The concept of ‘‘separation of procedures’’ and the ST-transformation are briefly reviewed together with the equivalence theorem that a d-dimensional quantum system with finite-range interactions is equivalent to the corresponding (d+1)-dimentional classical system with finite-range interactions. This theorem yields the introduction of the quantum transfer-matrix method. Thermo quantum dynamics is formulated using the quantum transfer-matrix method. This new formulation has the great merit that the thermal average for any observable Q in the thermodynamic limit is expressed as an expectation value over a temperaturedependent state vector in the single (conjugate) Hilbert space in the contrast to the usage of the double Hilbert space in thermo field dynamics.