Representations of Some Lie Algebras

Abstract

Comparison of the quantum theory with experiments is made by evaluating expectation values of observables. Evaluation of expectation values is carried in a c-number representation of the vector space of the states of the system by choosing a suitable basis. A basis that immediately suggests itself for investigating time-evolution of a system is the one spanned by the eigenvectors of the hamiltonian governing its motion. As has been pointed out in the last chapter, hamiltonians encountered frequently in quantum optics can be classified as elements of the Lie algebras or of their direct products. Hence, the problem of representation of a quantum optical system in terms of the eigenvectors of its hamiltonian reduces to that of finding eigenvectors of hermitian elements of the Lie algebras introduced in the last chapter. However, it will be seen that there are other bases which prove to be of not only mathematical interest but also of importance in understanding various physics aspects.