Renormalization group at finite temperature

The renormalization-group equation in quantum field theory at finite temperature is investigated. Owing to the freedom of the renormalization procedure, one can scale the temperature as well as the momentum in choices of renormalization points. The result is an extended version of the renormalization group at zero temperature. Its Lie differential form defines two types of sets of renormalization-group coefficients. Several examples of the applications include the high-momentum limit (deep-inelastic limit), the high-temperature limit, the low-temperature limit, and the critical behavior near a phase transition point