Representations of the Rotation and Lorentz Groups and Their Applications by I.M. Gelfand | (PDF) Free Download

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This monograph on the description and study of representations of the rotation group of three-dimensional space and of the Lorentz group features advanced topics and techniques crucial to many areas of modern theoretical physics. Prerequisites include a familiarity with the differential and integral calculus of several variables and the fundamentals of linear algebra. Suitable for advanced undergraduate and graduate students in mathematical physics, the book is also designed for mathematicians studying the representations of Lie groups, for whom it can serve as an introduction to the general theory of representation.The treatment encompasses all the basic material of the theory of representations used in quantum mechanics. The two-part approach begins with representations of the group of rotations of three-dimensional space, analyzing the rotation group and its representations. The second part, covering representations of the Lorentz group, includes an exploration of relativistic-invariant equations. The text concludes with three helpful supplements and a bibliography.

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Editorial Reviews: About the Author The authors of this book were Soviet mathematicians with expertise especially in mathematical physics.

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⭐I purchased this book, sight unseen, on account of the authors and the very reasonable price, but I would have appreciated a view of the contents prior to purchase.Note that the copyright is 1963 and the typography is not up to current standards. The book is 366 pages in length. There is no index, but the Table of Contents is nine pages in length.1. From the Table of Contents:The Group of Rotations in Three-dimensional spaceInfinitesimal Rotations and the Determination of the Irreducible Representations of the Group of RotationsSpherical Functions …Tensors and Tensor RepresentationsSpinors and Spinor Representations …Equations Invariant with respect to Rotation …The Lorentz Group …General Relativistic-Invariant EquationsEquations arising from Invariant Lagrangian Functions …The Determination of the Rest Mass and Spin of a ParticleThe Charge and Energy of Relativistic Particles2. From the Preface:The present book is devoted to a study of the rotation group of three-dimensional space and of the Lorentz group. The reader is assumed to be acquainted with the fundamentals of linear algebra…The theory of representations, in particular of the three-dimensional rotation group and the Lorentz group, is used extensively in quantum mechanics. In this book we have gathered together all the fundamental material which, in our view, is necessary to quantum mechanical applications.3. From Chapter 1, page 16. (Note that the typography of the book is better than the following typewritten extract.)Suppose we are given a unitary representation of the rotation group G. This means that there is associated with every element g a unitary matrix Tg = ||aik(g)|| such that to the product of the rotations g1 and g2 there corresponds the product of the matrices Tg1 and Tg2, i.e.,Tg1g2 = Tg1 Tg2.In particular, to the identity rotation e there corresponds the unit matrix E. For the parameters defining the rotation g we take the coordinates x1, x2, x3 of the vector, parallel to the axis of the rotation, whose length is equal to the angle of rotation…Expand T(x1, x2, x3) about the point x1 = x2 = x3 = 0 by Taylor’s theorem. ThenT(x1, x2, x3) = E + A1 x1 + A2 x2 + A3 x3 + …when E is the unit matrix, and A1, A2, and A3 are constant matrices, the partial derivatives of the matrix T(x1, x2, x3) with respect to x1, x2, x3 evaluated at the point x1 = x2 = x3 = 0. … The matrices A1, A2, and A3 are called the matrices of the infinitesimal rotations about the coordinates axes.We will now show that the matrices A1, A2, A3 completely determine the representation… To do this we take an arbitrary vector (x1, x2, x3) and consider two rotations about this vector: g(t x1, t x2, t x3) and g(s x1, s x2, s x3). The product of these two rotations is obviously a rotation about the same axis, and is given by the parameters (t + s) x1, (t + s) x2, (t + s) x3…. Since a representation preserves products, we have alsoT[(t + s) x1, (t + s) x2, (t + s) x3] = T(t x1, t x2, t x3) T(s x1, s x2, s x3).Differentiating both sides with respect to s and putting s = 0, we getd/dt T(t x1, t x2, t x3) = d/ds T(s x1, s x2, sx 3)|(s=0) T(t x1, t x2, t x3). But,d/ds T(s x1, s x2, s x3)|(s=0) = A1 x1 + A2 x2 + A3 x3.From this we obtain the following equation for the matrix X(t) = T(t x1, t x2, t x3):d/dt X(t) = (A1 x1 + A2 x2 + A3 x3) X(t).Apart from this, the fundamental conditionX(0) = T(0, 0, 0) = E must be satisfied. … Its solution…X(t) = exp[t(A1 x1 + A1 x2 + A3 x3)].

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