Minicourse on Spacetime Thermodynamics

This is a mincourse aimed at grad students and postdocs I'm teaching at the University of Maryland, March 21-23, March 30-April 1. A schedule, description, and the audio recordings can be found at the MCFP website. (Note: in order to get the complete video, you will need to either download it, or else use a Dropbox mobile app!)

This page is for posting Lecture Notes, Problems Sets, Errata, Open Questions, and a Bibliography.


Lecture 1, Problem Set #1

Note: During this lecture I mistakenly attributed the path integral derivation of Rindler wedge thermality. The correct reference is Unruh and Weiss (not Wald). This mistake has been corrected in the lecture notes. Also, the actual lecture ended around 1.6, the rest will be covered next time.

Lecture 2, Problem Set #2

Note: The recording of the lecture muddled the algebra of the Casini-Huerta proof of the c-theorem a bit, although the main ideas were all correct. It should be patched up in the notes, although you may prefer to get it directly from their paper (see bibliography).

Lecture 3, Problem Set #3.

Lecture 4

Lecture 5

Note: the proof of LM had some wrong signs, but I've fixed it in the scanned lecture notes. (But I made no effort to get the numerical coefficients right.) The notes also include a couple of pages of bonus material on the laws of black hole thermodynamics, not covered in class.

Lecture 6, followed by Discussion thereafter.

Note: My original lecture notes leave off at 5.2, but the last 5 pages of notes refer to a talk I gave on the quantum focussing conjecture. This goes into more detail than I discussed it during the Discussion, but it may provide some helpful context. I had no written notes for the "information puzzle" discussion, so please consult the video of the Discussion for that.

Open Questions (AFAIK)

A Dozen Open Questions

Pedagogical Bibliography (very incomplete)

1 Overviews

1.1 Entanglement Entropy (Review)

Matt Headrick, "Entanglement Entropy, Quantum Field Theory, and Holography". (His webpage has more material.)

Nabil Iqbal, "Entanglement Entropy in Field Theory and Gravity"

1.2 Renormalization of Black Hole Entropy (Reviews)

Appendix A of:
Raphael Bousso, Zachary Fisher, Stefan Leichenauer, and Aron C. Wall, "A Quantum Focussing Conjecture"

Sergey N. Solodukhin, "Entanglement entropy of black holes''

1.3 Information Problem (Review)

Daniel Harlow, "Jerusalem Lectures on Black Holes and Quantum Information"

2 Information Theory

2.1 Properties of Entropy

Alfred Wehrl, "General properties of entropy" (Review)

J. Aczél, B. Forte and C. T. Ng, Why the Shannon and Hartley Entropies Are 'Natural'

Imre Csiszár, "Axiomatic Characterizations of Information Measures"

W. Ochs, "A new axiomatic characterization of the von Neumann entropy"

2.2 Von Neumann Algebras / Algebraic QFT

The Wikipedia article is surprisingly good (click through for definitions). See also Tomita-Takesaki theory.

Alain Connes, Noncommutative Geometry, Ch. 5.

Haag, Local Quantum Physics (Springer Verlag, Berlin, 1992).

H. J. Borchers, "On revolutionizing quantum field theory with Tomita’s modular theory"

2.3 Relative Entropy

Huzihiro Araki, ``Relative Entropy of States of von Neumann Algebras'', part II.

3 Calculating Entropy

3.1 Replica Trick / Geometric Entropy

L. Susskind, J. Uglum, "Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory"

Curtis Callan, Frank Wilczek, "On Geometric Entropy"

C. Holzhey, F. Larsen, and F. Wilczek, "Geometric and renormalized entropy in conformal field theory"

S. N. Solodukhin, "The conical singularity and quantum corrections to entropy of black hole"

P. Calabrese and J. L. Cardy, "Entanglement entropy and quantum field theory"
"Entanglement entropy and quantum field theory: A Non-technical introduction"
"Entanglement entropy and conformal field theory"

J. Cardy, O. Castro-Alvaredo, and B. Doyon, "Form factors of branch-point twist fields in quantum integrable models and entanglement entropy"

3.2 Area Law

Rafael D. Sorkin, "On the Entropy of the Vacuum outside a Horizon"

Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, "Quantum source of entropy for black holes"

Mark Srednicki, "Entropy and Area"

J. Eisert, M. Cramer, M.B. Plenio, "Area laws for the entanglment entropy - a review"

3.3 Subleading Terms

Corner contributions in D = 3
H. Casini, M. Huerta, Universal terms for the entanglement entropy in 2+1 dimensions

Pablo Bueno, Robert C. Myers, Corner contributions to holographic entanglement entropy

Log Divergences in D = 4
Sergey N. Solodukhin, "Entanglement entropy, conformal invariance and extrinsic geometry"

Dmitri V. Fursaev, Alexander Patrushev, Sergey N. Solodukhin, "Distributional Geometry of Squashed Cones"

3.4 Holographic EE

Shinsei Ryu, Tadashi Takayanagi, "Holographic Derivation of Entanglement Entropy from AdS/CFT"

Shinsei Ryu, Tadashi Takayanagi, "Aspects of Holographic Entanglement Entropy"

Tatsuma Nishioka, Shinsei Ryu, Tadashi Takayanagi, "Holographic Entanglement Entropy: An Overview"

Matthew Headrick, Tadashi Takayanagi, "A holographic proof of the strong subadditivity of entanglement entropy"

Veronika E. Hubeny, Mukund Rangamani, Tadashi Takayanagi, "A Covariant Holographic Entanglement Entropy Proposal"

Aron C. Wall, "Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy"

Aitor Lewkowycz, Juan Maldacena, "Generalized gravitational entropy"

Thomas Faulkner, Aitor Lewkowycz, Juan Maldacena, "Quantum corrections to holographic entanglement entropy"

Netta Engelhardt, Aron C. Wall, "Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime"

4 Higher Curvature Corrections

4.1 Wald Entropy / Noether charge

R. M. Wald, "Black hole entropy is the Noether charge"

Ted Jacobson, Gungwon Kang, Robert C. Myers "On Black Hole Entropy"

V. Iyer and R. M. Wald, "Some properties of Noether charge and a proposal for dynamical black hole entropy"

V. Iyer and R. M. Wald, "A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes"

4.2 Nonstationary Surfaces

Dmitri V. Fursaev, Alexander Patrushev, Sergey N. Solodukhin, "Distributional Geometry of Squashed Cones"

Xi Dong, "Holographic Entanglement Entropy for General Higher Derivative Gravity"

Rong-Xin Miao, "Universal Terms of Entanglement Entropy for 6d CFTs"

Aron C. Wall, "A Second Law for Higher Curvature Gravity"

5 Edge Modes for Gauge Fields

5.1 Chern-Simons / 2+1 Gravity

S. Carlip, "The Statistical Mechanics of the (2+1)-Dimensional Black Hole"

5.2 Lattice Gauge Theory

P. V. Buividovich, M. I. Polikarpov, "Entanglement entropy in gauge theories and the holographic principle for electric strings"

William Donnelly, "Decomposition of entanglement entropy in lattice gauge theory"
"Entanglement entropy and nonabelian gauge symmetry"

Horacio Casini, Marina Huerta, Jose Alejandro Rosabal, "Remarks on entanglement entropy for gauge fields"

5.3 Gravity

William Donnelly, Laurent Freidel, "Local subsystems in gauge theory and gravity"

5.4 Maxwell: Kabat puzzle and its resolution

Dan Kabat, "Black Hole Entropy and Entropy of Entanglement"

William Donnelly, Aron C. Wall "Entanglement entropy of electromagnetic edge modes"
"Geometric entropy and edge modes of the electromagnetic field"

6 More Applications

6.1 RG flow monotonicity

D = 2: c-theorem
H. Casini, M. Huerta, "A c-theorem for the entanglement entropy"
"A finite entanglement entropy and the c-theorem"

G. Vidal, J. I. Latorre, E. Rico, A. Kitaev, "Entanglement in quantum critical phenomena"

D = 3: F-theorem
H. Casini, M. Huerta, "On the RG running of the entanglement entropy of a circle"

Tarun Grover, "Chiral Symmetry Breaking, Deconfinement and Entanglement Monotonicity"

Cesar A. Agon, Matthew Headrick, Daniel L. Jafferis, Skyler Kasko, "Disk entanglement entropy for a Maxwell field"

D = 4: a-theorem
Sergey N. Solodukhin, "The a-theorem and entanglement entropy".

6.2 Topological EE

Michael Levin, Xiao-Gang Wen, "Detecting topological order in a ground state wave function"

Tarun Grover, Yi Zhang, Ashvin Vishwanath, "Entanglement Entropy as a Portal to the Physics of Quantum Spin Liquids"

6.3 Relation to Confinement

Igor R. Klebanov, David Kutasov, Arvind Murugan, "Entanglement as a Probe of Confinement"

Aitor Lewkowycz, "Holographic Entanglement Entropy and Confinement"

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