A formalism has been developed, using Feynman's space-time formulation of nonrelativistic
quantum mechanics whereby the behavior of a system of interest, which is coupled to other
external quantum systems, may be calculated in terms of its own variables only. It is shown
that the effect of the external systems in such a formalism can always be included in a general
class of functionals (influence functionals) of the coordinates of the system only. The proper-
ties of influence functionals for general systems are examined. Then, specific forms of influence
functionals representing the effect of definite and random classical forces, linear dissipative
systems at finite temperatures, and combinations of these are analyzed in detail. The linear
system analysis is first done for perfectly linear systems composed of combinations of harmonic
oscillators, loss being introduced by continuous distributions of oscillators. Then approximately
linear systems and restrictions necessary for the linear behavior are considered. Influence func-
tionals for all linear systems are shown to have the same form in terms of their classical response
functions. In addition, a fluctuation-dissipation theorem is derived relating temperature and
dissipation of the linear system to a fluctuating classical potential acting on the system of
interest which reduces to the Nyquist-Johnson relation for noise in the case of electric circuits.
Sample calculations of transition probabilities for the spontaneous emission of an atom in free
space and in a cavity are made. Finally, a theorem is proved showing that within the require-
ments of linearity all sources of noise or quantum fluctuation introduced by maser-type
amplification devices are accounted for by a classical calculation of the characteristics of the
maser. © 1963 Academic Press