Quantum Field Theory

Quantum field theory is at the heart of modern physics and forms the backbone of the standard model, which is our current best understanding of the laws of particles and forces. The reputation of QFT is that it is very difficult to learn. Quantum field theory by Lewis H. Ryder is a solid modern pedagogical introduction to the ideas and techniques of QFT. The book assumes some familiarity with quantum mechanics and special relativity.

The readership is graduate students in theoretical physics. This book is clearly of a pedagogical nature and contains very detailed worked examples and proofs of statements.

The book consists of 11 chapters.

Chapter 1 gives a general overview of particle physics. The reader is exposed to the basic idea of QFT here and the 4 forces of nature. This chapter gives one a taste for the standard model though the book is not really a book about particle physics.

Chapter 2 introduces single-particle relativistic wave equations. Here the Klein-Gordon equation the Dirac equation & antiparticles, the Maxwell and Proco equations are covered. The importance of the Lorentz and Poincare groups in physics are outlined. The differential geometry of Maxwell’s equations is also presented.

Classical field theory is the topic of Chapter 3. Here the Lagrangian formulation and variational principles are reviewed. The Bohm-Aharonov effect is presented as is Yang-Mills theory from a geometric perspective.

Chapter 4 deals with the canonical quantisation and particles. Due to technical difficulties the Klein-Gordon and Dirac equations cannot be single particle equations. In this chapter the reader starts to deal with quantum field theory as a theory of “many particles”. The real and complex Klein-Gordon fields, Dirac fields and the electromagnetic field are dealt with via canonical quantisation.

From a modern perspective path-integrals are the root to quantisation of fields. In Chapter 5 the Feynman path-integral formulation of quantum mechanics is presented. Topics covered here include perturbation theory & S-matrices and Coulomb scattering. This chapter lays down the ideas of path-integrals ready for QFT.

Chapter 6 develops the path-integral quantisation and Feynman rules for scalar and Dirac fields. Topics covered include: generating functionals, functional integration, Green’s functions & propagators, interacting fields, fermions & anticommuting variables and S matrix formula.

Chapter 7 moves on to discuss path-integral quantisation of gauge fields. Topics covered include: gauge fixing, Faddeev-Popov ghosts, Feynman rules in the Lorentz gauge, the Ward-Takahashi identity in QED, the BRST transformations and the Slavnov-Taylor identities.

Spontaneous symmetry breaking and the Weinberg-Salam model are the topic of Chapter 8. Topics covered include: the Goldstone theorem, spontaneous breaking of gauge symmetries, superconductivity and the Weinberg-Salam model.

Chapter 9 covers the subject of renormalisation. Topics covered include: divergences in QFT, dimensional analysis, regularisation, loop expansions, counter-terms, the renormalisation group, 1-loop renormalisation of QED, renormalisabilty of QCD, asymptotic freedom, anomalies, and renormalisation of Yang-Mills theories with spontaneous symmetry breaking.

Chapter 10 introduces the notion of topological objects in field theory. Topics covered include: the sine-Gordon kink, the Dirac monopole, instantons and theta-vacua.

N= 1 supersymmety in 4 dimensions is the topic of Chapter 11. The theory is built at first in component form and then the power of superspace methods are exposed. Topics covered include the super-Poincare algebra, superspace & super fields, chiral super fields and the Wess-Zumino model.

Paperback: 507 pages
Publisher: Cambridge University Press; 2 edition (6 Jun 1996)
Language English
ISBN-10: 0521478146
ISBN-13: 978-0521478144

Mathematical Physics

Mathematical physics can be understood as the study of the mathematical structures behind physics. Mathematical Physics A Modern Introduction to it Foundations by Sadri Hassani gives a rather substantial introduction to mathematical physics. One novel feature is the short biographical accounts of the people who developed the mathematics featured in the book.

The key to the power of this book is that it discusses the two main pillars of modern mathematical physics: functional analysis and differential geometry. The level of the presentation is aimed at beginning postgraduate students in physics or mathematics.

The book is arranged into 9 sections.

Section 0 covers the mathematical preliminaries: sets, maps, metric spaces, cardanality and mathematical induction.

Section I covers finite dimensional vector spaces: vector space & linear transformations, operator algebra, the matrix representation and spectral decomposition.

Section II then moves on to describe infinite dimensional vector spaces. Topics covered here include: Hilbert spaces, generalised functions, orthogonal polynomials and Fourier analysis.

Complex analysis is the topic of Section III. Topics in this section include: complex calculus, calculus of residues as well as more advanced ideas like meromorphic functions, analytical continuation and the gamma & beta functions.

Section IV covers differential equations: separation of variables, second order linear differential equations, complex analysis of SOLDEs and integral transforms.

Hilbert spaces are the topic of Section V. Topics covered include: basic operator theory (bounded & compact operators and their spectra), integral equations and Sturm-Liouville systems.

Section VI introduces Green’s functions in one dimension and then goes on to discuss Green’s functions in multidimensions. Both the formalism and specific examples are discussed.

After this the book takes on a more geometric direction.

Section VII covers groups and manifolds. This section covers: elementary group theory, group representations, tensor algebra, differential manifolds and tensor calculus. Also covered is exterior calculus and basics of symplectic geometry.

Section VIII covers Lie groups and their applications. Here differential geometry is developed. Topics include: Lie groups & Lie algebras, differential geometry (vector fields, Riemannian metrics, covariant derivatives, geodesics, Killing vector fields), Lie groups and differential equations and the calculus of variations. Within this section general relativity is briefly discussed as is Noether’s theorem.

The book contains exercises and plenty of worked examples.

Hardcover: 1046 pages
Publisher: Springer; 1st edition (February 8, 1999)
Language: English
ISBN-10: 0387985794
ISBN-13: 978-0387985794