In the classical path approximation, the correlation function for electric dipole radiation is given by Pa the radiator. represents the average over electron states (see Eq. (47) of paper I): denote the dipole moment and the density matrix for The thermal average denoted by the subscript av where P(x) and W(v) are the position and velocity distribution functions for the electron perturbers (defined by Eqs.(37) to (40) in paper II). The time development operator for the system T (R, x, v, t) is the solution of the differential equation a and it may be written in an interaction representation defined by T_(R, ✯, v, t) = exp (−itH /h) U( a where (II. 8) (II. 9) a at in U ̧(t) = √(t) U ̧(t) (II. 10) and ≈ (t) = exp (itH/h)V(t) exp (-itH /h). (II. 11) It should be noted that V(t) is identical with V(t) in paper II except that we have not yet made the no quenching assumption which removes the unperturbed part H in the Hamiltonian H in Eq. (II. 11). Using the time ordering operator O, U (t) may be written in the form a a To evaluate the trace over atomic states in Eq. (II. 6), it is convenient a In paper II and III, the correlation function C(t) was evaluated for the case of no lower state interactions in order to keep the mathematics as simple as possible because one of the U(t) operators in Eq. (II. 13) may then be replaced by a unit operator. In this paper we will give a more general evaluation of C(t) which includes lower state interactions. For this purpose we introduce in the next section a more compact tetradic notation. Furthermore, it should be noted already at this stage that we will interchange the sequence of approximations with respect to paper II by deriving the generalized unified theory before making the no quenching approximation. This makes the results of the unified theory also useful for situations where the no quenching approximation cannot be made like, for example, microwave lines. III. THE TETRADIC NOTATION The purpose of the tetradic notation which we shall use is to write the product of the U (t) operators in Eq. (II. 13) in terms of a a single operator. To do this we first consider the product of the matrix elements (al Ala') and (B| BB') where A and B may be any arbitrary operator. This product may be written in terms of the direct product A&B according to (al Ala') (8|B|B') = (αB | AB❘a'B'), (III. 1) where the product states laß) = la) 18) are essentially the same as the states of Barangers "doubled atom" (Baranger, 1962). This direct product, AB, is a simple form of tetradic operator. If one of the operators A or B happens to be a unit operator I, we may conveniently denote this fact by means of superscripts and r according to That is, a superscript & denotes a "left" operator which operates only on the "left" subspace (in this case the la), la') subspace) and a superscript r denotes a "right" operator which operates on the "right" subspace. It is thus clear that any "left" operator will commute with any "right" operator: With this notation, the thermal average in Eq. (II. 13) can now be We have chosen to write (b!U (t)! c) as (cut) b) simply for con a a venience in the derivation given in later sections. Noting the definition of U(t) given in Eq. (II. 12), we define operators v (R, x, v, t) and a Since any "left" operator commutes with any "right" operator, we have u'*(R, X, v, t) U1(R, X, v, t) = 19 exp { - £ £}£}{} a a = (R, X, v, t) (R, X, v, t')dt (III. 7) where ✅(Ñ‚ X, v, t) = √2 (Ñ‚ X, V, t) - ↓1 *(‚ X, V, t). (III. 8) We have now succeeded in replacing the two U (t) operators by a more a a alba). (III. 9) (t) is formally the same as the operator U_(t); that is, it satisfies the same type of differential equation in — U(Ñ‚ X, v, t) = ỡ (Ñ‚ Î, v, t) U(R, X, v, t) . at (III. 10) This means that all of the line broadening formalism which has been developed for U(t), will be directly applicable to U(t). a To make the formal correspondence more complete we use the operators H, H, V' (R, x, v, t) and V¶(Ñ‚ X, v, t) to define the tetradics and ↑ (R, x, v, t) according to e e |