Usually complicated unitary operators in exponential form are hard to handle and the transformation stated by them are not physically clear until these operators are disentangled. By virtue of the technique of integration within an ordered product (IWOP) of operators, we can disentangle some complicated unitary operators and then reveal their physical role. Many new unitary operators can be found after new quantum mechanical representations are constructed by virtue of the IWOP technique. The unitary operators for permutation, Hilbert transform, Householder transform, and Hardmad transform can also be introduced. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics and provides a new route connecting classical transformations to quantum mechanical unitary operators. In this way, the symbolic method exhibits further beauty and elegance and can be increasingly used for developing various fields of quantum physics.
Off-Center Coherent-State Representation and an Application to Semiclassics
