Monique Combescure

This book presents various types of coherent states introduced and studied in the physics and mathematics literature and describes their properties together with application to quantum physics problems. It is intended to serve as a synthesis on coherent states and their applications for physicists and mathematicians. The basic mathematical structures of coherent states are first explained for the harmonic oscillator coherent states (also named canonical or Schrödinger coherent states).
Then we introduce the squeezed coherent states and explain their connections with the Weyl quantization, the symplectic group and quadratic Hamiltonians. The next step is to study the propagation of coherent states by arbitrary smooth Hamiltonian. This is applied to give a rigorous proof of the Gutzwiller trace formula. In a second part of the book we consider coherent states in various non Euclidean settings.
The z-torus coherent states are explained with application to quantization of cat maps and quantum ergodicity. Then we consider coherent states in the sense of Gilmore-Perelomov for the group SU(2) and SU(1, 1) ; coherent states for the hydrogen atom which are related to the group SO(4) and the unit sphere S3. In the last part of the book coherent states are introduced for infinite systems of bosons, with application of the Hepp's method to the classical limit of large systems.
Then we consider finite systems of fermions and we introduce super-symmetric systems with explicit examples using coherent states.