Quantum Field Theory: A Quantum Computation Approach

✍ Scribed by Yannick Meurice

• Publisher: IOP Publishing
• Year: 2021
• Tongue: English
• Leaves: 256
• Category: Library

✦ Synopsis

This book introduces quantum field theory models from a classical point of view. Practical applications are discussed, along with recent progress for quantum computations and quantum simulations experiments. New developments concerning discrete aspects of continuous symmetries and topological solutions in tensorial formulations of gauge theories are also reported.

Quantum Field Theory: A quantum computation approach requires no prior knowledge beyond undergraduate quantum mechanics and classical electrodynamics. With exercises involving Mathematica and Python with solutions provided, the book is an ideal guide for graduate students and researchers in high-energy, condensed matter and atomic physics.

Key Features

• Introduces models from a symmetry point of view at the classical level
• Includes the path-integral formulation used as the main quantization method
• The quantum models are defined on space-time lattices with emphasis on the time continuum limit
• Discrete tensor formulations are introduced from scratch
• Provides quantum computations that require practical setups and approximations

✦ Table of Contents

PRELIMS.pdf
Preface
Acknowledgements
Author biography
Yannick Meurice
CH001.pdf
Chapter 1 Introduction
1.1 Goals of the lecture notes
1.2 Classical electrodynamics and its symmetries
1.3 Field quantization
1.4 The need for discreteness in quantum computing
1.5 Symmetries and predictive models
References
CH002.pdf
Chapter 2 Classical field theory
2.1 Classical action, equations of motion and symmetries
2.2 Transition to field theory
2.3 Symmetries
2.4 The Klein–Gordon field
2.5 The Dirac field
2.6 Maxwell fields
2.7 Yang–Mills fields
2.8 Linear sigma models
2.9 General relativity
2.10 Examples of two-dimensional curved spaces
2.11 Mathematica notebook for geodesics
References
CH003.pdf
Chapter 3 Canonical quantization
3.1 A one-dimensional harmonic crystal
3.2 The infinite volume and continuum limits
3.3 Free KG and Dirac quantum fields in 3 + 1 dimensions
3.4 The Hamiltonian formalism for Maxwell’s gauge fields
CH004.pdf
Chapter 4 A practical introduction to perturbative quantization
4.1 Overview
4.2 Dyson’s chronological series
4.3 Feynman propagators, Wick’s theorem and Feynman rules
4.4 Decay rates and cross sections
4.5 Radiative corrections and the renormalization program
References
CH005.pdf
Chapter 5 The path integral
5.1 Overview
5.2 Free particle in quantum mechanics
5.3 Complex Gaussian integrals and Euclidean time
5.4 The Trotter product formula
5.5 Models with quadratic potentials
5.6 Generalization to field theory
5.7 Functional methods for interactions and perturbation theory
5.8 Maxwell’s fields at Euclidean time
5.9 Connection to statistical mechanics
5.10 Simple exercises on random numbers and importance sampling
5.11 Classical versus quantum
References
CH006.pdf
Chapter 6 Lattice quantization of spin and gauge models
6.1 Lattice models
6.2 Spin models
6.3 Complex generalizations and local gauge invariance
6.4 Pure gauge theories
6.5 Abelian gauge models
6.6 Fermions and the Schwinger model
References
CH007.pdf
Chapter 7 Tensorial formulations
7.1 Remarks about the discreteness of tensor formulations
7.2 The Ising model
7.3 O(2) spin models
7.4 Boundary conditions
7.5 Abelian gauge theories
7.6 The compact abelian Higgs model
7.7 Models with non-abelian symmetries
7.8 Fermions
References
CH008.pdf
Chapter 8 Conservation laws in tensor formulations
8.1 Basic identity for symmetries in lattice models
8.2 The O(2) model and models with abelian symmetries
8.3 Non-abelian global symmetries
8.4 Local abelian symmetries
8.5 Generalization of Noether’s theorem
References
CH009.pdf
Chapter 9 Transfer matrix and Hamiltonian
9.1 Transfer matrix for spin models
9.2 Gauge theories
9.3 U(1) pure gauge theory
9.4 Historical aspects of quantum and classical tensor networks
9.5 From transfer matrix functions to quantum circuits
9.6 Real time evolution for the quantum ising model
9.7 Rigorous and empirical Trotter bounds
9.8 Optimal Trotter error
References
CH010.pdf
Chapter 10 Recent progress in quantum computation/simulation for field theory
10.1 Analog simulations with cold atoms
10.2 Experimental measurement of the entanglement entropy
10.3 Implementation of the abelian Higgs model
10.4 A two-leg ladder as an idealized quantum computer
10.5 Quantum computers
References
CH011.pdf
Chapter 11 The renormalization group method
11.1 Basic ideas and historical perspective
11.2 Coarse graining and blocking
11.3 The Niemeijer–van Leeuwen equation
11.4 Tensor renormalization group (TRG)
11.5 Critical exponents and finite-size scaling
11.6 A simple numerical example with two states
11.7 Numerical implementations
11.8 Python code
11.9 Additional material
References
CH012.pdf
Chapter 12 Advanced topics
12.1 Lattice equations of motion
12.2 A first look at topological solutions on the lattice
12.3 Topology of U(1) gauge theory and topological susceptibility
12.4 Mathematica notebooks
12.5 Large field effects in perturbation theory
12.6 Remarks about the strong coupling expansion
References
APP1.pdf
Chapter
APP2.pdf
Chapter
Reference