A generalization of the notion of coherent state, developed by this author and others, is extended to the unitary groups. The construction of ”quasi-coherent states”, an over-complete set of states which transform according to a given irreducible representation of a compact Lie group G, is the goal. This had been done previously for \(G=SU(2)\); and here the same problem is solved for \(G=U(N)\) and SU(N). A kind of generating functional is involved, from which expectation values of G-invariant operators can be computed.
There are group integrations involved which are only carried out for the adjoint representation of U(N) and SU(N). Both real and complex representations are considered. Some physical applications to systems with one degree of freedom are discussed. These are based on the notion of a quasi-coherent state as representing a physical ”condensate”. A free quantum field in a singlet quasi-coherent state has an analog two- dimensional lattice gauge theory which exhibits critical behaviour. This critical behaviour is investigated for large N.