Introduction to quantum field theory with applications to quantum gravity / Iosif L. Buchbinder and Ilya Shapiro.

Format:

Book

Author/Creator:

Buchbinder, I. L., author.

Contributor:

Shapiro, Ilya, editor.

Series:

Oxford graduate texts.

Oxford scholarship online.

Oxford graduate texts

Oxford scholarship online

Language:

English

Subjects (All):

Quantum field theory.

Physical Description:

1 online resource (544 pages) : illustrations (black and white).

Edition:

First edition.

Place of Publication:

Oxford, England : Oxford University Press, [2021]

Summary:

This textbook presents a detailed introduction to the general concepts of quantum field theory, with special emphasis on principal aspects of functional methods and renormalisation in gauge theories, and includes an introduction to semiclassical and perturbative quantum gravity in flat and curved spacetimes.

Contents:

Cover

Introduction to Quantum Field Theory with Applications to Quantum Gravity

Copyright

Preface

Contents

Part I Introduction to Quantum Field Theory

1 Introduction

1.1 What is quantum field theory, and some preliminary notes

1.2 The notion of a quantized field

1.3 Natural units, notations and conventions

Comments

2 Relativistic symmetry

2.1 Lorentz transformations

2.2 Basic notions of group theory

2.3 The Lorentz and Poincar´e groups

2.4 Tensor representation

2.5 Spinor representation

2.6 Irreducible representations of the Poincaré group

Exercises

3 Lagrange formalism in field theory

3.1 The principle of least action, and the equations of motion

3.2 Global symmetries

3.3 Noether's theorem

3.4 The energy-momentum tensor

4 Field models

4.1 Basic assumptions about the structure of Lagrangians

4.2 Scalar field models

4.2.1 Real scalar fields

4.2.2 Complex scalar fields

4.2.3 Sigma model

4.3 Spinor field models

4.3.1 Equations of motion for a free spinor field

4.3.2 Properties of the Dirac spinors and gamma matrices

4.3.3 The Lagrangian for a spinor field

4.4 Models of free vector fields

4.4.1 Massive vector field

4.4.2 Massless vector fields

4.4.3 The gauge-invariant form of the Proca Lagrangian and Stüeckelberg fields

4.5 Scalar and spinor filelds interacting with an electromagnetic field

4.6 The Yang-Mills field

5 Canonical quantization of free fields

5.1 Principles of canonical quantization

5.2 Canonical quantization in field theory

5.3 Canonical quantization of a free real scalar field

5.4 Canonical quantization of a free complex scalar field

5.5 Quantization of a free spinor field

5.6 Quantization of a free electromagnetic field

Exercises.

Comments

6 The scattering matrix and the Greenfunctions

6.1 Particle interactions and asymptotic states

6.2 Reduction of the S-matrix to Green functions

6.3 Generating functionals of Green functions and the S-matrix

6.4 The S-matrix and the Green functions for spinor fields

6.4.1 Definition of the spinor field S-matrix and Green functions

6.4.2 Generating functional of spinor field Green functions

7 Functional integrals

7.1 Representation of the evolution operator by a functional integral

7.1.1 The functional integral for the one-dimensional quantum system

7.1.2 The functional integral for quantum system with a finite number of degrees of freedom

7.1.3 The functional integral in scalar field theory

7.2 Functional representation of Green functions

7.3 Functional representation of generating functionals

7.4 Functional integrals for fermionic theories

7.4.1 Grassmann algebra and anticommuting variables

7.4.2 Generating functional of spinor Green functions

7.5 Perturbative calculation of generating functionals

7.6 Properties of functional integrals

7.7 Techniques for calculating functional determinants

8 Perturbation theory

8.1 Perturbation theory in terms of Feynman diagrams

8.2 Feynman diagrams in momentum space

8.3 Feynman diagrams for the S-matrix

8.4 Connected Green functions

8.5 Effective action

8.6 Loop expansion

8.7 Feynman diagrams in theories with spinor fields

8.7.1 Feynman diagrams in the theory with Yukawa coupling

8.7.2 Feynman diagrams in spinor electrodynamics

9 Renormalization

9.1 The general idea of renormalization

9.2 Regularization of Feynman diagrams

9.2.1 Cut-off regularization

9.2.2 Pauli-Villars regularization

9.2.3 Dimensional regularization.

9.3 The subtraction procedure

9.3.1 The substraction procedure: one-loop diagrams

9.3.2 The Substraction procedure, and one-loop counterterms

9.3.3 The substraction procedure: two-loop diagrams

9.4 The superficial degree of divergences

9.5 Renormalizable and non-renormalizable theories

9.5.1 Analysis of the superficial degree of divergences

9.5.2 Theory for a real scalar field with the interaction λϕ3

9.5.3 A real scalar field with a λϕ4 interaction

9.5.4 Spinor electrodynamics

9.5.5 The fermi (or Nambu) model

9.5.6 The sigma model

9.6 The arbitrariness of the subtraction procedure

9.7 Renormalization conditions

9.8 Renormalization with the dimensional regularization

9.9 Renormalization group equations

9.9.1 Derivation of renormalization group equations

9.9.2 Renormalization group with scale-transformed momenta

9.9.3 The Solution to the renormalization group equation

9.9.4 The behavior of the running coupling constant

9.9.5 Renormalization group functions in dimensional regularization

10 Quantum gauge theories

10.1 Basic notions of Yang-Mills gauge theory

10.2 Gauge invariance and observables

10.3 Functional integral for gauge theories

10.3.1 The Faddeev-Popov method

10.3.2 Faddeev-Popov ghosts

10.3.3 Total action of the quantum gauge theory

10.4 BRST symmetry

10.4.1 BRST transformations and the invariance of total action

10.4.2 Two more BRST invariants

10.5 Ward identities

10.5.1 Ward identities in quantum electrodynamics

10.5.2 Ward identities for non-Abelian gauge theory

10.5.3 Ward identities and the renormalizability of gauge theories

10.6 The gauge dependence of effective action

10.7 Background field method

10.8 Feynman diagrams in Yang-Mills theory

10.9 The background field method for Yang-Mills theory.

10.10 Renormalization of Yang-Mills theory

10.10.1 Power counting in the Yang-Mills theory

10.10.2 Renormalization in the background field method

10.10.3 Renormalization in the Lorentz gauge

Part II Semiclassical and Quantum Gravity Models

11 A brief review of general relativity

11.1 Basic principles of general relativity

11.2 Covariant derivative and affine connection

11.3 The curvature tensor and its properties

11.4 The covariant equation for a free particle: the classical limit

11.5 Classical action for the gravity field

11.6 Einstein equations and the Newton limit

11.7 Some physically relevant solutions and singularities

11.7.1 The spherically-symmetric Schwarzschild solution

11.7.2 The standard cosmological model

11.8 The applicability of GR and Planck units

12 Classical fields in curved spacetime

12.1 General considerations

12.2 Scalar fields

12.2.1 Conformal symmetry of the metric-scalar model

12.3 Spontaneous symmetry breaking in curved space and induced gravity

12.4 Spinor fields in curved space

12.4.1 Tetrad formalism and covariant derivatives

12.4.2 The covariant derivative of a Dirac spinor

12.4.3 The commutator of covariant derivatives

12.4.4 Local conformal transformation

12.4.5 The energy-momentum tensor for the Dirac field

12.5 Massless vector (gauge) fields

12.6 Interactions between scalar, fermion and gauge fields

13 Quantum fields in curved spacetime: renormalization

13.1 Effective action in curved spacetime

13.2 Divergences and renormalization in curved space

13.2.1 The general structure of renormalization in the matter sector.

13.2.2 The general structure of renormalization in the vacuum sector

13.3 Covariant methods: local momentum representation

13.3.1 Riemann normal coordinates

13.3.2 Local momentum representation

13.3.3 Effective potential in curved spacetime: scalar fields

13.3.4 The fermion contribution to effective potential

13.4 The heat-kernel technique, and one-loop divergences

13.4.1 The world function σ(x, x') and related quantities

13.4.2 Coincidence limits of the first terms of the evolution operator

13.4.3 Separating the UV divergences

14 One-loop divergences

14.1 One-loop divergences in the vacuum sector

14.1.1 Scalar fields

14.1.2 Spinor fields

14.1.3 Massless vector fields

14.1.4 Massive vector fields (The Proca model)

14.2 Beta functions in the vacuum sector

14.3 One-loop divergences in interacting theories

14.3.1 A self-interacting scalar

14.3.2 Non-Abelian vector fields

14.3.3 Yukawa model

14.3.4 Standard Model-like theories

15 The renormalization group in curved space

15.1 The renormalization group based on minimal subtractions

15.2 The effective potential from a renormalization group

15.3 The global conformal (scaling) anomaly

16 Non-local form factors in flat and curved spacetime

16.1 Non-local form factors: simple example

16.2 Non-local form factors in curved spacetime

16.3 The massless limit and leading logs vs. the infrared limit

17 The conformal anomaly and anomaly-induced action

17.1 Conformal transformations and invariants

17.2 Derivation of the conformal anomaly.

17.2.1 Example of a derivation using a nonlocal form factor.

Notes:

This edition also issued in print: 2021.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

0-19-187466-3

0-19-257531-7

OCLC:

1240586819