The outward field represents the way the dissipated energy leaves the system. In principle, information on the measurement can be extracted through this output channel but it is usually lost and the line f is only a noise source in this case. We now present the electrical equations associated with the measurement device of fig. All equations are implicitly written in the frequency representation and the impedances are functions of frequency. This relation just means that the positive and negative frequency components correspond, respectively, to the annihilation and creation operators of quantum field theory.
Fields corresponding to different lines commute with each other.
The quantity is the energy per mode. It reduces to the zero point energy at the limit of zero temperature and to the classical result at the high temperature limit. We also assume that the fields incoming through the various ports are uncorrelated with each other as well as with amplifier noises. Now the output fields have their fluctuations determined by the transformation of the input fields through their interaction with the measurement device. Moreover, fluctuations are no longer independent in different ports.