Correlation identities and application

General correlation identities are derived in a unified way using the quantum analysis,(1-4) which include the well-known classical correlation identities(5-7) such as the Ising correlation identities.(5,6) In particular, quantum correlation identities are applied to the Heisenberg models and the transverse Ising model with local constants of motion. Integral identities are also presented in the spirit of the Gauss-Manin connection and they are applied to derive the generalized Bernoulli equation on the pressure and energy of ideal quantum gasses. There are many applications to condensed matter physics, which will be summarized in the present article together with an integrated list of papers on such applications in order to show how useful these correlation identities are.