# William Donnelly

I am a theoretical physicist working at the interface of quantum information theory and quantum gravity. I am interested in the question of what quantum phenomena such as entanglement tell us about the structure of spacetime and its dynamics; and conversely what gravity can teach us about entanglement.

### Bio

I am currently a postdoctoral research associate at Perimeter Institute. Previously I was a postdoc in the High Energy/Gravity Theory Group at UC Santa Barbara and in the Physics of Information Lab at the University of Waterloo. I completed my PhD in Physics under the supervision of Ted Jacobson in the Gravitation Theory Group at the University of Maryland. Before that I did my Masters in Applied Mathematics at the University of Waterloo, where my advisor was Achim Kempf. I have a Bachelor of Mathematics in Computer Science from the University of Waterloo, and have worked at a number of high-tech companies in the areas of computer graphics and parallel computing. Before that I was a curious kid who loved to build things out of Lego.

## Research

Here is a talk I gave at the Perimeter Institute, outlining some results and future directions of my research program: Pirsa: 16050020 - Entanglement of spacetime.

A complete list of my papers is available on Google Scholar. An incomplete list (containing only the physics papers) is available on arXiv or INSPIRE.

### Locality in quantum gravity

When dealing with a complicated system, it is useful to break it up into smaller subsystems. How do we describe such subsystems in a theory of gravity? And how do we put them together into larger systems? Laurent Freidel and I described local subsystems for gravity in [3], by introducing degrees of freedom at the boundary of a region of space. These degrees of freedom play the role of a reference system or screen. Our subsystems come with a new group of physical symmetries, which tell us about the structure of entanglement in gravity and provide a guide toward its quantization.

A related problem in quantum gravity is the formulation of physical observables. In quantum field theory we study correlation functions of local operators, but such objects don't exist in gravity. This provides an interesting source of tension between gravity and quantum theory. Steve Giddings and I constructed operators in perturbative gravity that are approximately local and reduce to the familiar local operators of quantum field theory in the weak-gravity limit [3]. We found the leading perturbative corrections to the commutation relations of these observables, which are proportional to the Newtonian potential.

In [2] we gave a precise statement of the non-locality of operators in perturbative gravity, and proved lower bounds on their non-local influence.

One class of observables that can be constructed in gravity are gravitational Wilson lines, which correspond to particles with gravitational field lines compressed into a line, rather than filling a volume. While these can be defined at the linear level, there is a question of whether a similar construction can exist nonperturbatively. Donald Marolf, Eric Mintun and I studied this problem in the simpler setting of gravity in 2+1 dimensions with a negative cosmological constant, and found precise limits to localizing gravitational field lines in a small angle [5].

In [1], Ben Michel, Don Marolf, Jason Wein and I considered a discrete model of holographic duality based on tensor networks. We added degrees of freedom to the edges of the tensor network, and showed how this leads to features expected from a bulk theory with perturbative gravitons.

Jason Pye, Achim Kempf and I studied a model of ultraviolet physics which has its origins in classical information theory, specifically Shannon's sampling theorem. We calculated the effect of this modified ultraviolet structure on commutation relations of fields and vacuum entanglement [6].

- William Donnelly, Ben Michel, Donald Marolf, and Jason Wien. Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers. JHEP04 (2017) 093.
- William Donnelly and Steven B. Giddings. Observables, gravitational dressing, and obstructions to locality and subsystems. Phys. Rev. D 94, 104038, 2016.
- William Donnelly and Laurent Freidel. Local subsystems in gauge theory and gravity. JHEP09 (2016) 102.
- William Donnelly and Steven B. Giddings. Diffeomorphism-invariant observables and their nonlocal algebra. Phys. Rev. D 93, 024030, 2016.
- William Donnelly, Donald Marolf, and Eric Mintun. Combing gravitational hair in 2+1 dimensions. Class. Quantum Grav. 33, 025010, 2015.
- Jason Pye, William Donnelly, and Achim Kempf. Locality and entanglement in bandlimited quantum field theory. Phys. Rev. D 92, 105022, 2015.

### Entanglement of quantum fields

My doctoral thesis was on the subject of entanglement entropy in gauge theory. One important aspect of the problem is that in gauge theory, regions of space are no longer independent as they are in scalar field theory. In technical terms, the Hilbert space does not factorize. In [9], I showed how to extend the Hilbert space of nonabelian lattice gauge theory so that it factorizes: the resulting entropy is a sum of entropy of local degrees of freedom and edge modes localized on the boundary. I have argued in [5] that this is the only definition of entanglement entropy consistent with the predictions of Euclidean field theory, such as the thermality of de Sitter space. The technical tools (in particular the use of spin network states) built on earlier work on entanglement entropy in loop quantum gravity [11].

Another issue arising in gauge theories is that the entanglement entropy calculated by standard Euclidean methods contains a "contact term" without any clear statistical interpretation. Aron Wall and I showed that the contact term depends on how one treats the infrared sector of the theory. In two spacetime dimensions, the contact term disappears when the infrared divergences are properly cured by the introduction of a minimal charge [8]. We went on to show how to properly treat the topological sector in any dimension [6]. Recently Aron and I found that the contact term and non-factorization both reflect the same underlying physics: the contact term is the statistical entropy of the edge modes [3,4].

Electromagnetism has a duality symmetry that swaps electric and magnetic fields. How does entanglement entropy behave under this symmetry? To study this issue, Ben Michel, Aron Wall and I studied entanglement entropy for the general class of abelian p-form gauge theories [1]. We showed how the partition function changes under duality at the quantum level, unifying and correcting some disparate results in the literature. We further showed how this anomaly affects the entanglement entropy when switching between the dual theories.

In [2], Gabriel Wong and I generalized these gauge theory results to string theory, using a two-dimensional model of gauge-gravity duality. We found that the edge modes in string theory can be understood as open strings coupled to a brane-like object at the entangling surface, which we call an entanglement brane.

- William Donnelly, Ben Michel, and Aron C. Wall. Electromagnetic Duality and Entanglement Anomalies. Phys. Rev. D 96, 045008, 2017.
- William Donnelly and Gabriel Wong. Entanglement branes in a two-dimensional string theory. JHEP09 (2017) 097.
- William Donnelly and Aron C. Wall. Geometric entropy and edge modes of the electromagnetic field. Phys. Rev. D 94, 104053, 2016.
- William Donnelly and Aron C. Wall. Entanglement entropy of electromagnetic edge modes. Phys. Rev. Lett. 114, 111603, 2015.
- William Donnelly. Entanglement entropy and nonabelian gauge symmetry. Class. Quantum Grav. 31, 214003, 2014.
- William Donnelly and Aron C. Wall. Unitarity of Maxwell theory on curved spacetimes in the covariant formalism. Phys. Rev. D 87, 125033, 2013.
- William Donnelly. Vacuum entanglement and black hole entropy of gauge fields. University of Maryland PhD thesis, 2012.
- William Donnelly and Aron C. Wall. Do gauge fields really contribute negatively to black hole entropy? Phys. Rev. D 86, 064042, 2012.
- William Donnelly. Decomposition of entanglement entropy in lattice gauge theory. Phys. Rev. D 85, 085004, 2012.
- William Donnelly. Entanglement entropy in quantum gravity. University of Waterloo MMath thesis, 2008.
- William Donnelly. Entanglement entropy in loop quantum gravity. Phys. Rev. D 77, 104006, 2008.

### Relativistic quantum information

A potential experimental signature of vacuum entanglement is "entanglement harvesting", whereby entanglement of a quantum field is transferred to localized particles. Recently my colleagues at the University of Waterloo and I considered a generalization of this protocol that we call "entanglement farming" that consists of repeated entanglement harvesting from the same quantum field. We showed that by iterating entanglement harvesting, we drive the cavity toward a state that enhances its ability to impart entanglement, and quantified the total amount of entanglement that can be extracted from a cavity quantum field using our protocol [2]. We went on to show that our setup acts as a sensitive detector of cavity vibrations: a quantum seismograph [1].

- Eric G. Brown, William Donnelly, Achim Kempf, Robert B. Mann, Eduardo Martín-Martínez, and Nicholas C. Menicucci. Quantum seismology. New J. Phys. 16, 105020, 2014.
- Eduardo Martín-Martínez, Eric G. Brown, William Donnelly, and Achim Kempf. Sustainable entanglement production from a quantum field. Phys. Rev. A 88, 052310, 2013.

### Lorentz-violating gravity

With my doctoral advisor Ted Jacobson, I studied several aspects of two Lorentz-violating gravity theories: Einstein-aether theory and Hořava gravity. In [3], we considered the lowest-order corrections of Einstein-aether theory to cosmology, and its impact on inflation. In [2], we corrected a misconception in the literature concerning the stability of Lorentz-violating theories and derived the criterion for stability for linearized perturbations in these theories. In [1], we derived the Hamiltonian formulation of Hořava gravity, giving the algebra of constraints and showing that the Hamiltonian is a sum of constraints plus a boundary term.

- William Donnelly and Ted Jacobson. Hamiltonian structure of Hořava gravity. Phys. Rev. D 84, 104019, 2011.
- William Donnelly and Ted Jacobson. Stability of the aether. Phys. Rev. D 82, 081501, 2010.
- William Donnelly and Ted Jacobson. Coupling the inflaton to an expanding aether. Phys. Rev. D 82, 064032, 2010.

### Computer graphics

As an undergraduate student in computer science I published several articles on computer graphics. The best known of these, with Andrew Lauritzen, introduced variance shadow maps, an approximate solution to the problem of shadow map aliasing. The technique was rapidly adopted by the computer game industry, and appears in several published games. Our paper inspired new variance-based approaches to other problems such as normal map anti-aliasing (LEAN mapping) and screen-space ambient occlusion. I also did research in real-time computer graphics as an intern for NVIDIA, and published several new techniques in the GPU Gems book series.

- William Donnelly and Andrew Lauritzen. Variance Shadow Maps. Proceedings of I3D 2006.
- William Donnelly. Per-Pixel Displacement Mapping with Distance Functions. GPU Gems 2, pp. 123-136, 2005.
- Hubert Nguyen and William Donnelly. Hair Animation and Rendering in the Nalu Demo. GPU Gems 2, pp. 361-380, 2005.
- William Donnelly and Joe Demers. Generating Soft Shadows using Occlusion Interval Maps. GPU Gems, pp. 205-215, 2004

## Other things

I have been a referee for Phys. Rev. D, Phys. Rev. X, Classical and Quantum Gravity, JHEP, and Phys. Rev. Letters.

I have an Erdős-Bacon number of 7 (for a rather loose definition of what constitutes a film).