Quantum Gravity and Quantum Information

Minimal Qualification Program

 

The minimal qualification program prepares the graduate students toward being qualified for the PhD research and future career in Loop Quantum Gravity. The program includes two aspects: The fundamentals and The research qualification. The students are expected to finish the qualifications in about 2 years but no more than 3 years.

This program shares similarity to some specialized master programs in Europe and in China, where the master students finish a set of specialized courses and start to research. The period of 2-3 years master studies often plays a crucial role for the students’ development, and it may even determine the levels of students’ future research, because the students have to make the trasition from their education phase to the research phase within these 2-3 years, and the master studies builds the inital condition for the students’ research life. The purpose of the minimal qualificaiton program is to help the students make the successful transition and have an excellent starting point for their research career.

 

The Fundamentals

The student needs to have solid knowledge in the subjects that are closely related to the research field. The successful candidate needs to pass the written exams (the Minimal Qualification Exams) for 3 core subjects and one of the 2 elective subjects. It is allowed that the student focuses on one subject for certain period and takes the exam of the subject right after. The following lists all the core and elective subjects, examination syllabi, and references. Some good online resources may be found in e.g. the PSI program (PIRSA video library) at Perimeter Institute for Theoretical Physics

Core Subject 1: General Relativity

Manifold and tensor field, metric tensor, covariant derivative, geodesics, curvature and its computation, differential forms, killing field, special relativity, Einstein equation and solutions, Linearized gravity, homogeneous and isotropic cosmology, the Schwarzschild spacetime, Reissner–Nordström spacetime, Kerr-Newman spacetime, black holes, dS and AdS spacetimes, initial value formulation of general relativity, asymptotic flatness, spinors, quantum field theory on curved spacetimes, Hawking and Unruh effects, black hole thermodynamics. Lagrangian and Hamiltonian formulation of gravity

References:

R. M. Wald, “General Relativity” Required Chapters: 1-6 and 10-14, appendix B, C, D, E.

One can find the chapters in C. Liang, “Differential Geometry and General Relativity” I, II, III (in Chinese) corresponding to the required chapters in Wald’s book, except for spinors and QFT on curved spacetime. There is an appendix on dS abd AdS.

Core Subject 2: Quantum Field Theory

Lagrangian and Hamiltonian Field Theory, Noether’s theorem, quantum Klein-Gordon field, quantum Dirac field, quantum Electromagnetic field, interactions and perturbative expansion, Feynman diagrams and Feynman rules, quantum electrodynamics, radiative corrections, Ward identity, functional methods, renormalization, Wilsonian renormalization group, spontaneous symmetry breaking, effective action, nonabelian gauge invariance, basics of Lie algebra, quantization of nonabelian gauge theory, anomalies, Higgs mechanism.

References:

M. E. Peskin and D. V. Schroeder, “Introduction to Quantum Field Theory” Required Chapters: 2-7, 9-12, 15, 16, 19.2, 20.1, 20.2.

L. H. Ryder, “Quantum Field Theory”

Core Subject 3: Basics of Loop Quantum Gravity

Lagrangian and Hamiltonian formulations of gravity, ADM formulation, connection formulation, quantum configuration space, cylindrical functions, kinematica Hilbert space, spin-network states, holonomy-flux algebra, area and volume operators, *-algebra and GNS construction, quantum Gauss constraint, intertwiners, solutions of quantum diffeomorphism constraint, quantum Hamiltonian constraint, master constraint, complexifer coherent state, reduced phase space quantization, LQG black hole entropy, EPRL spinfoam model, Loop Quantum Cosmology.

References:

M. Han, W. Huang, Y. Ma, “Fundamental Structure of Loop Quantum Gravity”, arXiv:gr-qc/0509064 Required Chapters,1-5, 6.1, 6.3, 7.2

Required video lecture series: Introduction to Loop Quantum Gravity (圈量子引力)

I. Agullo and P. Singh, “Loop Quantum Cosmology: A brief review”, arXiv:1612.01236

Thomas Thiemann, “Modern Quantum Canonical General Relativity”, Cambridge University Press 2008 (This book contains detailed discussions of almost all aspects in LQG)

Carlo Rovelli, Francesca Vidotto, “Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory”, Cambridge University Press 2014 (This book focuses on the Spin Foam approach of LQG, and discusses many very recent progresses)

Abhay Ashtekar, Jerzy Lewandowski, “Background Independent Quantum Gravity: A Status Report”, Class. Quant. Grav.21:R53,2004 (Pedagogical overview of the basics of LQG for non-experts)

Elective Subject 1: Mathematical Physics

Abstract measure theory, Hilbert spaces, Banach spaces, local convex spaces, bounded operators, unbunded operators, spectral theorem, C*-algebra and repesentation, Fourier transform, Lie group and Lie algebra, simply Lie algebras, roots and weights, highest weight representations, Lorentz group and algebra, unitary representation of Lorentz group.

References:

Reed M., Simon B. Vol 1. Methods of mathematical physics. Functional analysis (AP, 1980)(K)(T)(412s)

Appendices in Thomas Thiemann, “Modern Quantum Canonical General Relativity”, Cambridge University Press 2008

Fuchs J., Schweigert C. Symmetries, Lie algebras and representations (CUP, 1997)(ISBN 0521560012)

Elective Subject 2: Numerical Methods

(1) Root-finding methods: Bisection Method, Newton-Raphson Method, Secant Method; (2) Linear Algebraic Equation Solvers: Gaussian Elimination, LU Decomposition, Iterative Methods; (3) Numerical Integration: Trapezoidal Rule, Simpson’s Rule, Gaussian Quadrature; (4) Differential Equations: Euler’s Method, Runge-Kutta Methods; Monte Carlo Methods; (5) Fast Fourier Transform (FFT).

The students can choose C or Python to implement their codes.

References:

M. Hjorth-Jensen, “COMPUTATIONAL PHYSICS”

Lecture notes on Computational Physics by Prof. Wolfgang Tichy.

 

Research Qualification

The student must finish independently at least one research paper. The paper is expected to be published in a leading journal of physics.

The student will have an oral exam to report the research results at the end of this program. The student’s research paper, oral and written exams will be evaluated by an international committee, which decides the final grade of the student.