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The Theory of Spinors [Hardcover]

Elie Cartan
5.0 out of 5 stars  See all reviews (4 customer reviews)

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Book Description

March 15 1967 0262030209 978-0262030205 Revised edition

The French mathematician Élie Cartan (1869–1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities.
The book is divided into two parts. The first is devoted to generalities on the group of rotations in n-dimensional space and on the linear representations of groups, and to the theory of spinors in three-dimensional space. Finally, the linear representations of the group of rotations in that space (of particular importance to quantum mechanics) are also examined. The second part is devoted to the theory of spinors in spaces of any number of dimensions, and particularly in the space of special relativity (Minkowski space). While the basic orientation of the book as a whole is mathematical, physicists will be especially interested in the final chapters treating the applications of spinors in the rotation and Lorentz groups. In this connection, Cartan shows how to derive the "Dirac" equation for any group, and extends the equation to general relativity.
One of the greatest mathematicians of the 20th century, Cartan made notable contributions in mathematical physics, differential geometry, and group theory. Although a profound theorist, he was able to explain difficult concepts with clarity and simplicity. In this detailed, explicit treatise, mathematicians specializing in quantum mechanics will find his lucid approach a great value.

--This text refers to the Paperback edition.

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Most helpful customer reviews
Format:Paperback
This book is extremely easy to understand compared to other texts on the subject. It serves well to lead you to other books such as Hermann Weyl's text on Quantum Groups. The author has the most intuitive explanations that I have seen in any other works on the subject. This is extremely useful for me, since I was trained as an electrical engineer, and sometimes I feel that authors of mathematics text get lost in formal definitions in n space; although important I feel that relating them to basic understandable behavior in 3-space is especially useful, and done well in this text. It is my current thought that formulating a certain level of abstraction is necessary for the furthering of mathematics, like what is done with Geometric Algebra, but making these relations clear for myself at least in the physical world has been an important hard earned step. I believe this book to be simply outstanding and recommend it to anyone interested in linking group theory to Quantum Mechanics, especially Engineers who don't want to physically build anything but like math, like myself. On the subject, I used this book to understand the mathematics in the majority of texts on Quantum Mechanics, I approach the equations now, not as an abstract concepts but looking at their space behavior even for n space; Hopefully I will be able to exploit some useful properties of the lie group useful for Quantum Computation. This book is one of my current favorites; I recommend it, but especially to those interested in Quantum Mechanics or Group Theory from a non Mathematics background. Of course this is not a beginning text, but it can be used along side a beginning text such as "Algebraic Structures" or "A course on Group Theory" both from Dover. Read more ›
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5.0 out of 5 stars Review of theory of spinors Jan. 11 2000
By Reader
Format:Paperback
This is an excellent introductory book on spinors, the basic mathematical object used to represent particles with spin.
The author begins by defining the spinor as a form of a square root of a 3 dimensional null vector. Scalars, vectors and tensors are then described by their properties under simple geometrical transformations such as reflection and rotation. The author then represents vectors as 2x2 matrices. The transformational properties of spinors are defined by their relation to vectors and tensors under these same simple transformations. The author then shows how spinors are useful for finding the irreducible representations of the rotation group. These concepts are then extended to higher dimensional spinors. Specific applications are shown for Laplace's equation, the Dirac equation and to general relativity.
The is an introductory, inexpensive, brief and easy to read book. The book also covers a fair amount of ground. It is an excellent first book for the subject. It does not contain modern developments in the field or some elements of the current notational system for representing spinors. Yet, for me it was the first book that gave me a sense of really understanding the significance of the Dirac equation and quantum physic's concept of spin.
Was this review helpful to you?
5.0 out of 5 stars Review of theory of spinors Jan. 11 2000
By Reader
Format:Paperback
This is an excellent introductory book on spinors, the basic mathematical object used to represent particles with spin.
The author begins by defining the spinor as a form of a square root of a 3 dimensional null vector. Scalars, vectors and tensors are then described by their properties under simple geometrical transformations such as reflection and rotation. The author then represents vectors as 2x2 matrices. The transformational properties of spinors are defined by their relation to vectors and tensors under these same simple transformations. The author then shows how spinors are useful for finding the irreducible representations of the rotation group. These concepts are then extended to higher dimensional spinors. Specific applications are shown for Laplace's equation, the Dirac equation and to general relativity.
The is an introductory, inexpensive, brief and easy to read book. The book also covers a fair amount of ground. It is an excellent first book for the subject. It does not contain modern developments in the field or some elements of the current notational system for representing spinors. Yet, for me it was the first book that gave me a sense of really understanding the significance of the Dirac equation and quantum physic's concept of spin.
Was this review helpful to you?
By Lowell
Format:Paperback
This book is a great work on the subject of spinors. Since it is based solely on Elie Cartan's work, the termonology and wording reflects that of the transition coming out of the Post-Victorian math era (i.e., when quaternions and the more abstract mathematics where used for applied calculations in physics, which have been replaced with scalars and vectors). The geometrical definition layed down in the third chapter makes this a very comprehensive book for the newcomer to the subject. Quote page 42., "A spinor is thus a sort of "directed" or "polarized" isotropic vector; a rotation about an axis through an angle of 2pi changes the polarization of this isotropic vector."
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Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com: 4.4 out of 5 stars  7 reviews
45 of 48 people found the following review helpful
5.0 out of 5 stars Review of theory of spinors Jan. 11 2000
By Reader - Published on Amazon.com
Format:Paperback
This is an excellent introductory book on spinors, the basic mathematical object used to represent particles with spin.
The author begins by defining the spinor as a form of a square root of a 3 dimensional null vector. Scalars, vectors and tensors are then described by their properties under simple geometrical transformations such as reflection and rotation. The author then represents vectors as 2x2 matrices. The transformational properties of spinors are defined by their relation to vectors and tensors under these same simple transformations. The author then shows how spinors are useful for finding the irreducible representations of the rotation group. These concepts are then extended to higher dimensional spinors. Specific applications are shown for Laplace's equation, the Dirac equation and to general relativity.
The is an introductory, inexpensive, brief and easy to read book. The book also covers a fair amount of ground. It is an excellent first book for the subject. It does not contain modern developments in the field or some elements of the current notational system for representing spinors. Yet, for me it was the first book that gave me a sense of really understanding the significance of the Dirac equation and quantum physic's concept of spin.
9 of 9 people found the following review helpful
4.0 out of 5 stars translated from French Sept. 13 2009
By Roger Bagula - Published on Amazon.com
Format:Paperback
We have Weyl, Pauli, Dirac and Cartan to thank for our modern
theory of groups in physics. This book published in 1937
has none of the later Lie algebra representations of the Cartan generalization of groups
and thus, like Weyl's similar book may deceive the reader into thinking
he understands when he has only a rough and not very even
introduction to these groups. This book doesn't reach much higher than SU(2),
SO(3) and the Dirac U(1)*SU(2)*SU(2).
The standard model of physics deals with the symmetry breaking of SU(5)
( the Cartan A_4 group) to U(1)*SU(2)*SU(3). The Lie algebras and
irreducible Cartan representations of such higher symmetries
will demand the student read further than this text.
So this book is an historical introduction that gives the starting basis
for the mathematics needed by modern students in physics and chemistry.
11 of 12 people found the following review helpful
5.0 out of 5 stars Review of theory of spinors Jan. 11 2000
By Reader - Published on Amazon.com
Format:Paperback
This is an excellent introductory book on spinors, the basic mathematical object used to represent particles with spin.
The author begins by defining the spinor as a form of a square root of a 3 dimensional null vector. Scalars, vectors and tensors are then described by their properties under simple geometrical transformations such as reflection and rotation. The author then represents vectors as 2x2 matrices. The transformational properties of spinors are defined by their relation to vectors and tensors under these same simple transformations. The author then shows how spinors are useful for finding the irreducible representations of the rotation group. These concepts are then extended to higher dimensional spinors. Specific applications are shown for Laplace's equation, the Dirac equation and to general relativity.
The is an introductory, inexpensive, brief and easy to read book. The book also covers a fair amount of ground. It is an excellent first book for the subject. It does not contain modern developments in the field or some elements of the current notational system for representing spinors. Yet, for me it was the first book that gave me a sense of really understanding the significance of the Dirac equation and quantum physic's concept of spin.
15 of 18 people found the following review helpful
5.0 out of 5 stars Masterpiece based on the original lecture notes of Cartan Dec 20 2001
By Lowell - Published on Amazon.com
Format:Paperback
This book is a great work on the subject of spinors. Since it is based solely on Elie Cartan's work, the termonology and wording reflects that of the transition coming out of the Post-Victorian math era (i.e., when quaternions and the more abstract mathematics where used for applied calculations in physics, which have been replaced with scalars and vectors). The geometrical definition layed down in the third chapter makes this a very comprehensive book for the newcomer to the subject. Quote page 42., "A spinor is thus a sort of "directed" or "polarized" isotropic vector; a rotation about an axis through an angle of 2pi changes the polarization of this isotropic vector."
8 of 9 people found the following review helpful
5.0 out of 5 stars The most important book that I have ever read May 11 2009
By sheine - Published on Amazon.com
Format:Paperback|Verified Purchase
This is also the most difficult book that I have read, it is poorly written. Everybody that I have recommended it to has given up on it. There is scarcely a page in this book that when properly understood is not a key to understanding something in physics. The second chapter on multivectors is worth more than every book that I have read on vectors. Unless things have changed since I taught physics, most people who use vectors have a shallow knowledge of them. One example is the bivector, which we have been taught are axial vectors that are somehow different from regular vectors, polar vectors. Actually they define planes and the so-called axial vector is perpendicular to the plane. In standard physics, nothing happens along the axial vector, e.g. rotation or magnetic field, because the action takes place on the plane of the bivector. It is also interesting that each of the four Maxwell equations is a different type of multivector. And it seems to me that the search for a magnetic charge is futile because the Maxwell equation would require it to be a trivector, known to most people as a triple scalar product. This would require a single number to change sign when you go from a right to left-handed coordinate system.

This book is a case of very hard work paying off. What is really needed is a rewrite of this book in more comprehensible form.
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