Jonstraveladventures.github.io

own University of Cape Town e T Department of Mathematics and Applied Mathematics Cap Dissertation presented for the degree of

f Master of Science rsity o Entanglement entropy, e the Ryu-Takayanagi prescription, Univand conformal maps

Author: Alastair Grant-Stuart

Supervisors: Prof. Jeff Murugan Dr. Jonathan Shock

March 2017

n The copyright of this thesis vests in the author. No ow quotation from it or information derived from it is to be T published without full acknowledgement of the source. e The thesis is to be used for private study or nonap commercial research purposes only. of C Published by the University of Cape Town (UCT) in terms University of the non-exclusive license granted to UCT by the author.

Abstract

We define and explore the concepts underpinning the Ryu-Takayanagi prescription for entanglement entropy in a holographic theory. We begin by constructing entanglement entropy in finite-dimensional quantum systems, and defining the boundary at infinity of a bulk spacetime. This is sufficient for a na?ive application of the Ryu-Takayanagi prescription to some simple examples; nonetheless, we review the general theory of minimal submanifolds in Riemannian ambient manifolds in order to better characterise the objects involved in the prescription. Finally, we explore the symmetries of the the boundary theory to which the prescription applies, and thereby extend the aforementioned examples. Throughout, emphasis is placed on making explicit the mathematical structures that are taken for granted in the research literature.

Acknowledgements

I would like to thank Prof. Jeff Murugan and Dr. Jonathan Shock for introducing me to this field of research, and for their support, guidance and abundant patience. Additionally, I would like to thank S. Shajidul Haque for his assistance and discussions at the outset of this work.

For funding this project, I am grateful to the National Institute for Theoretical Physics.

For caring, I thank Emma Gibson.

Plagiarism declaration

I know the meaning of plagiarism and declare that all of the work in the dissertation, save for that which is properly acknowledged, is my own.

Contents

1 Introduction

3

1.1 Holographic entanglement entropy and the Ryu-Takayanagi

conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Aims and outline . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Entanglement, information theory and entropy

7

2.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Tensor products and direct sums of Hilbert spaces . . 7

2.1.2 Entangled states . . . . . . . . . . . . . . . . . . . . . 9

2.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Density operator formalism . . . . . . . . . . . . . . . 16

2.3.2 Reduced density operators and entanglement entropy 19

2.4 Entanglement entropy in quantum field theories . . . . . . . . 23

3 Conformal completions and the boundary at infinity

25

3.1 Conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Conformal completion of space and spacetime manifolds . . . 31

3.2.1 Conformal completion of Euclidean space Ed . . . . . 32 3.2.2 Conformal completion of Minkowski spacetime R1,d . . 34

3.2.3 Conformal completion of anti-de Sitter spacetime AdSd+2 37

4 Calculations of entanglement entropy using the Ryu-Takayanagi

prescription

43

4.1 Volume forms on (pseudo-)Riemannian manifolds . . . . . . . 44

4.2 Explicit calculations of area-minimising surfaces . . . . . . . . 46

4.2.1 The disc . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2 The infinite strip . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 The annulus . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1

5 Just enough minimal surface theory

57

5.1 Surfaces embedded in a Riemannian manifold . . . . . . . . . 57

5.1.1 Vectors tangent and normal to an embedded surface . 58

5.1.2 On the locality of covariant derivatives . . . . . . . . . 59

5.2 The Second Fundamental Form, mean curvature and minimal

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 General definitions . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Defining mean curvature relative to a particular unit

normal field . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.3 Submanifolds of codimension 1 . . . . . . . . . . . . . 64

5.3 Mean curvature in hyperbolic ambient manifolds. . . . . . . . 67

6 Conformal symmetry transformations

72

6.1 Conformal symmetries of a manifold . . . . . . . . . . . . . . 72

6.2 Conformal maps on the plane . . . . . . . . . . . . . . . . . . 74

6.3 Conformal maps on the Riemann sphere . . . . . . . . . . . . 77

6.4 Geometric properties of Mo?bius transformations . . . . . . . 83

6.5 Constructing particular Mo?bius transformations . . . . . . . . 85

6.5.1 Determining a Mo?bius transformation by its action on

specified points . . . . . . . . . . . . . . . . . . . . . . 86

6.5.2 Preserving the unit disc . . . . . . . . . . . . . . . . . 88

6.6 Examples of specific Mo?bius transformations, and applications

to entanglement entropy . . . . . . . . . . . . . . . . . . . . . 90

6.6.1 The half-plane and the circle . . . . . . . . . . . . . . 90

6.6.2 Disjoint discs and the annulus . . . . . . . . . . . . . . 91

7 Conclusion

99

A Spherical coordinates in Ed

102

2

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download