, hyperbolic black holes, and boson stars are fundamental tools in this quest.Next, I explore a few examples of direct applications of the correspondence as a model building device. I discuss how AdS/CFT can be used to construct quasi realistic strongly coupled physical systems ranging from relativistic fluids to plasmas and high temperature superconductors. Finally, I compare some of the results obtained in this thesis with known standard field theory results.

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Applications of the gauge/gravity duality (2013)

While varied applications of gauge/gravity duality have arisen in literature from studies of condensed matter systems including superconductivity to studies of quenched Quantum Chromodynamics (QCD), this thesis focuses on applications of the duality to holographic QCD-like ﬁeld theories and to inflationary model that uses a QCD-like ﬁeld theory.In particular the first half of the thesis examines a holographic QCD-like ﬁeld theory with scalar quarks closely related to the Sakai-Sugimoto model of holographic QCD. The behaviour of baryons and mesons in the model is examined to find a continuous mass spectrum for the mesons, and a baryon that can identified with a topological charge. It then slightly modifies the theory to further study the behaviour of holographic ﬁeld theories.The second half of the thesis presents a novel model for early Universe inflation, using an SU(N) gauge ﬁeld theory as the inflaton. The inflation model is studied at both weak coupling and strong coupling using the gauge/gravity duality. The robustness of model’s predictions to exciting multiple inflationary fields beyond the single ﬁeld of its original proposal, and its possible role in breaking the supersymmetry of the Minimal Supersymmetric Standard Model is also explored.

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Holographic effective theories for strongly coupled physics (2009)

In this thesis we summarize three of our work using the gauge/gravity dualities to build effective theories for strongly coupled phenomena. First, we study the thermodynamics of large N pure 2+1 dimensional Yang-Mills theory on a small spatial S². By studying the effective action for the Polyakov loop order parameter, we show analytically that the theory has a second order deconfinement transition. Our results together with extrapolation of lattice QCD results imply a critical radius in the phase diagram where the deconfinement transition switches from second order to first order. We show that the point at the critical radius and temperature can be either a tricritical point with universal behavior or a triple point separating three distinct phases.Second, we study a model of holographic QCD at zero temperature and finite chemical potential. We find that as the baryon chemical potential is increased, the system transitions to a nuclear matter phase characterized by a condensate of instantons on the probe D-branes in the string theory dual. The electrostatic interactions between the instantons cause the condensate to expand towards the UV with increasing chemical potential, giving a holographic version of the expansion of the Fermi surface. We also find possible explanation of the ``chiral density wave'' instability in large N QCD. We argue that the model can be used to make semi-quantitative predictions of the binding energy per nucleon for nuclear matter in QCD.Third, we consider an Abelian Higgs model placed in an AdS black hole background. Such model has been shown to exhibit superconductor like transitions. In the superconducting phase the system shows infinite DC conductivity. This suggests the possibility of turning on a time independent supercurrent. In this paper we study such supercurrent solutions and the associated phase diagram. We find a critical point in the phase diagram where the second order superconducting transition becomes first order. Supercurrent solutions are well studied in condensed matter systems. We find some qualitative agreement with known results.

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Aspects of SU(2|4) symmetric field theories and the Lin-Maldacena geometries (2008)

Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theoryon R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S².We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state.

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Constraints on geometry from causal holographic information and relative entropy (2017)

In this thesis we find constraints to asymptotically anti de-Sitter space dual to holographic conformal field theory states using the holographic duality. A recent conjecture involving the causal holographic information surface propsed that for smooth asymptotically anti de-Sitter spacetimes that obey the null energy condition, the area of the Ryu-Takayanagi surface will always be less than or equal to the area of the causal holographic information surface. This conjecture is explored in three dimensional spacetimes that are dual to translation invariant states on the boundary conformal field theory in two dimensions. A series expansion of the Ryu-Takayanagi surface and causal holographic information surface is derived, and is used to translate the constraint between the ar- eas of the two surfaces into a constraint on the asymptotic structure of such geometries near the conformal boundary. The translated constraints are compared to the constraints given by the null energy condition - and it is found that the first two leading order constraints are the same. We then outline some preliminary results of an ongoing project whose goal is to understand the dual of relative entropy of holographic states defined on null cone regions on the conformal boundary. We derive the modular Hamiltonian for vacuum states defined on null cone regions in a conformal field theory using known results for modular Hamiltonians on null planes. We also derive the Ryu- Takayanagi surface associated with such null cone regions. Using these results, it is argued that, for null cones whose base is cut by a constant time cut, will not give new constraints beyond what is already known for ball shaped regions.

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Tensor networks for dynamic spacetimes (2017)

Tensor networks give simple representations of complex quantum states. They have proven useful in the study of condensed matter systems and conformal fields, and recently have provided toy models of AdS/CFT. Underlying the tensor network - AdS/CFT connection is the association of a graph geometry with the tensor network. This geometry is most easily understood as containing only spatial directions. In the context of the AdS/CFT correspondence this limits tensor network toy models to describing static spacetimes. Here we look to extend tensor network models of AdS/CFT by capturing the geometry of a dynamic spacetime in a network description. We review the role of tensor networks in our understanding of AdS/CFT to motivate this extension, before proposing a network picture that captures key features of AdS/CFT.

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The strong subadditivity of holographic entanglement entropy; from boundary to bulk (2017)

One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement.

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Second order relative entropy in holographic theories (2016)

Recently, there has been growing recognition that the tools from quantum information theory might be well-suited to studying quantum gravity in the context of the gauge/gravity correspondence. It is exploring this connection that is the main motivation for the work in this thesis. In particular, we focus on holographic field theories which possess classical spacetime duals. The aim is that certain conditions on the classical duals will narrow down the types of field theories that can be holographic. This will give a better understanding of the limitations and robustness of the gauge/gravity correspondence. We do so by computing the canonical energy for general perturbations around anti-de Sitter spacetime, which is dual to quantum Fisher information in the field theory. We go on to prove the positivity of canonical energy and discuss the addition of matter fields. We further show that our result can be interpreted as an interaction between scalar fields living in an auxiliary de Sitter spacetime. We concluded with a summary of progress and future challenges for this program.

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Simplifying Plasma Balls and Black Holes with Nonlinear Diffusion (2014)

The AdS / CFT dictionary, while still incomplete, hints at deep connections between thermal field theories and the dynamics of black holes. Without specifying a Lagrangian, we develop a non-standard approximation for field theories dominated by thermal noise in order to show that many black hole features are universal. Our model is a nonlinear partial differential equation which may be derived, as it was last year, by examining random equilibration of energy on a collection of sites. An extension pairing energy with other conserved quantities is also proposed. For typical holographic gauge theories, the linear versions of our models show that Hagedorn densities of states are associated with long lived lumps of deconfined plasma. With the help of numerical and mathematical results, we show that the nonlinear diffusion properties are more subtle and discuss the implications for using these models to study unsolved problems in holography.

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This is a small sample of students and/or alumni that have been supervised by this researcher. It is not meant as a comprehensive list.