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- Lie Groups, Lie Algebras, and Some of Their Applications
By Robert Gilmore. Only a century has elapsed since , when Marius Sophus Lie began his research on what has evolved into one of the most fruitful and beautiful branches of modern mathematics—the theory of Lie groups. These researches culminated twenty years later with the publication of landmark treatises by S.
Lie Groups, Lie Algebras, and Some of Their Applns.
What could be better than a circle or a sphere? The Sun and the planets were supposed to circle the Earth. It took a long time to get to the apparently less than perfect ellipses! Of course most shapes in the natural world display little or no symmetry, but many are almost symmetric. An orange is close to a perfect sphere; Humans are almost symmetric about their vertical axis, but not quite, and ancient man must have been aware of this. Could this lack of exact symmetry have been viewed as a sign of imperfection, imperfection that humans need to atone for?
It must have been clear that highly symmetric objects were special, but it is a curious fact that the mathematical structures which generate symmet- rical patterns were not systematically studied until the nineteenth century. That is not to say that symmetry patterns were unknown or neglected, wit- ness the Moors in Spain who displayed the seventeen different ways to tile a plane on the walls of their palaces!
In physics, the study of crys- tals elicited wonderfully regular patterns which were described in terms of their symmetries. In the Twentieth Century, with the advent of Quantum Mechanics, symmetries have assumed a central role in the study of Nature. In Cosmological terms, this means that our Universe emerged from the Big Bang as a highly symmetrical structure, although most of its symmetries are no longer evident today.
Like an ancient piece of pottery, some of its parts may not have survived the eons, leaving us today with its shards. This is a very pleasing concept that resonates with the old Greek ideal of perfection.
Did our universe emerge at the Big Bang with perfect symmetry that was progressively shattered by cosmological evo- lution, or was it born with internal defects that generated the breaking of its symmetries? It is a profound question which some physicists try to answer today by using conceptual models of a perfectly symmetric universe, e. Some symmetries of the Natural world are so commonplace, that they are difficult to identify.
The outcome of an experiment performed by un- dergraduates should not depend on the time and location of the bench on which it was performed. Their results should be impervious to shifts in time and space, as consequences of time and space translation invariances, respectively. But there are more subtle manifestations of symmetries. Better yet, you can perform experiments whose outcomes are the same as if you were standing still!
Today, you can leave your glass of wine on an airplane at cruising altitude without fear of spilling. The great genius that he was elevated this to his principle of Rela- tivity: the laws of Physics do not depend on whether you are at rest or move with constant velocity! However if the velocity changes, you can feel it a little turbulence will spill your wine.
Our experience of the everyday world appears complicated by the fact that it is dominated by frictional forces; in a situation where their effect can be neglected, simplicity and symmetries in some sense analogous concepts are revealed. According to Quantum Mechanics, Physics takes place in Hilbert spaces. Bizarre as this notion might be, we have learned to live with it as it con- tinues to be verified whenever experimentally tested.
Surely, this abstract identification of a physical system with a state vector in Hilbert space will eventually be found to be incomplete, but in a presently unimaginable way, which will involve some other weird mathematical structure. That Nature uses the same mathematical structures invented by mathematicians is a pro- found mystery hinting at the way our brains are wired.
Whatever the root cause, mathematical structures which find natural representations in Hilbert spaces have assumed enormous physical interest. Since physicists are mainly interested in how groups operate in Hilbert spaces, we will focus mostly in the study of their representations.
Mathe- matical concepts will be introduced as we go along in the form of scholia sprinkled throughout the text. Our approach will be short on proofs, which can be found in many excellent textbooks. From representations, we will focus on their products and show how to build group invariants for possible physical applications.
We will also discuss the embeddings of the represen- tations of a subgroup inside those of the group. Numerous tables will be included. This book begins with the study of finite groups, which as the name in- dicates, have a finite number of symmetry operations.
The smallest finite group has only two elements, but there is no limit as to their number of elements: the permutations on n letters forms a finite group with n!
Finite groups have found numerous applications in physics, mostly in crystallography and in the behavior of new materials. In elementary particle physics, only small finite groups have found applications, but in a world with extra dimensions, and three mysterious families of elementary particles, this situation is bound to change.
Notably, the sporadic groups, an exceptional set of twenty six finite groups, stand mostly as mathematical curiosities waiting for an application. We then consider continuous symmetry transformations, such as rota- tions by arbitrary angles, or open-ended time translation, to name a few. Continuous transformations can be thought of as repeated applications of in- finitesimal steps, stemming from generators.
Typically these generators form algebraic structures called Lie algebras. Our approach will be to present the simplest continuous groups and their associated Lie algebras, and build from them to the more complicated cases. Special attention will be devoted to exceptional groups and their representations. In particular, the magic square and magic triangle will be discussed.
We will link back to finite groups, as most can be understood as subgroups of continuous groups. Group-theoretic aspects of the Standard Model and Grand-Unification are presented as well.
The algebraic construction of the five Exceptional Lie algebras is treated in detail. Two generalizations of Lie algebras are also dis- cussed, Super-Lie algebras and their classification, and infinite-dimensional affine Kac-Moody algebras. I would like to thank Professors L. Brink, and J. Patera, as well as Drs. Belyaev, Sung-Soo Kim, and C. Luhn, for their critical reading of the manuscript, and many useful suggestions.
Fi- nally, I owe much to my wife Lillian, whose patience, encouragements, and understanding made this book possible. Related Papers.
By Christoph Schweigert. By Lars Brink. By Karl-Georg Schlesinger. By Frederique Harmsze. Super-Ehlers in any dimension. By Alessio Marrani. Download pdf. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up.
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Du kanske gillar. Alice in Quantumland Robert Gilmore Inbunden. Ladda ned. Spara som favorit. Lie group theory plays an increasingly important role in modern physical theories. Many of its calculations remain fundamentally unchanged from one field of physics to another, altering only in terms of symbols and the language.
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Lie Groups, Lie Algebras, and Some of Their Applications
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