1 comments about Loops, Knots, Gauge Theories and Quantum Gravity.

1. The search for a theory of quantum gravity has occupied the time of a large number of researchers for over half a century, but as yet there is hardly any agreement on the conceptual foundations of such a theory, and this situation is aggravated by the lack of any experimental results that would drive its construction. Such evidence is absolutely necessary, for the lack of it sometimes leads the researcher into making wild speculations, which even though they are interesting from a mathematical standpoint, cannot be distinguished from alternatives that also have no experimental foundation. This situation has not prevented researchers from trying to find a theory of quantum gravity, and some of them proceed by analogy to what is done in quantum field theories that are well understood, such as quantum electrodynamics. Other researchers, particularly those that have a sophisticated mathematical background, have chosen string theory as the best candidate for a quantum theory of gravity.
   
   As is readily apparent in the forward to this book, the authors favor the first approach, believing that quantum gauge theories, of which quantum electrodynamics is a primary example, offer the best hope for guidance in constructing a viable quantum gravity. They emphasize though a very particular aspect of these theories, namely that the requirement for gauge invariance forces one to view the "Wilson loops" as being the entities of primary importance. But more importantly, the authors assert that that Wilson loops allow one to gain information in the non-perturbative realm of quantum field theory. Calculations in non-perturbative quantum field theory are notoriously difficult, even though some progress has been made in the area of lattice gauge theories, so any insight the authors can offer in this regard is of utmost importance. Hence this book should be viewed as a study of quantum observables on the loop space. The authors hope that these observables, called `Schwinger functions' in the perturbative realm, will along with the differential equations and boundary conditions that determine them, will give a viable theory of quantum gravity.
   
   The differential geometry of gauge theories is usually done using the formalism of principal fiber bundles. Classical gauge fields are viewed as sections of these bundles, and the results of non-trivial field interactions are compared from point to point by the use of parallel transport along curves defined in the base spaces of these bundles. This comparison is done with a `connection' on the bundle, and for a closed curve the failure of an entity to return to its original value after traversing the curve is taken to be a sign of non-trivial interactions or "curvature". Principal fiber bundles of course have an associated Lie group and elements of this group act on objects to parallel transport them along the closed curves. These group elements are thus dependent on the curve, and are called `holonomies'.
   
   This is the classical picture, but what happens to this scenario in the quantum realm, and in this realm is it plausible to view it as a theory of quantum gravity? The authors spend the first six chapters discussing the loop group, and its use in the quantization of classical electrodynamics and classical Yang-Mills theory, as compared with what is done in the usual Hamiltonian formalism. The `quantum loop representation' plays a central role in their exposition, which is motivated by essentially two different approaches, one of which is essentially a Fourier transform of wavefunctions of the connection, while the other involves the quantization of a non-canonical algebra. In both cases the quantization procedure involves coming to grips with a constrained system, which as is well known is very challenging and the loop representation cannot be expected to be a panacea in this regard.
   
   The trick involves the identification of the physical states taking into account diffeomorphism invariance and the Hamiltonian constraint. The authors do this for pure quantum gravity (no matter fields) using the Ashtekar formalism and `point-splitting' methods, reinforcing the idea of course that one is not going to escape the need for regularization, as is the case for all successful quantum field theories so far. The inclusion of matter fields is done for (uncharged) Weyl fermions, with the geometric interpretation that the Weyl part of the Hamiltonian is a translation operator in much the same way as the Hamiltonian in the case of pure gravity. The authors believe take this to mean that the loop representation for quantum gravity predicts the Dirac equation for fermions, but unfortunately they do not elaborate on this in much detail at all.
   
   If loop quantization is to bring about a "unified" field theory in some sense then it must be able to show how the loops from one theory can be combined with the loops from another. The authors do this for the case of general relativity and electrodynamics, wherein a loop representation is introduced that is based on a single loop that accounts for the information of these two interacting theories. As expected, this involves enlarging the symmetry group SU(2) to U(2). They show that the wavefunctions for the unified loop representation depend on two loops, but that there is no effective distinction between the two loops. Generalizations to the case of Yang-Mills + general relativity are alluded to in the text but not discussed in any depth.

1 comments about Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos (Interdisciplinary Applied Mathematics).

1. As it promises in its title, this book is concerned with developing a practising knowledge of bifurcation analysis. The book starts with a very simple introduction, and takes off from there. The content is lively and straightforward. Concepts are introduced in plain language, without the technical complexity of formal mathematical jargon. Few theorems are proved, so the concepts remain in focus. Computational techniques are an important concern, and their problems. All in all, an entertaining and at once informative book to learn using a hands-on-approach. It manages to cover a fairly large amount of material, for a thorough introduction. Useful for people who are primarily interested in applications.

No comments about Complexity And Criticality (Imperial College Press Advanced Physics Texts).

1 comments about A Unified Grand Tour of Theoretical Physics, 2nd edition.

1. A conducted grand tour of the fundamental theories which shape our modern understanding of the physical world. This book covers the central themes of spacetime geometry and the general-relativistic account of gravity; quantum mechanics and quantum field theory; gauge theories and the fundamental forces of nature, statistical mechanics and the theory of phase transitions. The basic structure of each theory is explained in explicit mathematical detail with emphasis on conceptual understanding rather than on the technical details of specialized applications. Straightforward accounts are given of the standard models of particle physics and cosmology, and some of the more speculative ideas of modern theoretical physics are examined. This book is unique in bringing together the diverse areas of physics which are usually treated as independent. Designed to be accessible to final year undergraduates in physics and mathematics and to provide first year graduate students with a broad introductory view of theoretical physics, it will also be of interest to scientists and engineers in other disciplines who need an account of the subject at a level intermediate between semi-popular and technical research.

No comments about Reviews of Nonlinear Dynamics and Complexity: Volume 1 (Annual Reviews of Nonlinear Dynamics and Complexity).

No comments about Introduction to Relativistic Continuum Mechanics (Lecture Notes in Physics).

2 comments about Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics).

1. In 1990, I came across Hestenes' book New Foundations for Classical Mechanics in a university bookstore, and immediately purchased it. In 1999, I finally purchased the paperback version of CA to GC (Kluwer, please keep the paperback in print!). An engineer by training, I was previously only familiar with college vector analysis and with indicial tensor notation. These two books revolutionized my understanding of the algebraic representation of geometric information. Amazon.com now has the "look inside" feature in place for this book, so you can check the table of contents.

   Science proceeds both by discovery and by a process of recasting what has been learned in simpler and clearer form. It is the fruits of this latter process that are presented in CA to GC, though some new mathematical results are also introduced. The authors have succeeded admirably in recasting large areas of intermediate to advanced mathematics in a powerful unified algebraic language of exceptional clarity. The authors show how the traditional languages of complex numbers, quaternions, matrices, vectors, tensors, spinors and differential forms are all subsumed by the elegant language of Clifford algebra, and their calculi by Clifford analysis. Quite apart from the pleasure that the clarity of Clifford algebra/analysis affords, its value also lies in making it easier to understand what has already been discovered, and thus extending the mathematical grasp of the human mind. I have often seen the terms "breakthrough" and "groundbreaking" applied to paltry advances in science, mostly by the innovators themselves, but surely Clifford algebra/analysis is deserving of such an appellation. It has been long in gestation, but its time has come.

   In CA to GC, the authors present a tour-de-force of mathematical exposition, the writing displaying the same perspicuity and precision that marks all of Hestenes' writing. While further-refined versions of much of the material of the book can now be downloaded in the form of pdf files from Hestenes' website, this book will go down in history as a classic of unifying mathematical exposition. The university student should begin with the New Foundations for Classical Mechanics book, but CA to GC should be read by every mathematician, physicist and engineering scientist. The reader contemplating learning Clifford algebra/analysis should also take a look at the rapidly growing amount of information online, and at other books on the subject. It is an honor for me to be the first reviewer of this book on Amazon.com.




2. i have been working a few years in geometric calculus and i believe this book should be in every house of every geometrist and every person that is intersted in geometric concepts with physics applications

No comments about Inverse Problems & Inverse Scattering of Plane Waves.

No comments about Quantum Analogues: From Phase Transitions to Black Holes and Cosmology (Lecture Notes in Physics).

1 comments about A Modern Approach to Critical Phenomena.

1. The studies of the critical behaviors for a system near its phase
   transition point constitutes an important subfield of the
   researches in statistical mechanics and condensed matter physics.
   It has acquired a renewed interest due to the discovery of quantum
   critical behaviors in correlated electron systems. The development
   of the theory of critical phenomena has a tremendous impact on
   both statistical mechanics and quantum field theory, and the
   associated renormalization group idea has now become a basic
   language in thinking about many fundamental problems in condensed
   matter physics. No wonder that there exist many books in the
   market which address this issue. Among them, I have to mention the
   classic book by S. K. Ma and the more recent little, but
   refreshingly clear, book by John Cardy. There is also an
   overwhelming monograph by J. Zinn-Justin. In my opinion, Igor
   Herbut's book stands out from its pedagogy and its modern flavors.
   To be more precise, although the contents of this book is, by its
   very nature, challenging, this book is accessible to any motivated
   graduate students with a solid background in quantum mechanics and
   statistical mechanics. Moreover, unlike some old monographs, this
   book not only teach you the phi-4 theory, but also contains nice
   discussions of the superconducting transition, the nonliear-sigma
   model, the KT transition, and the charge-vortex duality. In the
   end of the book, it gives the readers a brief but useful
   discussion of the quantum phase transition. On the technical side,
   this book is devoted almost exclusively to the momentum-shell RG
   approach which is a powerful tool and it is still widely used in
   the research literature. However, I would like to recommend the
   uninitiated readers to read this book in companion with a standard
   field theory textbook, such as chapter 10-13 of Peskin's book, to
   learn some different perspectives on the renormalization group, in
   particular, the Callen-Symanzik equation and the related stuff.
   
   As far as I can tell, the only missing "standard" topic in this
   book about the critical phenomena is a systematic discussion of
   the large-N calculation of the critical exponent. In the mean
   time, since most of the book is devoted to the classical phase
   transitions, this book deals exclusively with bosonic field
   theories. The author may like to consider to add more discussions
   about critical properties of fermonic systems or even the
   fermion-gauge coupled systems in the chapter about quantum phase
   transition (in the 2nd edition?) of this book. Of course, nobody
   promise you a rose garden. As a whole, this book serves as a
   useful reference to bridge the gap between the usual graduate
   course and the research literature. After reading this book, the
   student can go on studying the more specialized monograph, such as
   Sachdev's book, and more importantly, starting doing their own
   research works. I highly recommend this book to anyone interested
   in the field theoretical approach to condensed matter physics.