Coherent States

The important role of coherent states [1, 2] in the contemporary quantum physics (especially , in quantum optics) is well-known. In particular, coherent states are essentially used the definition and evaluation of functional integrals [3]. The concept of coherent states has appeared , useful also in, the mathematical physics, in the theory of representations and in some other branches of mathematics. Introduced for boson oscillator (the Heisenberg group) the coherent states now are defined for wide class of quantum physical systems (including quantum fields), and also for systems, connected to other groups (including the supergroups). The coherent states can be defined for quantum groups, and also for various deformations (generalizations) of the exponents. The extensive and rather full bibliography of the theoretical researches concerning with coherent states, squeezed states and their numerous generalizations, is contained in [5]. There are several definitions of coherent states 1. as eigenstates of annihilation operator: a|z = z|z, z ∈ C (Barut-Girardello coherent states); 2. as outcome of action of unitary displacement operator D(z) = e za † −z * a on the fixed vector of the Hilbert space (usually the Fock vacuum |0): |z = D(z)|0 (Perelomov coherent states); 3. as the states, minimizing the Heisenberg (or Schrödinger-Robertson) uncertainty relations ; 4. as the states, satisfying to small number of the natural conditions [4] (Klauder-Gazeau coherent states).