arXiv subjects

William Donnelly

Publications and source records attributed to William Donnelly.

35 records · Page 2Linked to original sources

arXiv2016

Entanglement branes in a two-dimensional string theory

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large $N$. The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the Modular hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an E-brane.

hep-th↗

arXiv2016

Local subsystems in gauge theory and gravity

We consider the problem of defining localized subsystems in gauge theory and gravity. Such systems are associated to spacelike hypersurfaces with boundaries and provide the natural setting for studying entanglement entropy of regions of space. We present a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary. In Yang-Mills theory the new degrees of freedom are a choice of gauge on the boundary, transformations of which are generated by the normal component of the nonabelian electric field. In general relativity the new degrees of freedom are the location of a codimension-2 surface and a choice of conformal normal frame. These degrees of freedom transform under a group of surface symmetries, consisting of diffeomorphisms of the codimension-2 boundary, and position-dependent linear deformations of its normal plane. We find the observables which generate these symmetries, consisting of the conformal normal metric and curvature of the normal connection. We discuss the implications for the problem of defining entanglement entropy in quantum gravity. Our work suggests that the Bekenstein-Hawking entropy may arise from the different ways of gluing together two partial Cauchy surfaces at a cross-section of the horizon.

hep-th↗

arXiv2016

Diffeomorphism-invariant observables and their nonlocal algebra

Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics. These can be thought of as operators that create a particle, together with its inseparable gravitational field, and reduce to usual field operators of quantum field theory in the weak-gravity limit; they include both Wilson-line operators, and those creating a Coulombic field configuration. We also describe operators creating the field of a particle in motion; as in the electromagnetic case, these are expected to help address infrared problems. An important characteristic of the quantum theory of gravity is the algebra of its observables. We show that the commutators of the simple observables of this paper are nonlocal, with nonlocality becoming significant in strong field regions, as predicted previously on general grounds.

hep-th↗

arXiv2016

Entanglement entropy of electromagnetic edge modes

The vacuum entanglement entropy of Maxwell theory, when evaluated by standard methods, contains an unexpected term with no known statistical interpretation. We resolve this two-decades old puzzle by showing that this term is the entanglement entropy of edge modes: classical solutions determined by the electric field normal to the entangling surface. We explain how the heat kernel regularization applied to this term leads to the negative divergent expression found by Kabat. This calculation also resolves a recent puzzle concerning the logarithmic divergences of gauge fields in 3+1 dimensions.

hep-th↗

arXiv2016

Combing gravitational hair in 2+1 dimensions

The gravitational Gauss law requires any addition of energy to be accompanied by the addition of gravitational flux. The possible configurations of this flux for a given source may be called gravitational hair, and several recent works discuss gravitational observables (`gravitational Wilson lines') which create this hair in highly-collimated `combed' configurations. We construct and analyze time-symmetric classical solutions of 2+1 Einstein-Hilbert gravity such as might be created by smeared versions of such operators. We focus on the AdS$_3$ case, where this hair is characterized by the profile of the boundary stress tensor; the desired solutions are those where the boundary stress tensor at initial time $t=0$ agrees precisely with its vacuum value outside an angular interval $[-α,α]$. At linear order in source strength the energy is independent of the combing parameter $α$, but non-linearities cause the full energy to diverge as $α\to 0$. In general, solutions with combed gravitational flux also suffer from what we call displacement from their naive location. For weak sources and large $α$ one may set the displacement to zero by further increasing the energy, though for strong sources and small $α$ we find no preferred notion of a zero-displacement solution. In the latter case we conclude that naively-expected gravitational Wilson lines do not exist. In the zero-displacement case, taking the AdS scale $\ell$ to infinity gives finite-energy flux-directed solutions that may be called asymptotically flat.

hep-th↗

arXiv2015

Locality and entanglement in bandlimited quantum field theory

We consider a model for a Planck scale ultraviolet cutoff which is based on Shannon sampling. Shannon sampling originated in information theory, where it expresses the equivalence of continuous and discrete representations of information. When applied to quantum field theory, Shannon sampling expresses a hard ultraviolet cutoff in the form of a bandlimitation. This introduces nonlocality at the cutoff scale in a way that is more subtle than a simple discretization of space: quantum fields can then be represented as either living on continuous space or, entirely equivalently, as living on any one lattice whose average spacing is sufficiently small. We explicitly calculate vacuum entanglement entropies in 1+1 dimension and we find a transition between logarithmic and linear scaling of the entropy, which is the expected 1+1 dimensional analog of the transition from an area to a volume law. We also use entanglement entropy and mutual information as measures to probe in detail the localizability of the field degrees of freedom. We find that, even though neither translation nor rotation invariance are broken, each field degree of freedom occupies an incompressible volume of space, indicating a finite information density.

quant-ph↗

arXiv2015

Geometric entropy and edge modes of the electromagnetic field

We calculate the vacuum entanglement entropy of Maxwell theory in a class of curved spacetimes by Kaluza-Klein reduction of the theory onto a two-dimensional base manifold. Using two-dimensional duality, we express the geometric entropy of the electromagnetic field as the entropy of a tower of scalar fields, constant electric and magnetic fluxes, and a contact term, whose leading order divergence was discovered by Kabat. The complete contact term takes the form of one negative scalar degree of freedom confined to the entangling surface. We show that the geometric entropy agrees with a statistical definition of entanglement entropy that includes edge modes: classical solutions determined by their boundary values on the entangling surface. This resolves a longstanding puzzle about the statistical interpretation of the contact term in the entanglement entropy. We discuss the implications of this negative term for black hole thermodynamics and the renormalization of Newton's constant.

hep-th↗

arXiv2014

Quantum Seismology

We propose a quantum mechanical method of detecting weak vibrational disturbances inspired by the protocol of entanglement farming. We consider a setup where pairs of atoms in their ground state are successively sent through an optical cavity. It is known that in this way it is possible to drive that cavity toward a stable fixed-point state. Here we study how that fixed-point state depends on the time interval between pairs of atoms and on the distance between the cavity's mirrors. Taking advantage of an extremely precise resonance effect, we find that there are special values of these parameters where the fixed-point state is highly sensitive to perturbations, even harmonic vibrations with frequencies several orders of magnitude below the cavity's natural frequency. We propose that this sensitivity may be useful for high precision metrology.

quant-ph↗

arXiv2014

Entanglement entropy and nonabelian gauge symmetry

Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang-Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.

hep-th↗

arXiv2013

Sustainable entanglement production from a quantum field

We propose a protocol by which entanglement can be extracted repeatedly from a quantum field. In analogy with prior work on entanglement harvesting, we call this protocol entanglement farming. It consists of successively sending pairs of unentangled particles through an optical cavity. Using non-perturbative Gaussian methods, we show that in certain generic circumstances this protocol drives the cavity field towards a non-thermal metastable state. This state of the cavity is such that successive pairs of unentangled particles sent through the cavity will reliably emerge significantly entangled. We calculate thermodynamic aspects of the harvesting process, such as energies and entropies, and also the long-term behavior beyond the few-mode approximation. Significant for possible experimental realizations is the fact that this entangling fixed point state of the cavity is reached largely independently of the initial state in which the cavity was prepared. Our results suggest that reliable entanglement farming on the basis of such a fixed point state should be possible also in various other experimental settings, namely with the to-be-entangled particles replaced by arbitrary qudits and with the cavity replaced by a suitable reservoir system.

quant-ph↗

arXiv2013

Unitarity of Maxwell theory on curved spacetimes in the covariant formalism

Quantum field theory in curved spacetime may be defined either through a manifestly unitary canonical approach or via the manifestly covariant path integral formalism. For gauge theories, these two approaches have produced conflicting results, leading to the question of whether the canonical approach is covariant, and whether the path integral approach is unitary. We show the unitarity of the covariant U(1) Maxwell theory, defined via the Wick rotation of a Euclidean path integral. We begin by gauge-fixing the path integral, taking care with zero modes, large gauge transformations, and nontrivial bundles. We find an extra geometric factor in the partition function that has been overlooked in previous work, coming from the zero mode of the gauge symmetry, which affects the entropy and stress-energy tensor. With this extra factor, the covariant calculation agrees with the canonical result for ultrastatic manifolds, and in D = 2. Finally, we argue that if there exists a unitary (but not necessarily covariant) canonical formulation, then the covariant formulation must also be unitary, even if the two approaches disagree.

hep-th↗

arXiv2012

Do gauge fields really contribute negatively to black hole entropy?

Quantum fluctuations of matter fields contribute to the thermal entropy of black holes. For free minimally-coupled scalar and spinor fields, this contribution is precisely the entanglement entropy. For gauge fields, Kabat found an extra negative divergent "contact term" with no known statistical interpretation. We compare this contact term to a similar term that arises for nonminimally-coupled scalar fields. Although both divergences may be interpreted as terms in the Wald entropy, we point out that the contact term for gauge fields comes from a gauge-dependent ambiguity in Wald's formula. Revisiting Kabat's derivation of the contact term, we show that it is sensitive to the treatment of infrared modes. To explore these infrared issues, we consider two-dimensional compact manifolds, such as Euclidean de Sitter space, and show that the contact term arises from an incorrect treatment of zero modes. In a manifestly gauge-invariant reduced phase space quantization, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy.

hep-th↗

arXiv2012

Decomposition of entanglement entropy in lattice gauge theory

We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.

hep-th↗

arXiv2012

Coupling the inflaton to an expanding aether

We consider a Lorentz-violating theory of inflation consisting of Einstein-aether theory with a scalar inflaton coupled bilinearly to the expansion of the aether. We determine the conditions for linearized stability, positive energy, and vanishing of preferred-frame post-Newtonian parameters, and find that all these conditions can be met. In homogeneous and isotropic cosmology, the inflaton-aether expansion coupling leads to a driving force on the inflaton that is proportional to the Hubble parameter. This force affects the slow-roll dynamics, but still allows for a natural end to inflation.

gr-qc↗

arXiv2012

Stability of the aether

The requirements for stability of a Lorentz violating theory are analyzed. In particular we conclude that Einstein-aether theory can be stable when its modes have any phase velocity, rather than only the speed of light as was argued in a recent paper.

hep-th↗

arXiv2012

Hamiltonian structure of Horava gravity

The Hamiltonian formulation of Horava gravity is derived. In a closed universe the Hamiltonian is a sum of generators of gauge symmetries, the foliation-preserving diffeomorphisms, and vanishes on shell. The scalar constraint is second class, except for a global, first-class part that generates time reparametrizations. A reduced phase space formulation is given in which the local part of the scalar constraint is solved formally for the lapse as a function of the 3 metric and its conjugate momentum. In the infrared limit the scalar constraint is linear in the square root of the lapse. For asymptotically flat boundary conditions the Hamiltonian is a sum of bulk constraints plus a boundary term that gives the total energy. This energy expression is identical to the one for Einstein-aether theory which, for static spherically symmetric solutions, is the usual Arnowitt-Deser-Misner energy of general relativity with a rescaled Newton constant.

hep-th↗

arXiv2008

Entanglement Entropy in Loop Quantum Gravity

The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. We show that the entanglement entropy may be defined in loop quantum gravity, and compute its value for spin network states. The entanglement entropy for an arbitrary region of space is expressed as a sum over punctures where the spin network intersects the region's boundary. Our result agrees asymptotically with results previously obtained from the isolated horizon framework, and we give a justification for this agreement. We conclude by proposing a new method for studying corrections to the area law and its implications for quantum corrections to the gravitational action.

gr-qc↗