Teaching - Fabio Anza, PhD

Introduction to Loop Quantum Gravity

Between November and December 2019 I was invited by the University of Palermo to give a set of lectures on the basics of Loop Quantum Gravity–One of the leading theoretical frameworks to describe the quantum aspects of the spacetime.

Introduction

While originally targeted to Master and Ph.D. students, the course was restructured to accommodate the high degree of variety in the audience: from Bachelor students in Physics to Full Professors. Due to the broad interest observed, I decided to focus on:

The conceptual aspects of the search for a quantum theory of gravity.
How Loop Quantum Gravity provides a well-defined and coherent description of a Quantum Geometry.
The interplay between classical General Relativity, in its Hamiltonian formulation, and the Quantum description of geomety provided by Loop Quantum Gravity.
Eventually, I discuss three applications of Loop Quantum Gravity: (1) the resolution of the cosmological singularity; (2) The principle of maximum acceleration and the resolution of the singularity inside a black hole; (3) the Entropy of a Black Hole.

Aspects of the dynamics have only been mentioned, but not discussed in details. Both for time constraints and technical reasons. Below you can find the arguments treated in the lectures. Also, please bear in mind that the duration of the lectures is not homogeneous. We have had 1h, 1.5h and 2h lectures.

Lectures

Lecture 1 – Motivation: Quantum chunks of Space.

What to expect from a quantum geometry: Probabilistic description; Fuzziness; Discreteness. An empirical rule for the quantization of geometry. Example: The quantum tetrahedron. Quanta of area and volume.

Lecture 2 – Introduction to Quantum Gravity – Part A

Ontological road to Quantum Gravity: Covariant Fields, Quantum Fields and General-Covariant Quantum Fields. The world today: Quantum Mechanics, Standard Model, General Relativity, Renormalization. Formulating the problem: Unification vs Completion. The Bronstein argument, the Planck scale and GR as renormalized Quantum Gravity.

Lecture 3 – Introduction to Quantum Gravity – Part B

The history of LQG and its relation with other approaches. The empirical ground for LQG: Gravitational Waves confirm GR at high curvature. No breaking of Lorentz Symmetry. No sign of Supersymmetry. A discussion on modern science, String Theory, and my brief experience as a particle physicist. A philosophical digression: How do we choose the best approach to an unsolved problem? The falsification of theories on the ground of empirical data; The Bayesian perspective and the issue with the domain of applicability/reliability of a theory.

Lecture 4 – Some useful math: Group Theory and SU(2)

Introduction to Group Theory: Group law; Representations and Irreps; Lie groups and Lie algebras; Lie groups and algebras as manifolds and tangent spaces. SU(2) and its algebra su(2). Haar measure. The Hilbert space of square-integrable functions on SU(2): L₂[SU(2)]. Peter-Weyl theorem, part 1, 2, 3. Preliminary example: basis for L₂[U(1)]. Basis for L₂[SU(2)]: the Wigner matrices. Spinors and the relation between SL(2,C) and SU(2). Recoupling theory and an example: the 4-valent Intertwiner.

Lecture 5 – The Kinematics of LQG

Discussion on the basic quantization strategy: Discretize, Quantize, Limit. Example from QED/QCD. Discussion on SU(2) and Lorentz: Partial gauge-fixing. Introducing triangulations, graphs and their combinatorial data: Nodes and Links. Gauge-dependent and gauge-invariant Hilbert space: L₂[SU(2)^L/SU(2)^N]. Holonomy and Left-invariant vector field operators. SU(2) gauge, Gauss theorem and the geometric interpretation as collections of polyhedra. Connection to the classical variables: Tetrads, triads and the area vector.

References

C. Rovelli, F. Vidotto – Covariant Loop Quantum Gravity

R. Gambini, J. Pullin – A first course in Loop Quantum Gravity

P. Menotti – Lectures on Gravitation, ArXiv:1703.05155

Dona, Speziale – Introductory lectures to Loop Quantum Gravity, ArXiv:1007.0402

G. Cicogna – Metodi matematici della fisica

M. Maggiore – A modern introduction to quantum field theory

Lecture 6 – Regge calculus and the “classical” Kinematics

Classical discretization of General Relativity: Regge Calculus. Examples in 2D, 3D, 4D. The variables of Regge geometries. The notion of Deficit angle in 2D, 3D and 4D. discretization. Discretization of the Einstein-Hilbert action of GR: Regge Action. Discrete Equations of motion. Geometric interpretation of the Quantum Kinematics Variables.

Lecture 7 – Hamiltonian General Relativity, discretization and the Holonomy-Flux algebra.

Elements of Dirac’s theory of constrained Hamiltonian systems. How to do the Canonical analysis of a Gauge theory: the example of electromagnetism. ADM formalism for GR. The EH action with ADM variables: Constraints and their physical meaning. Dynamics without time: the example of a classical particle in a potential. Tetrads formulation of GR, Palatini and Holst. Canonical variables in the Holst action. Discretization, smearing and the Holonomy-Flux algebra. Classical phase space T*SU(2)^L//T*SU(2)^N and its relation with the Kinematics of LQG.

Lecture 8 – Algebra of Observables of the quantum geometry: Spin Networks and Geometric Operators

The Hilbert space of a 4-valent intertwiners at fixed representations. Area and Volume operators. The volume spectrum of a quantum tetrahedron. The Hilbert space of LQG and the algebra of observables on a fixed graph Η_Γ. Decomposition via the Intertwiner Hilbert space K_n. Wigner-3j symbols. Dimension of Intertwiner space. Geometric operators: Area, Volume and Dihedral angle. CSCOs on Η_Γ.

Lecture 9 – Three application: Black Hole entropy; Big-Bounce in Loop Quantum Cosmology; Maximal acceleration

Brief discussion on some Dynamical aspects: SL(2,C), Simplicity Constraints, 4D Discretization and the Y_γ-map. Empirical approaches to maximal acceleration. Maximal acceleration in LQG, the paper by Vidotto and Rovelli. Implications for the resolution of singularities in GR. Brief summary of classical cosmology and its formulation with Ashtekar variables. LQG corrections to the Friedman equation: the Big-Bounce. Comments on the model-independence of the bounce. Brief summary of the thermodynamics of black holes. Entropy and Area. Hawking’s temperature. The Frodden-Gosh-Perez notion of local energy. Test particles close to a horizon with Rindler coordinates. Unruh’s temperature and Bekenstein-Hawking entropy. Kinematical derivation of the entropy of a black hole: the punctured surface at thermal equilibrium. Quick sketch of the Dynamical derivation for the entropy of a black hole, using the simplicity constraint. Summary and closing remarks.