Representations of the Rotation and Lorentz Groups and Their Applications
Representations of the Rotation and Lorentz Groups and Their Applications

Author: I. M. Gelfand

Publisher: Courier Dover Publications

Published: 2018-04-18

Total Pages: 385

ISBN-13: 0486823857

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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.

The Rotation and Lorentz Groups and Their Representations for Physicists
The Rotation and Lorentz Groups and Their Representations for Physicists

Author: K. Srinivasa Rao

Publisher: John Wiley & Sons

Published: 1988

Total Pages: 380

ISBN-13: 9780470210444

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Here is a detailed, self-contained work on the rotation and Lorentz groups and their representations. Treatment of the structure of the groups is elaborate and includes many new results only recently published in journals. The chapter on linear vector spaces is exhaustive yet clear, and the book highlights the fact that all results of the orthosynchronous proper Lorentz group may be obtained from those of the rotation group via complex quaternions. The approach is unified, and special properties and exceptional cases are addressed.

Group Theory and General Relativity
Group Theory and General Relativity

Author: Moshe Carmeli

Publisher: World Scientific

Published: 2000

Total Pages: 416

ISBN-13: 9781860942341

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This is the only book on the subject of group theory and Einstein's theory of gravitation. It contains an extensive discussion on general relativity from the viewpoint of group theory and gauge fields. It also puts together in one volume many scattered, original works, on the use of group theory in general relativity theory. There are twelve chapters in the book. The first six are devoted to rotation and Lorentz groups, and their representations. They include the spinor representation as well as the infinite-dimensional representations. The other six chapters deal with the application of groups -- particularly the Lorentz and the SL(2, C) groups -- to the theory of general relativity. Each chapter is concluded with a set of problems. The topics covered range from the fundamentals of general relativity theory, its formulation as an SL(2, C) gauge theory, to exact solutions of the Einstein gravitational field equations. The important Bondi-Metzner-Sachs group, and its representations, conclude the book The entire book is self-contained in both group theory and general relativity theory, and no prior knowledge of either is assumed. The subject of this book constitutes a relevant link between field theoreticians and general relativity theoreticians, who usually work rather independently of each other. The treatise is highly topical and of real interest to theoretical physicists, general relativists and applied mathematicians. It is invaluable to graduate students and research workers in quantum field theory, general relativity and elementary particle theory.

Physics of the Lorentz Group
Physics of the Lorentz Group

Author: Sibel Baskal

Publisher: Morgan & Claypool Publishers

Published: 2015-11-01

Total Pages: 173

ISBN-13: 1681740621

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This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincaré sphere on polarization optics.

Rotations, Quaternions, and Double Groups
Rotations, Quaternions, and Double Groups

Author: Simon L. Altmann

Publisher: Courier Corporation

Published: 2013-04-09

Total Pages: 315

ISBN-13: 0486317730

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This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.

Theory and Applications of the Poincaré Group
Theory and Applications of the Poincaré Group

Author: Young Suh Kim

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 346

ISBN-13: 9400945582

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Special relativity and quantum mechanics, formulated early in the twentieth century, are the two most important scientific languages and are likely to remain so for many years to come. In the 1920's, when quantum mechanics was developed, the most pressing theoretical problem was how to make it consistent with special relativity. In the 1980's, this is still the most pressing problem. The only difference is that the situation is more urgent now than before, because of the significant quantity of experimental data which need to be explained in terms of both quantum mechanics and special relativity. In unifying the concepts and algorithms of quantum mechanics and special relativity, it is important to realize that the underlying scientific language for both disciplines is that of group theory. The role of group theory in quantum mechanics is well known. The same is true for special relativity. Therefore, the most effective approach to the problem of unifying these two important theories is to develop a group theory which can accommodate both special relativity and quantum mechanics. As is well known, Eugene P. Wigner is one of the pioneers in developing group theoretical approaches to relativistic quantum mechanics. His 1939 paper on the inhomogeneous Lorentz group laid the foundation for this important research line. It is generally agreed that this paper was somewhat ahead of its time in 1939, and that contemporary physicists must continue to make real efforts to appreciate fully the content of this classic work.

Applications of Finite Groups
Applications of Finite Groups

Author: J. S. Lomont

Publisher: Academic Press

Published: 2014-05-12

Total Pages: 359

ISBN-13: 1483268969

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Applications of Finite Groups focuses on the applications of finite groups to problems of physics, including representation theory, crystals, wave equations, and nuclear and molecular structures. The book first elaborates on matrices, groups, and representations. Topics include abstract properties, applications, matrix groups, key theorem of representation theory, properties of character tables, simply reducible groups, tensors and invariants, and representations generated by functions. The text then examines applications and subgroups and representations, as well as subduced and induced representations, fermion annihilation and creation operators, crystallographic point groups, proportionality tensors in crystals, and nonrelativistic wave equations. The publication takes a look at space group representations and energy bands, symmetric groups, and applications. Topics include molecular and nuclear structures, multiplet splitting in crystalline electric fields, construction of irreducible representations of the symmetric groups, and reality of representations. The manuscript is a dependable source of data for physicists and researchers interested in the applications of finite groups.

The Application of Group Theory in Physics
The Application of Group Theory in Physics

Author: Grigoriĭ I︠A︡kovlevich Li︠u︡barskiĭ

Publisher: Reader's Digest Young Families

Published: 1960

Total Pages: 392

ISBN-13:

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Elements of the theory of groups -- Some specific groups -- The theory of group representations -- Operations with group representations -- Representations of certain groups -- Small oscillations of symmetrical systems -- Second order phase transitions -- Crystals -- Infinite groups -- Representations of the rotation groups in two and three dimensions and of the full orthogonal group -- Clebsch-Gordon and Racah coefficients -- The Schrödinger equation -- Equations invariant under the Euclidean group of motions in space -- Absorption and Raman scattering of light -- Representations of the Lorentz group -- Relativistically invariant equations -- Nuclear reactions.