The Algebra of Logic

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Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. This means that the action of a particular rotation on vectors the quantum Hilbert space is only defined up to a sign. The other point of view is geometrical. One can explicitly construct the spinors, and then examine how they behave under the action of the relevant Lie groups.

This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of spinors, such as Fierz identities , are needed. The language of Clifford algebras [3] sometimes called geometric algebras provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras.

It largely removes the need for ad hoc constructions. In detail, let V be a finite-dimensional complex vector space with nondegenerate bilinear form g. It is an abstract version of the algebra generated by the gamma or Pauli matrices. If n is odd, this Lie algebra representation is irreducible. Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details. Spinors form a vector space , usually over the complex numbers , equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations see diagram.

The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected , but the simply connected spin group is its double cover.

So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter its tangent, normal, binormal frame actually gives the rotation , then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle above.

The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product , or Minkowski space with its Lorentz metric.

In the latter case, the "rotations" include the Lorentz boosts , but otherwise the theory is substantially similar. The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time. To obtain the spinors of physics, such as the Dirac spinor , one extends the construction to obtain a spin structure on 4-dimensional space-time Minkowski space. Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO 3,1 symmetry, and then builds the spin group at each point.

The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fibre bundle , the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation , or the Weyl equation on the fiber bundle. These equations Dirac or Weyl have solutions that are plane waves , having symmetries characteristic of the fibers, i. Such plane-wave solutions or other solutions of the differential equations can then properly be called fermions ; fermions have the algebraic qualities of spinors.

By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another. There does not seem to be any a priori reason why this would be the case. The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. Weyl spinors are insufficient to describe massive particles, such as electrons , since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed.

The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor.

The situation for condensed matter physics is different: one can can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials. In , an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.

One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups , and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah—Singer index theorem , and to provide constructions in particular for discrete series representations of semisimple groups.

The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article. The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space".

Several ways of illustrating everyday analogies have been formulated in terms of the plate trick , tangloids and other examples of orientation entanglement. Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah 's statement that is recounted by Dirac's biographer Graham Farmelo:. No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. Spinors were first applied to mathematical physics by Wolfgang Pauli in , when he introduced his spin matrices.

Spinor spaces were represented as left ideals of a matrix algebra in , by G. Juvet [15] and by Fritz Sauter. In Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. As a result, it admits a conjugation operation analogous to complex conjugation , sometimes called the reverse of a Clifford element, defined by. In this situation, a spinor [22] is an ordinary complex number.

Geometrical Vectors - Gabriel Weinreich - Google книги

An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:. Consider, for example, the implication this has for plane rotations. In general, because of logarithmic branching , it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued. In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements that is just the ring of complex numbers is identical to the space of spinors.

So, by abuse of language , the two are often conflated. One may then talk about "the action of a spinor on a vector. But in dimensions 2 and 3 as applied, for example, to computer graphics they make sense. For a long time after it was proposed by Einstein in , GR was counted as a shining achievement that lay somewhat outside the mainstream of interesting research.

Increasingly, however, contemporary students in a variety of specialties are finding it necessary to study Einstein's theory. In addition to being an active research area in its own right, GR is part of the standard syllabus for anyone interested in astrophysics, cosmology, string theory, and even particle physics.

There is no shortage of books on GR, and many of them are excellent. Indeed, approximately thirty years ago witnessed the appearance of no fewer than three books in the subject, each of which has become a classic in its own right: those by Weinberg , Misner, Thorne, and Wheeler , and Hawking and Ellis Each of these books is suffused with a strongly-held point of view advocated by the authors. This has led to a love-hate relationship between these works and their readers; in each case, it takes little effort to find students who will declare them to be the best textbook ever written, or other students who find them completely unpalatable.

For the individuals in question, these judgments may very well be correct; there are many different ways to approach this subject. The present book has a single purpose: to provide a clear introduction to general relativity, suitable for graduate students or advanced undergraduates.

I have attempted to include enough material so that almost any one-semester introductory course on GR can find the appropriate subjects covered in the text, but not too much more than that.

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In particular, I have tried to resist the temptation to write a comprehensive reference book. The only goal of this book is to teach you GR. An intentional effort has been made to prefer the conventional over the ideosyncratic.


  1. Spinor - Wikipedia.
  2. Participatory Dialogue: Towards a Stable, Safe and Just Society for All.
  3. Chicago Lectures in Physics.
  4. The Formation and Early Evolution of Stars: From Dust to Stars and Planets;
  5. Journal of the Optical Society of America?
  6. Geometrical Vectors | Mathematical Association of America.

If I can be accused of any particular ideological bias, it would be a tendency think of general relativity as a field theory, a point of view which helps one to appreciate the connections between GR, particle physics, and string theory. At the same time, there are a number of exciting astrophysical applications of GR black holes, gravitational lensing, the production and detection of gravitational waves, the early universe, the late universe, the cosmological constant , and I have endeavored to include at least enough background discussion of these issues to prepare students to tackle the current literature.

The primary question facing any introductory treatment of general relativity is the level of mathematical rigor at which to operate. There is no uniquely proper solution, as different students will respond with different levels of understanding and enthusiasm to different approaches. Recognizing this, I have tried to provide something for everyone. I have not shied away from detailed formalism, but have also attempted to include concrete examples and informal discussion of the concepts under consideration. Much of the most mathematical material has been relegated to appendices.

Some of the material in the appendices is actually an integral part of the course for example, the discussion of conformal diagrams , but an individual reader or instructor can decide just when it is appropriate to delve into them; signposts are included in the body of the text.

Surprisingly, there are very few formal prerequisites for learning general relativity; most of the material is developed as you go along. Certainly no prior exposure to Riemannian geometry is assumed, nor would it necessarily be helpful. It would be nice to have already studied some special relativity; although a discussion is included in Chapter One, its purpose is more to review the basics and and introduce some notation, rather than to provide a self-contained introduction.

Chicago Lectures in Physics

Beyond that, some exposure to electromagnetism, Lagrangian mechanics, and linear algebra might be useful, but the essentials are included here. The structure of the book should be clear. The first chapter is a review of special relativity and basic tensor algebra, including a brief discussion of classical field theory.

The next two chapters introduce manifolds and curvature in some detail; some motivational physics is included, but building a mathematical framework is the primary goal. General relativity proper is introduced in Chapter Four, along with some discussion of alternative theories. The next four chapters discuss the three major applications of GR: black holes two chapters , perturbation theory and gravitational waves, and cosmology.

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Each of these subjects has witnessed an explosion of research in recent years, so the discussions here will be necessarily introductory, but I have tried to emphasize issues of relevance to current work. These three applications can be covered in any order, although there are interdependencies highlighted in the text. Discussions of experimental tests are sprinkled through these chapters. Chapter Nine is a brief introduction to quantum field theory in curved spacetime; this is not a necessary part of a first look at GR, but has become increasingly important to work in quantum gravity and cosmology, and therefore deserves some mention.

On the other hand, a few topics are scandalously neglected; the initial value problem and cosmological perturbation theory come to mind, but there are others. Fortunately there is no shortage of other resources. The appendices serve various purposes: there are discussions of technical points which were avoided in the body of the book, crucial concepts which could have been put in various different places, and extra topics which are useful but outside the main development. Since the goal of the book is pedagogy rather than originality, I have often leaned heavily on other books listed in the bibliography when their expositions seemed perfectly sensible to me.

When this leaning was especially heavy, I have indicated it in the text itself. It will be clear that a primary resource was the book by Wald , which has become a standard reference in the field; readers of this book will hopefully be well-prepared to jump into the more advanced sections of Wald's book. These notes are available on the web for free, and will continue to be so; they will be linked to the website listed below. Perhaps a little over half of the material here is contained in the notes, although the advantages of owning the book several copies, even should go without saying.

Countless people have contributed greatly both to my own understanding of general relativity and to this book in particular too many to acknowledge with any hope of completeness. Some people, however, deserve special mention.

Ted Pyne learned the subject along with me, taught me a great deal, and collaborated with me the first time we taught a GR course, as a seminar in the astronomy department at Harvard; parts of this book are based on our mutual notes. Nick Warner taught the course at MIT from which I first learned GR, and his lectures were certainly a very heavy influence on what appears here. Neil Cornish was kind enough to provide a wealth of exercises, many of which have been included at the end of each chapter. And among the many people who have read parts of the manuscript and offered suggestions, Sanaz Arkani-Hamed was kind enough to go through the entire thing in great detail.

Apologies are due to anyone I may have neglected to mention. My friends who have written textbooks themselves tell me that the first printing of a book will sometimes contain mistakes. The website will also contain other relevant links of interest to readers. This set of lecture notes on general relativity has been expanded into a textbook, Spacetime and Geometry: An Introduction to General Relativity , available for purchase online or at finer bookstores everywhere.

The notes as they are will always be here for free. These lecture notes are a lightly edited version of the ones I handed out while teaching Physics 8. Each of the chapters is available here as PDF. Constructive comments and general flattery may be sent to me via the address below. What is even more amazing, the notes have been translated into French by Jacques Fric mirror. Je ne parle pas francais, mais cette traduction devrait etre bonne.

Dates refer to the last nontrivial modification of the corresponding file fixing typos doesn't count. Note that, unlike the book, no real effort has been made to fix errata in these notes, so be sure to check your equations. In a hurry? Can't be bothered to slog through lovingly detailed descriptions of subtle features of curved spacetime? While you are here check out the Spacetime and Geometry page -- including the annotated bibilography of technical and popular books, many available for purchase online.

Special Relativity and Flat Spacetime 22 Nov ; 37 pages the spacetime interval -- the metric -- Lorentz transformations -- spacetime diagrams -- vectors -- the tangent space -- dual vectors -- tensors -- tensor products -- the Levi-Civita tensor -- index manipulation -- electromagnetism -- differential forms -- Hodge duality -- worldlines -- proper time -- energy-momentum vector -- energy-momentum tensor -- perfect fluids -- energy-momentum conservation. Manifolds 22 Nov ; 24 pages examples -- non-examples -- maps -- continuity -- the chain rule -- open sets -- charts and atlases -- manifolds -- examples of charts -- differentiation -- vectors as derivatives -- coordinate bases -- the tensor transformation law -- partial derivatives are not tensors -- the metric again -- canonical form of the metric -- Riemann normal coordinates -- tensor densities -- volume forms and integration.

Curvature 23 Nov ; 42 pages covariant derivatives and connections -- connection coefficients -- transformation properties -- the Christoffel connection -- structures on manifolds -- parallel transport -- the parallel propagator -- geodesics -- affine parameters -- the exponential map -- the Riemann curvature tensor -- symmetries of the Riemann tensor -- the Bianchi identity -- Ricci and Einstein tensors -- Weyl tensor -- simple examples -- geodesic deviation -- tetrads and non-coordinate bases -- the spin connection -- Maurer-Cartan structure equations -- fiber bundles and gauge transformations.

Gravitation 25 Nov ; 32 pages the Principle of Equivalence -- gravitational redshift -- gravitation as spacetime curvature -- the Newtonian limit -- physics in curved spacetime -- Einstein's equations -- the Hilbert action -- the energy-momentum tensor again -- the Weak Energy Condition -- alternative theories -- the initial value problem -- gauge invariance and harmonic gauge -- domains of dependence -- causality. More Geometry 26 Nov ; 13 pages pullbacks and pushforwards -- diffeomorphisms -- integral curves -- Lie derivatives -- the energy-momentum tensor one more time -- isometries and Killing vectors.

Weak Fields and Gravitational Radiation 26 Nov ; 22 pages the weak-field limit defined -- gauge transformations -- linearized Einstein equations -- gravitational plane waves -- transverse traceless gauge -- polarizations -- gravitational radiation by sources -- energy loss. The Schwarzschild Solution and Black Holes 29 Nov ; 53 pages spherical symmetry -- the Schwarzschild metric -- Birkhoff's theorem -- geodesics of Schwarzschild -- Newtonian vs.

Cosmology 1 Dec ; 15 pages homogeneity and isotropy -- the Robertson-Walker metric -- forms of energy-momentum -- Friedmann equations -- cosmological parameters -- evolution of the scale factor -- redshift -- Hubble's law. This page collects any mistakes that people have been able to find in the book. Dates refer to when the addition was made to this page, not necessarily when it was sent to me. We provide solutions within hours to all customers contacting us. Yet most textbooks cover this topic by merely repeating the introductory-level treatment based on a limited algebraic or analytic view of the subject.

By contrast, Geometrical Vectors introduces a more sophisticated approach, which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear coordinates by carefully separating those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.

Written in an informal and personal style, Geometrical Vectors provides a handy guide for any student of vector analysis. Clear, carefully constructed line drawings illustrate key points in the text, and a set of problems is provided at the end of each chapter except the Epilogue to deepen understanding of the material presented.

Pertinent physical examples are cited to show how geometrically informed methods of vector analysis may be applied to situations of special interest to physicists. This text introduces an approach for mastering the concepts of vectors and vector analysis which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear co-ordinates.

It separates those relationships which are topologically invariant from those which are not. Written in an informal and personal style, this text provides a guide for any student of vector analysis. Clear line drawings illustrate key points in the text, and problem sets as well as physical examples are provided. Convert currency. Add to Basket. Condition: New. New copy - Usually dispatched within 2 working days.

Geometrical Methods for Physics

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