# 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

# A duality between the Ricci and energy-momentum tensors

I don’t claim this to be new, the earliest reference I can find is Baez and Bunn , though I am sure the idea is older than that.   The claim is that there is a kind of duality in the Einstein field equations between the Ricci tensor and the energy-momentum tensor. That is one can in essence switch the roles of the Ricci tensor and the energy-momentum tensor in the field equations.  I will assume familiarity with the  tensors and the field equations.

Lets see how this works. Recall the Einstein field equations in 4d

$$R_{\mu \nu}{-} \frac{1}{2}g_{\mu \nu}R + g_{\mu \nu} \Lambda = \kappa T_{\mu \nu}$$,

here $$\kappa = \frac{8 \pi \: G}{c^{4}}$$ is the gravitational constant

and of course  $$R:= R_{\lambda}^{\:\: \lambda}$$.

The field equations imply that

$$R_{\mu}^{\:\: \mu} {-} \frac{1}{2} g_{\mu}^{\:\: \mu}R_{\lambda}^{\:\: \lambda} + g_{\mu}^{\:\: \mu} \Lambda = \kappa T_{\mu}^{\:\: \mu}$$.

We assume that we are in 4d thus

$$g_{\mu}^{\:\: \mu} =4$$.

One could consider other dimensions, but things work out clearer in 4d and anyway this is where classical general relativity is formulated.

Thus we arrive at

$${-}R_{\mu}^{\:\: \mu} = \kappa T_{\mu}^{\:\: \mu} {-} 4 \Lambda$$.

Now using this result in the field equations produces

$$R_{\mu \nu} = \kappa \left( T_{\mu \nu}{-} \frac{1}{2}g_{\mu \nu}T_{\lambda}^{\:\: \lambda}\right) + g_{\mu\nu} \Lambda$$.

Now divide by the  gravitational constant to write the field equations as

$$T_{\mu \nu} {-} \frac{1}{2} g_{\mu \nu}T_{\lambda}^{\:\: \lambda} + g_{\mu \nu} \left( \frac{\Lambda}{\kappa}\right) = \left(\frac{1}{\kappa}\right)R_{\mu \nu}$$.

Comparing the above with the original form of the field equations we see that we have a kind of duality given by

$$R \rightarrow T$$

$$T \rightarrow R$$

$$\Lambda \rightarrow \frac{\Lambda}{\kappa}$$

together with the inversion of the gravitational constant,

$$\kappa \rightarrow \kappa^{-1}$$.

I some sense we have done nothing. Both forms of the Einstein field equations are equally valid and describe exactly the same physics. The difference, as I see it is that the second form, this “dual form”, is better from a geometric perspective.

In particular the Ricci curvature tensor has a clear geometric origin.  Via the   Raychaudhuri equation, the Ricci tensor (for a Lorentzian signature metric) measures  the degree to which near by test particles will tend to converge or diverge.

Then one can then paraphrase the Einstein field equations as

The degree test particles tend to converge or diverge  in time is determined by the matter content  + the cosmological constant.

I am not aware of any such nice interpretation of the Einstein tensor.

Another interesting point is that one gets at the vacuum equations very quickly with this “dual form”.  Just “turn off” T.

The real question here is “does this duality have a deeper meaning?”. This I really do not know.  It would also be interesting to understand if any technical issues can be addressed via this “duality” and how this really helps us understand gravity.

My literature hunts needs to continue…

References

 John C. Baez & Emory F. Bunn. The Meaning of Einstein’s Equation. Amer. Jour. Phys. 73 (2005), 644-652. Also available as  arXiv:gr-qc/0103044.

Update

The alternative form of the field equations is also presented in Wolfgang Rindler’s Essential Relativity, revised second edition, 1977.  So the idea is old and I am sure to be found in other books.

# 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