The French mathematician Élie Cartan (1869–1951) used to be one of many founders of the fashionable idea of Lie teams, a subject matter of critical value in arithmetic and in addition one with many purposes. during this quantity, he describes the orthogonal teams, both with actual or advanced parameters together with reflections, and likewise the comparable teams with indefinite metrics. He develops the speculation of spinors (he chanced on the final mathematical type of spinors in 1913) systematically by way of giving a merely geometrical definition of those mathematical entities; this geometrical foundation makes it really easy to introduce spinors into Riemannian geometry, and especially to use the assumption of parallel shipping to those geometrical entities.

The booklet is split into components. the 1st is dedicated to generalities at the crew of rotations in *n*-dimensional house and at the linear representations of teams, and to the speculation of spinors in 3-dimensional house. eventually, the linear representations of the crowd of rotations in that area (of specific significance to quantum mechanics) also are tested. the second one half is dedicated to the speculation of spinors in areas of any variety of dimensions, and especially within the house of detailed relativity (Minkowski space). whereas the elemental orientation of the booklet as a complete is mathematical, physicists may be specially attracted to the ultimate chapters treating the functions of spinors within the rotation and Lorentz teams. during this connection, Cartan exhibits tips to derive the "Dirac" equation for any crew, and extends the equation to common relativity.

One of the best mathematicians of the 20 th century, Cartan made extraordinary contributions in mathematical physics, differential geometry, and crew idea. even though a profound theorist, he used to be in a position to clarify tricky suggestions with readability and ease. during this specified, specific treatise, mathematicians focusing on quantum mechanics will locate his lucid procedure an exceptional value.