Category Archives: Book Reviews

Einstein Relatively Simple by Egdall


Both special and general relativity have a reputation for being very complicated theories to understand. While it is true that one needs some mathematical machinery to really master these theories, Egdall does a great job in showing that it is possible with a bare minimum of high school mathematics to get an appreciation of the main ideas.

Edgall does not completely stay away from some mathematics, though as it is isolated somewhat from the main text, the mathematically shy should not be scared away from this book. The style is lighthearted and is full of thought experiments illustrated by short entertaining stories.

The author does an excellent job in highlighting the main features of special and general relativity in a way suitable for the lay reader to understand. Moreover, the development of the theory is presented in a chronological/historical context by trying to describe Einstein’s trail of thought and how he was influenced by the various problems with physics found in the late 1800’s. We get a good overview of Einstein the man from this book.

However, like all popular science books, there are the odd statements like “…the photon’s perspective” and “..spacetime curvature has energy” which any physicist will question. That said, I do consider the book well-written, entertaining and a useful introduction to the ideas of relativity for the lay person.

The book is divided into two chapters, an appendix with more mathematical details and a rather extensive list of notes with sources. Part I deals with special relativity describing the initial development of the theory starting from the conflict between classical mechanics and electromagnetic theory. The strange consequences of Einstein’s special relativity including the idea of space-time, time dilation, length contraction and his famous equation E=mc^2 are discussed.

Part II introduces us to general relativity which is the most accurate theory of gravity know to science. The fundamental idea is that gravity should be viewed as space-time curvature and some of the consequences of this are discussed in this book. Some aspects of modern cosmology and the big bang theory are also briefly discussed.

Note: This book is based on lay courses in modern physics Edgall teaches at Lifelong Learning Institutes at several universities in South Florida.

Paperback: 300 pages
Publisher: World Scientific Publishing Company (February 24, 2014)
Language: English
ISBN-10: 9814525596
ISBN-13: 978-9814525596

Einstein Relatively Simple

Natural Operations in Differential Geometry by Kolar, Michor and Slovak


Natural Operations in Differential Geometry by Kolar, Michor and Slovak is a monograph that covers the modern way of thinking in differential geometry in terms of natural bundles and natural operators. The book stresses the role of naturality and functoriality in differential geometry.

The book covers a lot of material including fibered manifolds, connections, Jet bundles and Weil bundles.

I have found the book to be continually useful for a number of years now as it covers a lot of important modern material, especially jets and Weil bundles. It is a must for any serious differential geometer, but it is probably not the book to introduce you to the subject.

Chapter 1 “MANIFOLDS AND LIE GROUPS” covers manifolds, submersion & immersions, vector fields & flows, Lie groups and homogeneous spaces. This forms a general introduction to the “bread and butter” of differential geometry.

Chapter 2 “DIFFERENTIAL FORMS” introduces vector bundles, differential forms, derivations on the algebra of differential forms and the Frolicher-Nijenhuis bracket.

Chapter 3 “BUNDLES AND CONNECTIONS” deals with general fibre bundles and their connections. The definition of a general connection on a fibre bundle is given in terms of a one-form with values in the vertical vector bundle of the fibre bundle. In this chapter they discuss curvature and the special case of principal fibre bundles.

Chapter 4 “JETS AND NATURAL BUNDLES” introduces us to the notion of jets of functions between manifolds. Jets play an important and fundamental role in the theory of natural bundles which are defined as functors from the category of smooth manifolds to fibre bundles such that local diffeomorphisms of the manifold become bundle automorphisms.

Chapter 5 “FINITE ORDER THEOREMS” the authors develop a general framework for the theory of geometric objects and operators and to reduce local geometric considerations to finite order problems. A generalisation of the Peetre theorem is given for nonlinear operators.

Chapter 6 “METHODS FOR FINDING NATURAL OPERATORS” the authors present some general procedures useful for finding some equivariant maps and use these to solve concrete geometric problems.

Chapter 7 “FURTHER APPLICATIONS” discusses the Frolicher-Nijenhuis bracket, Jet functors and some topics from Riemannian geometry.

Chapter 8 “PRODUCT PRESERVING FUNCTORS” introduces us to the notion of a Weil algebra and a Weil functor. An important result here is that all product preserving natural functors are Weil functors.

Chapter 9 “BUNDLE FUNCTORS ON MANIFOLDS” here the authors discuss more general bundle functors, that is the ones that do not preserve products.

Chapter 10 “PROLONGATION OF VECTOR FIELDS AND CONNECTIONS” covers the prolongation of vector fields and connections to Weil bundles.

Chapter 11 “GENERAL THEORY OF LIE DERIVATIVES” discusses the notion of a generalised Lie derivative of a function between two manifolds. A major advantage advantage of this approach is that it enables one to study the Lie derivatives of the morphisms of fibered manifolds.

Chapter 12 “GAUGE NATURAL BUNDLES AND OPERATORS” the authors generalise some of the earlier presented notions to fiber bundles associated with and abstract principal bundle with an arbitrary structure group G.

You can find the book and some more details about the chapters here.

Hardcover: 434 pages
Publisher: Springer; 1993 edition (February 4, 1993)
Language: English
ISBN-10: 3540562354
ISBN-13: 978-3540562351

Isaac Newton and his Apple (Dead Famous)


We have all heard of Sir Issac Newton, that is right the guy who got hit on the head by a falling apple, well as the story goes…

In Isaac Newton and his Apple, Kjartan Poskitt describes in an entertaining and graphic way (illustrations by Phillip Reeve) the life and times of Newton.

The book, aimed children 9 and above, described Newton’s troubled life, his not very pleasant demeanor, his feud with Hooke and arguments with Leibniz. The book is scattered with other historical facts that help set the scene of Newton’s life.

And of course, it describes his mathematical genius and his catchy titled book Philosophiæ Naturalis Principia Mathematica. Interestingly, Isaac Newton and his Apple does contain a little proper mathematics. The ideas of Newton’s laws and even the dreaded calculus are presented with some mathematics. This should not scare anyone, you can, without spoiling the story too much, skip the mathematics.

Paperback: 192 pages

Publisher: Hippo; 1 edition (15 Oct 1999)

Language: English

ISBN-10: 0590114069

ISBN-13: 978-0590114066

Paradox by Jim Al-Khalili


Through out the development, and indeed the popularisation of physics apparent paradoxes have had a prominent place. These paradoxes are usually the product of applying false logic to physics and/or a misunderstanding of what the mathematical theories are really telling us about the Universe.

Prof. Jim Al-Khalili in his latest book, Paradox: The Nine Greatest Enigmas in Physics, takes us on a mind-bending tour of some of the perplexing puzzles that have shaped our understanding of the natural world.

This book is about my own personal favourite puzzles and conundrums in science, all of which have famously been referred to as paradoxes, but which turn out not to be paradoxes at all when considered carefully and viewed from the right angle.

Jim Al-Khalili


Photo by Andy Miah

Some of the paradoxes covered are well known. For example, Schrödinger’s famous cat, Olbers’ paradox and the time-travelers favorite, the grandfather paradox.

In resolving our paradoxes we will have to travel to the furthest reaches of the Universe and explore the very essence of space and time. Hold on tight.

Jim Al-Khalili

In my opinion Al-Khalili presents a rather readable and light-hearted review of some rather interesting issues in physics. It was an enjoyable read overall.

Outline of the book

Chapter 1, The Game Show Paradox, discusses the now infamous Monty Hall problem. Without giving the game away too much, one has to think about the role of prior knowledge in probability theory. This is more a problem in basic probability theory than physics, but it illustrates the need for careful thinking.

Achilles and the Tortoise is the title of chapter 2. Here the topic is Zeno’s paradoxes-“motion is an illusion”. Clearly, motion is not an illusion, so what is going wrong? What false logic and/or mathematics is being applied here?

Chapter 3 deals with Olber’s paradox: why is the night sky dark? The resolution takes us to the forefront of cosmology. How does this support the Big Bang?

Maxwell’s demon is the title of chapter 4. Maxwell imagined a hypothetical experiment describing how to violate the Second Law. In his thought experiment a container is divided into two parts by an insulating wall, with a door that can be opened and closed by some entity known as “Maxwell’s demon“. The demon is able to view the molecules and opens the door only for “hot molecules”. Thus, one side of the container heats up and the other cools, causing a decrease in entropy in violation of the Second Law. How is this resolved?

The idea of length contraction and simultaneity in special relativity are the subject of chapter 5. The Pole and the Barn Paradox, which is always used in university classes is discussed and resolved.

Chapter 6, The Twin Paradox, deals with the very famous paradox of relativistic twins. How can they age differently? What is time dilation and is it real?

Time travel and the paradoxes it creates have fascinated people for a long time. Do we know if time travel is allowed or not anyway? Chapter 7, The Grandfather Paradox, discusses the issues here and suggests some resolutions to this. What happens if you went back in time and killed your grandfather? Maybe we live in a multiverse and time travel means we enter a parallel world? Maybe the laws of physics just don’t allow time travel? Non-one really know and this is an unresolved paradox.

In chapter 6, The Paradox of Laplace’s Demon, the idea of determinism is discussed along with the technical issue of chaos. Determinism says that if we know all the positions and momenta of all the particles in the Universe at a given time, then we can use classical mechanics to predict exactly the state of the Universe at some later time. Any being that could do this would loose freewill as everything is totally predetermined, this is the demon. This is not compatible with quantum mechanics (as we know it), but even more basic than this classical systems do exhibit chaos- “sensitivity to initial conditions”. Can we have determinism, but not predictability?

The Paradox of Schrödinger’s Cat is the subject of chapter 9. According to quantum mechanics, is the cat when we are not actually looking at him alive and dead at the same time? What is the role of the observer? How come, if everything is made of quantum particles do macroscopic objects behave classically? Not everything here is that well understood and quantum decoherence, that is many quantum objects becoming a classical object, is an ongoing area of research, both mathematically and experimentally.

Are we alone in the Universe? As the Universe is large and old there must be other forms of life out there? Some of these must be intelligent, capable of space flight and given enough time colonise the Galaxy? In chapter 10, Fermi’s Paradox Al-Khalili asks these questions. Fermi highlighted the contradiction between high estimates of the probability of the existence of an extraterrestrial civilization and the lack of evidence that they exist.

The book finishes off in chapter 11 entitled Remaining Questions. Al-Khalili lists some of the open problems in science that today are at the forefront of human knowledge. One should take away from this chapter the sense that despite what we do know about the Universe, questions remain.

UK publication

Hardcover: 256 pages
Publisher: Bantam Press (12 April 2012)
Language: English
ISBN-10: 0593069293
ISBN-13: 978-0593069295

US publication

Paperback: 256 pages
Publisher: Broadway; Reprint edition (October 23, 2012)
Language: English
ISBN-10: 0307986799
ISBN-13: 978-0307986795


Space, time, and gravity: the theory of the big bang and black holes, by Wald

The general theory of relativity has a reputation of being very difficult to comprehend. This is especially true for the layperson or undergraduate students. Space, time, and gravity: the theory of the big bang and black holes by Robert M. Wald plots a very accessible course through the core phenomena of general relativity without any unnecessary mathematical detail.

The book is based on lectures that Wald gave in Chicago 1977. The 1992 version is updated due to the great advancements in observation cosmology since the first edition.

The book consists of 10 main chapters, an appendix (in the 1992 edition) and a list of suggestions for further reading.

Chapter 1 describes the geometry of space and time from a Newtonian “everyday” perspective. The ideas of spce-time diagrams, the principle of relativity and absolute simultaneity are given here. In the next chapter it shown how these ideas need modifying following Einstein.

Chapter 2 introduces special relativity. The principle concepts of Einstein’s axioms, relative simultaneity, the space-time interval, Lorentz contraction and time dilation are neatly presented. Wald emphasises the geometric way of thing which is paraphrased as “it is all in the metric”. With this in mind, geodesics in Minkowski space-time are discussed and the conclusion is drawn in special relativity the space-time is flat .

General relativity is the subject of Chapter 3. Wald discusses why Newtonian gravity cannot be directly incorporated in to special relativity. The key trouble is the notion of relative simultaneity and instantaneous interactions. The solution is that equivalence principle strongly suggest that space-time is not flat. Wald then moves on to discuss the Einstein field equations, the bending of light, and the gravitational red shift.

Chapter 4 discusses the implications of general relativity for cosmology. Principally it is argued that the Universe must have started from a singular point and that the “Big Bang” is rather unavoidable in the the context of classical general relativity. Wald describes the large scale structure of the Universe, the “Big Bang” and Hubble’s law. In the final part he discusses singularity theorems and questions if we should take singularities seriously? Only quantum gravity will really shed light on this…

Chapter 5 presents a walk through the evolution of the Universe. Evidence for why we think the Big Bang is a good theory is presented here. For example the CMBR and the cosmic abundance of deuterium.

Stella evolution up to white dwarfs and neutron stars is the subject of Chapter 6. Wald discusses stellar birth, the continued evolution of stars, teh electron degeneracy pressure and white dwarfs, supernovae and neutron stars.

The gravitational collapse to form black holes is the topic of Chapter 7. The basic structure of black holes is presented as is the cosmic censorship hypothesis and the no-hair theorem. Due to the singularity theorems of Penrose and Hawking, and other works it is generally accepted that complete gravitational collapse will always result in a black hole.

Chapter 8 discusses the Penrose process of energy extraction from a rotating black hole. Wald discusses the notion of ennergy-momentum in special and general relativity. The complications of what we mean by energy in general relativity are highlighted. The Ergosphere of a rotating black hole is described as is the Penrose process and the energy extraction limits.

The astrophysics of black holes is the subject of Chapter 9. Wald discusses the proposed mechanisms of black hole production: collapse of a massive enough star, collapse of a cluster of stars and primordial black holes. Wald then discusses gravitational lensing, gravitational radiation and X-ray sources.

Chapter 10 discusses particle creation near a black hole and Hawking radiation. The consequence for conservation laws of black hole evaporation are discussed as is the strong link with thermodynamics.

We have learnt a lot since the 1977 edition and also the 1992 edition about observation cosmology and the deep link between black holes and thermodynamics. That said, the book is a wonderful expose to these deep ideas in a way that is rather accessible.

As a note, I own the first edition from 1977 and so based the above review on that. From what I can tell the second edition is very similar but the numbers are updated.

Paperback: 164 pages
Publisher: University of Chicago Press; 2nd Revised edition edition (1 Mar 1992)
Language English
ISBN-10: 0226870294
ISBN-13: 978-0226870298

Quantum Field Theory A Modern Introduction by M. Kaku

Quantum Field Theory: A Modern Introduction

Quantum field theory is a many faceted subject and represent our deepest understanding of the nature of forces and matter. Quantum field Theory A Modern Introduction by Michio Kaku gives a rather wide overview of many essential ideas in modern quantum field theory.

The readership is graduate students in theoretical physics who already have some exposure to quantum mechanics and special relativity.

The book is divided into three parts.

Part 1 Quantum Fields and Renormalization

Chapter 1 gives a historic overview of quantum field theory. Topics here include: a review of the strong, weak and gravitational interaction, the idea of gauge symmetry, the action principle and Noether’s theorem.

Symmetries and group theory are the subjects of Chapter 2. Topics include: representations of U(1), SO(2), SO(3) and SU(2), spinors, the Lorentz group, the Poincare group and supersymmetry.

Chapter 3 moves on to the quantum theory of spin-0 and spin 1/2 fields. The emphasis here is on canonical quantisation. Topics covered here include: the Klein-Gordon field, propagator theory, Dirac spinors and Weyl neutrinos.

Quantum electrodynamics is the topic of Chapter 4. Again the emphasis is on canonical quantisation. Topics include: Maxwell’s equations, canonical quantisation in the Coulomb gauge, Gupta-Bleuler quantisation and the CPT theorem.

Chapter 5 describes the machinery of Feynman diagrams and the LSZ reduction formula. Topics here include: cross sections, propagator theory, the LSZ reduction formulas, teh time evolution operator, Wick’s theorem and Feynman rules.

The final chapter of part 1, Chapter 6 describes the renormalization of quantum electrodynamics. Topics here include: nonrenormalizable & renormalizable theories, the renormalization of phi-4 theory, regularisation, the Ward-Takahashi identites and overlapping divergences. The renormalization of QED is then broken down into fours steps.

Part 2 Gauge Theory and the Standard Model

Chapter 8 introduces path integrals which are now fundamental in particle theory. Topics here include: path integrals in quantum mechanics, from first to second quantisation, generators of connected graphs, the loop expansion, integration over Grassmann variables and the Schwinger-Dyson equations.

Chapter 9 covers gauge theory. Topics here include: local symmetry, Faddeev-Popov gauge fixing, the Coulomb gauge and the Gribov ambiguity.

The Weinberg-Salam model is the subject of Chapter 10. Topics here include: broken symmetries, the Higgs mechanism, weak interactions and the Coleman-Weinberg mechanism.

Chapter 11 discusses the standard model of particle physics. Topics here include: the quark model, QCD, jets, current algebra, mixing angles & decays and the Kobayashi-Maskawa matrix.

Chapter 12 discusses anomalies and the Ward identities. Topics here include: the Ward-Takahashi identity, the Slavonov-Taylor identities, BRST symmetry & quantisation, anomalies and Fujikawa’s method.

Chapter 12 covers the remormalization of gauge theories. Topics include: counterterms, dimensional regularization and BPHZ renormalization.

The modern perspective of QFT is based on Wilson’s renormalization group. Chapter 14 introduces the reader to this concept in the context of QCD. Topics here include: deep inelastic scattering, neutrino sum rules, the renormalisation group, asympptotic freedom and the Callan-Symanzik relation. The renormalization of QCD is presented via renormalization groups methods.

Part 3 Nonperturbative Methods and Unification

Chapter 15 introduces lattice gauge theory which allows questions in quantum field theory to be numerically tackled on computers. Topics here include: the Wilson lattice, scalars & fermions on the lattice, the strong coupling approximations, Monte Carlo simulations and the renormalization group.

Topological objects in field theory are the topic of Chapter 16. Topics include: solitons, monopoles, instantons & tunneling and Yang-Mills instantons & the theta vacua.

Chapter 17 discusses phase transitions and critical phenomena. Topics covered include: critical exponents, the Ising model, the Yang-Baxter relations, the mean-field approximation and scaling & the renormalisation group.

The idea of unification is the subject of Chapter 18. Topics include: unification & running coupling constants, SU(5), anomaly cancellation, the hierarchy problem, SO(10), technicolor, preons & subquarks and supersymmetry and strings.

Chapter 19 discusses quantum gravity. This chapter is about attempting to construct a perturbative theory of quantum general relativity. Topics include: the equivalence principle, vierbeins & spinors, GUTs & cosmology, the cosmological constant, Kaluza-Klein theory and counter terms in quantum gravity.

Supersummetry is the subject of Chapter 20. Topics covered here include: supersymmetric actions, superspace methods, Feynman rules, nonrenormalization theorems, finite field theories, super groups and supergravity.

Chapter 21 introduces the superstring. Topics include: quantisation of the bosonic string, teh four superstring theories, higher loops, string phenomenology, light-cone string field theory and the BRST action.

The book contains exercises.

Paperback: 804 pages
Publisher: OUP USA; New Ed edition (6 Oct 1994)
Language English
ISBN-10: 0195091582
ISBN-13: 978-0195091588

Topology and geometry for physicists, by C. Nash & S. Sen

Topology and Geometry for Physicists

Geometry and topology are now a well established tools in the theoretical physicists tool kit. Topology and geometry for physicists by C. Nash & S. Sen gives a very accessible introduction to the subject without getting bogged down with mathematical rigour.

Examples from condensed matter physics, statistical physics and theoretical high energy physics appear throughout the book.

However, one obvious topic missing is general relativity. As the authors state, good books on geometry & topology in general relativity existed at the time of writing.

The first 8 chapters present the key ideas of topology and differential geometry.

Chapter 1 discusses basic topology. Topics include homomorphisms, homotopy, the idea of topological invariants, compactness and connectedness. The reader is introduced to “topological thinking”.

Manifolds are the subject of Chapter 2. Topics include: the definition of manifolds, orientablilty, calculus on manifolds and differential structures.

Chapter 3 discusses the fundamental group. Topics include: the definition of the fundamental group, simplexes, triangulation and the fundamental group of a product of spaces.

Chapter 4 moves on to the homology group. Topics include: the definition of homology groups, relative homology, exact sequences, the Kunneth formula and the Poincare-Euler formula.

The higher homotopy groups are the subject of Chapter 5. Topics covered include: the definition of higher homotopy groups, the abelian nature of higher homotopy groups and the exact homotopy sequence.

The de Rham cohomology of a manifold is the subject of Chapter 6. Topics include: Poincare lemma, calculation of de Rham cohomology for simple examples, the cup product and a comparison of homology with cohomology.

Chapter 7 presents the core concepts of differential geometry. Topics here include: fibre bundles, sections, the Lie derivative, connections on bundles, curvature, parallel transport, geodesics, the Yang-Mills connection and characteristic classes.

Chapter 8 outlines Morse theory. Topics include: the Morse inequalities and the Morse lemma. Connection with physics is established via symmetry breaking selection rules in crystals.

The next two chapters look at application in physics of some of the ideas presented earlier in the book.

Defects and homotopy theory is the subject of Chapter 9. Topics include: planar spin in 2d, ordered mediums and the stability of defects theorem.

Chapter 10 discusses instantons and monopoles in Yang-Mills theory. Topics here include: instantons, instanton number & the second Chern class, instantons in terms of quaternions, twistor methods, monopoles and the Aharanov-Bohm effect.

Paperback: 311 pages
Publisher: Academic Press Inc; New edition edition (Jun 1987)
Language English
ISBN-10: 0125140819
ISBN-13: 978-0125140812

The book has also been reprinted by Dover Books in 2011.

Paperback: 311 pages
Publisher: Dover Publications Inc.; Reprint edition (17 Feb 2011)
Language English
ISBN-10: 0486478521
ISBN-13: 978-0486478524

An Introduction to Noncommutative Differential Geometry and its Physical Applications, by J. Madore

An Introduction to Noncommutative Differential Geometry and its Physical Applications Approaches to quantum gravity suggest that the very small scale structure of space-time becomes inherently “fuzzy”. This suggestion leads to a new perspective on geometry known as noncommutative geometry. The principle idea is to replace the sheaf of functions on a manifold with some noncommutative algebra and treat this as if it were the functions on some “space”. An Introduction to Noncommutative Differential Geometry and its Physical Applications by J. Madore presents an overview of noncommutative differential geometry assessable to physicists and geometers alike.

There are two main approaches to noncommutative differential geometry. The “mathematics” approach is based on Connes’ ideas of spectral triples. The “physics” approach is deformation and quantised coordinate rings. This books focuses on the “physics” approach and Connes’ spectral triples are mentioned in passing.

The readership is graduate students and researchers in mathematical or theoretical physics interested in noncommutative geometry and modifications to space-time. The prerequisites are some familiarisation with quantum mechanics and differential geometry. Knowing quantum field theory and maybe some string theory would help motivate reading the book, though this is not essential. Noncommutative geometry is a mathematical subject in its own right.

Chapter 1 is an introduction. This chapter gives the basic idea of noncommutative geometry as the reformation of the theory of manifolds in terms of the algebra of functions and then generalise this to more general algebras.

The next 5 chapters lay down the mathematics of noncommutative geometry.

Chapter 2 gives a review of differential manifolds in the language of the structure sheaf. Topics covered include: differential forms, vector fields, connections, metrics and de Rham cohomology. The reader is expected to be familiar with these topics, but not necessarily formulated algebraically in terms of the coordinate ring.

Matrix geometry is the subject of Chapter 3. Matrix geometry can be thought of as a finite dimensional noncommutative geometry and thus all calculations reduce to algebra. One of the main issues in noncommutative differential geometry is how to define differential forms and vector fields. The subtleties are introcuded in matrix geometry ready for more general algebras. Topics include: vector fields as derivations, differential calculi dual to the derivations, differential algebras & universal calculus, metrics and connections.

More general noncommutative geometries are the subject of Chapter 4. Topics include: general algebras, quantised coordinate rings, Poisson structures as “classical limits”, topological algebra (operator algebras) and Hopf algabras.

Chapter 5 discusses vector bundles and K-theory. Topics here include: classical vector bundles as projective modules, matrix analogues thereof and Fredholm modules.

Chapter 6 moves on to cyclic homology. Topics here include: universal calculus, Morita equivalence and the Loday-Quillen theorem.

The next chapters take on a more physics flavour.

Modifications of space-time are the subjects of Chapter 7. Topics here include: noncommutative space-time and “fuzzy” physics.

Chapter 8 discusses some extensions of space-time. Topics here include: the spinning particle, noncommutative electodynamics and Kaluza-Klein theory.

The book is full of examples often well motivated by physics. Each chapter contains short notes that point to the original literature and suggest further reading. The book contains a very health set of references that include arXiv numbers where possible.

Paperback: 380 pages
Publisher: Cambridge University Press; 2 edition (August 13, 1999)
Language: English
ISBN-10: 0521659914
ISBN-13: 978-0521659918

Local Quantum Physics, by R. Haag

Local Quantum Physics: Fields, Particles, Algebras Quantum field theory is the theory that describes all the known forces apart from gravity. However, the standard approach to quantum field theory via path-integrals is fraught with mathematical difficulties. One mathematical approach is to remove the fields as the primary objects and consider the algebras of observables as the fundamental objects of interest. Local Quantum Physics by Rudolf Haag introduces the reader to the ideas of constructive field theory and algebraic field theory.

Haag is a major player in algebraic field theory and this book gives his perspectives on the subject. The main mathematical tools employed in algebraic field theory are nets of c*-algebras. The book does not assume the reader is an expert in operator algebra, though some familiarity with quantum mechanics, quantum field theory and special relativity is a prerequisite. The book is not aimed at experts in constructive or algebraic field theory and so serves as a great introduction to the subject. The reader will be exposed to the main concepts and main theorems of algebraic field theory.

The readership is graduate students and researchers in mathematical physics interested in rigorous approaches to quantum field theory. The book may also be of interest to mathematicians working in operator algebra who would like to understand how to make contact with advanced physics.

In the following I will be referring to the second revised and enlarged edition of the book.

The book is divided into 8 chapters each containing between 3 and 5 sections.

Chapter I lays down the background of quantum field theory. Topics here include: basic concepts of quantum mechanics, locality in relativistic theories, Poincare invariant quantum field theory, the action principle and basic quantum field theory like canonical quantisation, free fields and gauge invariance.

General quantum field theory is the topic of Chapter II. Here one encounters the ideas of constructive field theory via the Wightman axioms. Topics discussed here include: the Wightman axioms, generating functionals, time ordered functions, covariant perturbation theory via Feynman diagrams, asymptotic configurations and particles, the S-matrix and the LSZ-formulation. The CPT theorem, spin-statistics theorem and analytical properties of the S-matrix are also discussed.

Chapter III moves on to algebraic field theory and the algebra of local observables. Topics here include: operator algebras (von Neumann, c* and w*), factors, positive linear forms and states, the GNS construction, nets of algebras of local observables and vacuum states. This chapter discusses the “guts” of algebraic field theory.

The next two chapters discuss some of more advanced aspects of algebraic field theory. That is superselction rules and the KMS-states.

Superselection sectors and symmetry is the topic of Chapter IV. Topic discussed include: charge superselction sectors, the DHR-anaysis, the Buchholz-Fredenhagen analysis, low dimensional space-time and braid statistics.

Chapter V moves on to thermal states and modular automorphisms. Topics here include Gibbs ensembles, the KMS condition, the Tomita-Takesaki theorem, equilibrium states, modular automorphisms of local algebras and nuclearity.

Chapter VI discusses the particle picture of quantum field theory. Topics here include: asymptotic particle cofigurations, particles & infraparticles and the physical state space of QED.

The interpretations and mathematical formalism of quantum physics is the topic of Chapter VII. Topics here include: the Copenhagen interpretation, the classical approximation, “quantum logic” and the EPR-effect.

The final part of the book, Chapter VIII is a concluding chapter which presents a retrospective look at algebraic field theory and presents some future challenges. Topics discussed here include: a comparison with Euclidean quantum field theory, supersymmetry and general relativity. The book presents the challenges faced by gravity, for instance QFT on curved space-times, Hawking radiation and the possibility of quantum gravity.

The book contains a healthy bibliography as well as an author index with references. This allows the reader to chase up the original literature, if desired.

Paperback: 390 pages
Publisher: Springer; 2nd. rev. and enlarged ed. edition (5 Aug 1996)
Language English
ISBN-10: 3540610499
ISBN-13: 978-3540610496

Supermanifolds: theory and apllications by A. Rogers

Supermanifolds: Theory and Applications

Supermanifolds are a useful geometric construction with applications in theoretical physics as well as pure mathematics. Supermanifolds: theory and applications by Alice Rogers describes the various approaches to the theory of supermanifolds and presents a unifying picture. Indeed it is not often necessary to specify exactly what definition of a supermanifold one is employing, though subtleties may need handling.

The readership is graduate students and researchers in mathematical physics, theoretical physics, differential or algebraic geometry who wish to learn about supermanifolds and their applications in theoretical physics. The power of this book is in the unifying approach and comparison between the different definitions of supermanifolds.

Chapter 1 is an introduction and overview.

Chapter 2 introduces superalgabra. Topics covered include: superalgabras & their morphisms, super matrices and super Lie algebras.

The notion of superspace is the subject of Chapter 3. Topics include: real Grassmann algabras, topology of superspace and complex superspaces.

Chapter 4 discusses functions of anticommuting variables. Topics here include: differentiation, Taylor expansion & Grassmann analytic continuation, the inverse function theorem and superholomorphic functions.

Supermanifolds in the DeWitt concrete approach are the subject of chapter 5. Topics include: the topology of supermanifolds, the body of a supermanifold and complex manifolds.

Chapter 6 discusses more on the geometry of DeWitt supermanifold, though the ideas pass to supermanifolds understood as locally super-ringed spaces. Topics discussed include: functions between supermanifolds, tangent vectors & vector fields, induced maps and integral curves.

Chapter 7 presents the algebro-geometric approach to supermanifolds. That is supermanifolds understood in terms of locally ringed spaces. Topics discussed include: the definition of supermanifolds in the language of locally ringed spaces, local coordinates and morphisms.

Chapter 8 discusses the different approaches to supermanifolds. Topics include: Batchelor’s theorem, split supermanifolds and a comparison between the concrete and algebro-geometric approach.

Super Lie groups are the subject of Chapter 9. Topics include: definitions & examples, super Lie groups and Lie algebras, super Lie group actions and the exponential map.

Chapter 10 moves on to tensors and forms. Topics include: tensors, Berezin densities, exterior differential forms and super forms.

The subtle issue of integration on supermanifolds is the topic of Chapter 11. Topics include: the Berezin integral, integration on compact supermanifolds, Voronov’s theory of integration of super forms and integration of exterior forms.

Chapter 12 moves on to geometric structures on supermanifolds. Topics here include: fibre bundles, the frame bundle, Riemannian metrics, even and odd sympelctic structures.

Chapter 13 links some of the supergeometric ideas with supersymmetry. Topics covered here include: the superspace formulation, superfields, supergravity and embeddings.

Super Riemannian surfaces are the subject of Chapter 14. Such objects are vital in superstring theory. Topics here include: the supergeometry of spinning strings, supermoduli spaces, contour integration and fields on super Riemannian surfaces.

Path integrals on supermanifolds are discussed in Chapter 15. Topics include: path integrals with fermions, fermionic Brownian motion, Stochastic calculus in superspace and Brownian motion on supermanifolds.

Chapter 16 discusses BRST quantisation. Topics include: symplectic reduction, BRST cohomology & quantisation.

Chapter 17 find applications of supermanifolds to differential geometry. Topics include: differential forms, spinors and the Atiyah–Singer index theorem.

Hardcover: 264 pages
Publisher: World Scientific Publishing Company (April 18, 2007)
Language: English
ISBN-10: 9810212283
ISBN-13: 978-9810212285