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Group And Representation Theory (Hardcover)
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Group And Representation Theory (Hardcover)
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This volume goes beyond the understanding of symmetries and
exploits them in the study of the behavior of both classical and
quantum physical systems. Thus it is important to study the
symmetries described by continuous (Lie) groups of transformations.
We then discuss how we get operators that form a Lie algebra. Of
particular interest to physics is the representation of the
elements of the algebra and the group in terms of matrices and, in
particular, the irreducible representations. These representations
can be identified with physical observables.This leads to the study
of the classical Lie algebras, associated with unitary, unimodular,
orthogonal and symplectic transformations. We also discuss some
special algebras in some detail. The discussion proceeds along the
lines of the Cartan-Weyl theory via the root vectors and root
diagrams and, in particular, the Dynkin representation of the
roots. Thus the representations are expressed in terms of weights,
which are generated by the application of the elements of the
algebra on uniquely specified highest weight states. Alternatively
these representations can be described in terms of tensors labeled
by the Young tableaux associated with the discrete symmetry Sn. The
connection between the Young tableaux and the Dynkin weights is
also discussed. It is also shown that in many physical systems the
quantum numbers needed to specify the physical states involve not
only the highest symmetry but also a number of sub-symmetries
contained in them. This leads to the study of the role of
subalgebras and in particular the possible maximal subalgebras. In
many applications the physical system can be considered as composed
of subsystems obeying a given symmetry. In such cases the reduction
of the Kronecker product of irreducible representations of
classical and special algebras becomes relevant and is discussed in
some detail. The method of obtaining the relevant Clebsch-Gordan
(C-G) coefficients for such algebras is discussed and some relevant
algorithms are provided. In some simple cases suitable numerical
tables of C-G are also included.The above exposition contains many
examples, both as illustrations of the main ideas as well as well
motivated applications. To this end two appendices of 51 pages - 11
tables in Appendix A, summarizing the material discussed in the
main text and 39 tables in Appendix B containing results of more
sophisticated examples are supplied. Reference to the tables is
given in the main text and a guide to the appropriate section of
the main text is given in the tables.
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