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

Low dimensional contact supermanifolds

I have been interested in contact structures on supermanifolds. I though it would be useful, and fun to examine a low dimensional example to illustrate the definitions. Let \(M\) be a supermanifold. We will understand supermanifolds to be “manifolds” with commuting and anticommuting coordinates.

For manifolds there are several equivalent definitions. The one that is most suitable for generalisation to supermanifolds is the following:

Definition A differential one form \(\alpha \in \Omega^{1}(M)\) is said to be a contact form if

  1. \(\alpha\) is nowhere vanishing.
  2. \(d\alpha\) is nondegenerate on \(ker(\alpha)\)

This needs a little explaining. First we have to think about the grading here. Naturally, any one form decomposes into the sum of  even and odd parts. To simplify things it makes sense to consider homogeneous structures, so we have even and odd differential forms. Due to the natural grading of differentials as fibre coordinates on antitangent bundle a Grassmann odd  form will be known as an even contact structure and vice versa. The reason for this will become clearer later.

 

The definition of a nowhere vanishing one form is that there exists vector fields \(X \in Vect(M)\) such that \(i_{X}\alpha =1\). Again, via our examples this condition will be made more explicit.

The kernel of a one form is defined as the span of all the vector fields that annihilate the one  form.  Thus we have

\(ker(\alpha) = \{X \in Vect(M)| i_{X}( \alpha)=0 \}\).

The condition of nondegeneracy  on \(d\alpha\) is that \(i_{X}(d \alpha)=0\) implies that  \(X=0\). That is there are non-nonzero vector fields in the kernal of the contact form that annihilate the exterior derivative of the contact form.

On to simple examples. Consider the supermanifold \(R^{1|1}\) equipped with natural coordinates \((t, \tau)\). here \(t\) is an even or commuting coordinate and \(\tau\) is an odd or anticommuting coordinate.

 

I claim that the odd one form \(\alpha_{0} = dt + \tau d \tau\) is an even contact structure.

First due to our conventions, \(dt\) is odd and \(d \tau\) is even,  so the above one form is homogeneous and  odd.

Next we see that if we consider \(X = \frac{\partial}{\partial t}\) then the nowhere vanishing condition holds. Maybe more intuitively we see that considering when \(t = \tau =0\)  the one form does not vanish.

The kernel is given by

\(ker(\alpha_{0}) = Span\left \{   \frac{\partial}{\partial \tau} {-} \tau \frac{\partial}{\partial t} \right \}\).

That is we have a single odd vector field as a basis of the kernel. That is we have one less even vector field as compared to the tangent bundle. Thus we have a codimension \((1|0)\) distribution.

Then \(d \alpha_{0}= d \tau d \tau\) , so it is clear that the nondegeneracy condition holds.

Thus I have proved my claim.

This is just about the simplest even contact structure you can have.

The odd partner to this is given by

\(\alpha_{1} = d \tau {-} \tau dt\)

This is clearly an even one form that is nowhere vanishing. The kernel is given by

\(ker(\alpha_{1}) = Span\left\{ \frac{\partial}{\partial t} {-} \tau \frac{\partial}{\partial \tau}  \right\}\).

Thus we have a codimension \((0|1)\) distribution.

The nondegeneracy condition also follows directly.

For those who know a little contact geometry compare these with the standard contact structure on \(R^{3}\).

There is a lot more to say here, but it can wait.  For those of you that cannot wait, see Grabowski’s preprint arXiv:1112.0759v2 [math.DG]. Odd contact structures are also discussed in my preprints arXiv:1111.4044 and arXiv:1101.1844.

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

Moon picture

moon pic

Here is a picture of the Moon I took on the 3rd February 2012. The picture was taken using my 7MP Advent digital camera (“point and click”) directly through the eyepiece of my Bresser Skylux NG 70-700 retractor. I used a Moon filter and 20mm eyepiece.

The results are ok. I will post more as I take them.

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

One month without coffee!

Well, as a new years resolution I decided to go without coffee or caffeinated drinks.  It is now just about a month without caffeine, or at least with nothing like the dose previously consumed.

I am not sure what effect this has had on my mathematics.

Paul  Erdös once remarked “in Hungary many mathematicians drink strong coffee”.

Alfréd Rényi stated “a mathematician is a machine which turns coffee into theorems”.

So, I think we can claim a link between coffee and mathematics!

Anyhow,  as of February I will be doing more mathematics in coffee houses.

 

 

 

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

Gauge field theory and complex geometry, by Y.I. Manin

Gauge Field Theory and Complex Geometry

Complex differential geometry, holomorphic bundle theory and supergeometry are all important topics in modern theoretical physics. Gauge field theory and complex geometry by Yuri Manin gives a detailed technical account complex geometry and supergeometry aimed at mathematical physicists interested in quantum field theory.

Manin gives a great account of superalgabras and supergeometry, a subject he has made large contrbuitions to.

The readership is graduate students and resarchers in mathematical physics interested in geometric constructions in physics.

The book starts with an introduction to geometric structures in quantum field theory. Topics here include: the Feynman path-integral, the Lagrangian of QED, Fermions, exchange bosons & connections, space-time & gravity and twistors. Much of the introduction should be familiar to the readers.

Chapter 1 covers Grassmannians, connections and integrability. Topics covered include: Grassmannians, Flag spaces, distributions & connections and Grassmannian spinors.

Chapter 2 discusses the Randon-Penrose transformations. Topics include: complex space-time, instantons, instantons & modules over a Grassmann algebra, curvature on the space of null geodesics, the flow of Yang-Mills field on the space of null geodeics and Green’s functions of the Laplace operator.

Superalgebras and introduced in Chapter 3. Topics covered include: the sign rule, tensors over a supercommutative ring, the supertrace & superdeterminant and scalar products.

Chapter 4 moves on to supergeometry and supermanifolds. Topics include: definition of supermanifolds, elimentary structures on supermanifolds, supergrassmannians, connections, the Frobenius theorem, the Berezin integral, volume forms, differential & pseudodifferential forms and Lie algebras of vector fields.

Chapter 5 moves onto geometric structures of supersymmetry and gravitation. Topics here include: supertwistors, superfields & components, monads on superspaces, flag superspaces and the geometry of simple supergravity.

An appendix on recent developments written by S. A. Merkulov is included. A. discusses developments in twistor theory. B. further discusses the geometry of supermanifolds.

Hardcover: 358 pages
Publisher: Springer; 2nd edition (June 27, 1997)
Language: English
ISBN-10: 3540613781
ISBN-13: 978-3540613787

Supersymmetric Methods in Quantum and Statistical Physics by G. Junker

Supersymmetric Methods in Quantum and Statistical Physics Supersymmetry is a very important topic in high energy physics, from both theoretical and phenomenological points of view. What is less well-known is the fact that supersymmetry is also a very powerful mathematical idea in quantum mechanics and statistical physics. Supersymmetric Methods in Quantum and Statistical Physics by Georg Junker gives an introduction to supersymmetric quantum mechanics and supersymmetric methods in statistical physics.

The book assumes some familiarity with quantum mechanics, though no knowledge of quantum field theory or supersymmetry is assumed.

The main topics are the Witten model, supersymmetric classical mechanics, shape-invariant potentials and exact solutions, supersymmetry in classical stocastic dynamics and supersymmetry in the Pauli & Dirac equations. Once supersymmetric mechanics was seen as a “test model” for theoretical high energy physics, but the ideas have now reached wider interest in physics.

Chapter 1 starts with a general introduction to supersymmetry. This chapter exposes the reader to basic ideas of supersymmetry in field theory and introduces supersymmetric quantum mechanics.

Supersymmetric quantum mechanics is the topic of Chapter 2. Here one is introduced to supersymmetry in quantum mechanics rather generally and then Witten’s N=2 supersymmetric model which is the main focus of most approaches to supersymmetric quantum mechanics.

In Chaper 3 the Witten model is discussed in further detail. Topics here include: Witten parity, the SUSY potential, zero-energy states & good SUSY and the asymptotic behaviour of the SUSY potential. Examples are included.

In Chapter 4 supersymmetric classical mechanics is reviewed. Topics here include: pseudoclassical mechanics, supersymmetric classical mechanics, the classical dynamics and quantisation via the canonical approach and path-integrals.

Exact solutions of the eigenvalue problem is the subject of Chapter 5. Here the reader is introduced to shape-invariant potentials and exact solutions. Examples and a comparison to the factorisation method are presented.

Chapter 6 discusses path-integration in further detail. Topics include: the WKB approximation, quasi-classical quantisation conditions and quasi-clasical eigenfunctions. Exactly solvable examples are presented as well as some numerical investigations.

Supersymmetry in classical stochastic dynamics is the subject of Chapter 7. Topics include: supersymmetry in the Fokker-Planck & Lengevin equations and the implications of good/broken supersymmetry.

Chapter 8 discusses supersymmetry in the Pauli & Dirac equations.

The final chapter, Chapter 8 gives concluding remarks and an overview. Here a list of textbooks covering supersymmetry in high energy physics is give. A table of applications of supersymmetry in theoretical physics is also included.

A rather large set of references is given.

Hardcover: 172 pages
Publisher: Springer (September 30, 1996)
Language: English
ISBN-10: 3540615911
ISBN-13: 978-3540615910

Random thoughts on mathematics, physics and more…