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The Theory of Spinors
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.
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.
192 pages, Paperback
First published March 15, 1967
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December 31, 2018
wish I had found it a decade ago. I've always been afraid to read the masters, thinking I must not be smart enough. In fact lecons des spineurs was so highly cited
Cartan, son of a blacksmith, wrote his original dissertation in the late 1800's on a similar topic. By the 1930's he had spent a LOT of time mastering the themes he is remembered for.
In addition to the title topic, this is a good book on tensors, Lie theory, representations, special relativity, general relativity, and other things.
Terry Gannon wrote that interesting mathematics is not about the greatest generality---it's about finding hidden structure, hidden gems. Cartan's work is a good example of this. Even toddlers know what a rotation of an object in 3-dimensional space is. But 99.9999% of us do not see as deeply as Cartan did into what sounds like a non-topic.
Google will tell you that a tensor is a multi-dimensional array. Which is not true. It's a multilinear map. That definition, available in highly refined form in Spivak DG1, isn't easy to learn about. So if you're interested in matrices you should add this to your library to get a broader view.
It's $5. More or less, all quants should buy a copy.
Cartan, son of a blacksmith, wrote his original dissertation in the late 1800's on a similar topic. By the 1930's he had spent a LOT of time mastering the themes he is remembered for.
In addition to the title topic, this is a good book on tensors, Lie theory, representations, special relativity, general relativity, and other things.
Terry Gannon wrote that interesting mathematics is not about the greatest generality---it's about finding hidden structure, hidden gems. Cartan's work is a good example of this. Even toddlers know what a rotation of an object in 3-dimensional space is. But 99.9999% of us do not see as deeply as Cartan did into what sounds like a non-topic.
Google will tell you that a tensor is a multi-dimensional array. Which is not true. It's a multilinear map. That definition, available in highly refined form in Spivak DG1, isn't easy to learn about. So if you're interested in matrices you should add this to your library to get a broader view.
It's $5. More or less, all quants should buy a copy.
Displaying 1 of 1 review