# Quantum Field Theory by L.H. Ryder 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 by S. Hassani 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

# General Theory of Relativity by P.A.M. Dirac General Theory of Relativity (Physics Notes) General relativity is via it’s formulation a theory heavily based on differential geometry. General Theory of Relativity by Dirac gives a 68 page introduction to the mathematical theory of general relativity. The style of the book is “no nonsense” and “uncluttered”. The book is based on series of lectures given by Dirac at Florida State University in 1973.

The book is aimed at advanced undergraduates, though it will be well suited for beginning graduate students and researchers who want a quick overview of the structure of general relativity.

Chapter 1 to 14 cover topics of differential geometry: parallel displacement, Christoffel symbols, geodesics, covariant derivatives and curvature tensors. All the basics of geometry required for general relativity.

Chapters 15 to 35 deal with the “guts” of general relativity. Topics covered include Einstein’s law of gravity, the Newtonian approximation, gravitational red shift, the Schwarzschild metric & black holes, harmonic coordinates, the field equations with matter, the gravitational action principle, the pesudo energy-tensor for gravity, gravitational waves and the cosmological constant.

Paperback: 68 pages
Publisher: Princeton University Press (January 8, 1996)
Language: English
ISBN-10: 069101146X
ISBN-13: 978-0691011462

# Advanced General Relativity by J. Stewart Advanced General Relativity General relativity is one of the cornerstones of modern physics, describing gravitational phenomena in geometric manner. Advanced General Relativity by John Stewart provides an introduction to some of the more advanced mathematical aspects of the theory. The readership is graduate students and researchers who already have some knowledge of general relativity.

Chapter 1 outlines the theory of differential manifolds, the tangent and cotangent spaces, tensor algebra, Lie derivatives, connections, geodesics and curvature tensors. All these geometric ideas are the “bread and butter” of general relativity.

Chapter 2 is where the more advanced topics start. The notion of spinors is introduced here. The Petrov classification and the Newman-Penrose formulation are presented.

Chapter 3 deals with asymptotic properties of space-time. Basically one would expect the space-time far away from an isolated source of gravity to be flat. This chapter deals with the Bondi and ADM mass, asymtopia for Minkowski space-time, asymptotic simplicity and conformal transformations.

Chapter 4 deals with the characteristic initial value problem in general relativity. The idea is to reformulate general relativity as the temporal evolution of 3-spaces.

Two appendices are included. The first deals with Dirac spinors and the second with the Newman-Penrose formalism.

Paperback: 240 pages
Publisher: Cambridge University Press; New Ed edition (26 Nov 1993)
Language English
ISBN-10: 0521449464
ISBN-13: 978-0521449465

# Mathematical Physics by R. Geroch Mathematical Physics (Lectures in Physics) Mathematics is the language of physics. Thus physicists need some background in mathematics. Within mathematics category theory is a collection of unifying ideas that really gets at the heart of mathematical structures. Mathematical Physics by Robert Geroch provides a good grounding in the basic mathematical structures required in physics from a categorical perspective. This book consists of 56 short chapters.

Chapers 1 to 24 discuss algebraic categories: groups, vector spaces, associative algebras, Lie algebras and representations. The main ideas of category theory are laid down in these chapters via motivating examples.

Chapters 25 to 42 take on a topological flavour. Topics about topological spaces include continuous mappings,compactness, connectedness, homotopy, homology, topological groups and topological vector spaces.

Chapters 43 to 56 combine algebra and topology by discussing measure spaces, distributions and Hilbert spaces. Topics here include bounded operators, the spectral theorem, not necessarily bounded operators and self-adjoint operators.

Paperback: 358 pages
Publisher: University Of Chicago Press (September 15, 1985)
Language: English
ISBN-10: 0226288625
ISBN-13: 978-0226288628

# Geometry, Topology and Physics by M. Nakahara Geometry, Topology and Physics Geometry and topology have become integral in the theoretical physicists tool kit. Ideas from geometry and topology are now fundamental in condensed matter physics, gravitational physics and particle theory. Nakahara’s book Geometry, Topology and Physics provides an assessable introduction to these ideas.

The book is a massive expansion and revision of lectures given by Nakahara at the School of Mathematical and Physical Sciences, the University of Sussex back in 1986.

Chapters 1 and 2 provide a basic review of the physical and mathematical ideas need as a preliminary to the rest of the book. Most of this is familiar territory to a beginning graduate student in mathematical or theoretical physics.

Chapters 3 to 8 introduce some fundamental ideas in geometry and topology: homology groups, homotopy groups, manifolds, de Rham cohomology, Riemannian geometry and complex manifolds. Examples of the applications of these constructions in physics are given throughout these chapters.

Chapters 9 to 12 cover the topics of fibre bundles, connections & curvature, characteristic classes and the Atiyah-Singer index theorem. All these topics are fundamental in understanding gauge theories that form the backbone of the standard model.

The final chapters 13 and 14 describe anomalies in gauge theories and the bosonic string respectively. In a sense, these chapters bring all the ideas laid out earlier in the book together to describe some quite advanced mathematical physics.

Paperback: 596 pages
Publisher: Taylor & Francis; 2 edition (4 Jun 2003)
Language English
ISBN-10: 0750306068
ISBN-13: 978-0750306065

# Higher contact-like structures and supersymmetry In my latest preprint “Higher contact-like structures and supersymmetry” I provide a novel geometric view of N=1 supersymmety in terms of a polycontact structure on superspace. The preprint can be found at arXiv:1201.4289v1 [math-ph]

The conception of the idea to describe supersymmetry in terms of some contact-like structure came from understanding SUSY mechanics in terms of a contact structure. See my preprint “Contact structures and supersymmetric mechanics” arXiv:1108.5291v2 [math-ph] and an earlier blog entry here.

# Young Researchers in Mathematics Conference 2012 Royal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee. The Young Researchers in Mathematics Conference is an annual event that aims to involve post-graduate and post-doctoral students at every level. It is a chance to meet and discuss research and ideas with other students from across the country.

I will be attending the Young Researchers in Mathematics Conference 2012 to be held at Bristol University 2nd-4th April.  I have offered to give a talk and right now awaiting confirmation that my talk has been accepted. My talk would fit into the Geometry and Topology tract.

I will post more details in due course.

YRM2012

# Mathematical Reviews

I have been invited to be a reviewer for Mathematical Reviews, which is run by the American Mathematical Association.  I have of course accepted.

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What the video below to find out more.

# Mathematics the langauge of Physics

It is a rather indisputable fact for physicists that mathematics really is the correct language   of physics.  Without mathematics one could not formulate physical theories and then make prediction to be tested against nature.  Indeed, the formulation of physical theories has required the development of new mathematics.  Theoretical physics is really the construction of mathematical models to describe nature.

Even the experimentalist cannot avoid mathematics.  One has a lot of analysis of results and statistics  to preform in order to make sense of the experiments.

It is rather clear then, that without mathematics one will not go very far in physics. Any understanding of nature is going to be rather superficial without some mathematics.

A little deeper than this I believe that mathematics is more than just a language for physics, or indeed all science. The structures, patterns and rules of mathematics can guide one in constructing/analysing theories. The notion of symmetry is so fundamental in modern theoretical physics and at its heart is group theory.  Understanding physics can be driven my mathematical beauty. Given a new theory the first question to ask is what are the symmetries?

One has to ask why mathematics is the language of the physical sciences? Can we understand why mathematics has been just so useful and powerful in structuring our understanding of the Universe?

Eugene Wigner in 1960 wrote an article The Unreasonable Effectiveness of Mathematics in the Natural Sciences which was published in Communications on Pure and Applied Mathematics.  Wigner argues that mathematics has guided many advances in the physical sciences and that this suggests some deep link between mathematics and physics far beyond mathematics simply being a language.

A very extreme version of this deep interconnection is Max  Tegmark’s mathematical universe hypothesis, which basically states that all mathematics is realised in nature.  What this hypothesise also suggests is that the Universe really is mathematical. We uncover this mathematical structure rather than impose it on nature. This would explain Wigner’s “unreasonable effectiveness”.

We are now close to having to think about the philosophy of mathematics and in particular Platonism. I am certainly no big thinker on philosophy and so will postpone discussion about the philosophy of mathematics.

I would not go as far as to say I believe in Tegmark’s hypothesis, but it is for sure an interesting and provocative idea.  It certainly makes one think about the relation between mathematics, physics  and the nature of our Universe.