Quantum fluctuation in thermal vacuum state for mesoscopic distributed parameter circuits

In this paper we consider a non-dissipative distributed parameter circuit at a finite temperature T. We find the unitary operator for diagonalizing the Hamiltonian of the uniform periodic transmission line. The unitary operator is expressed in a coordinate representation. Thermal field dynamic is used in our discussion. It is shown that distributing parameter circuits and quantum fluctuations, which also have distributing properties, are related to both the circuit parameters and the positions and the model of signals and temperature T. The higher the temperature, the more quantum noise the circuit exhibits. The research will be helpful to miniaturize intergreate circuits and electric components. It will be also significant for the futher study of the qualitities of mesoscopic system.