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The states |a, n) are a complete set and are stationary under the unperturbed Hamiltonian Ho, which is independent of time. When the interaction Hi is turned on, the states are no longer stationary. The question we ask is: "How do these states evolve in time?" In practice, this may mean "How do excited states |a, 0) decay into other states \b,n) where n photons are emitted?"

To answer this question, we must find the time translation operator for the full Hamiltonian (3.5), which will be written

This Hamiltonian depends on time. We assume that Hj{t) is switched on at time t — to so that

We found the time translation operator for a Hamiltonian which is independent of time in Sec. 1.8, and we therefore know Uo(t,tQ) corresponding to H0 (it is just the product of Ua and a similar expression for the field). The total time translation operator will be defined to be

It is therefore sufficient to find an equation for 17/. Since Hi need not commute with Ho, and since it depends on t, the form of C/j depends on the definition (3.9). Note that our definition differs in important ways from that given in, for example, Fetter and Walecka (1971). From the definition (3.9) the interaction time translation operator is unitary, but note that f/j-(tj,¿2)i^/(¿2> ¿3) / Ui(ti,h) [see Prob. 3.5].

Now consider any physical observable represented by the operator 0(t). Under the full time translation operator it evolves according to

Under the free, noninteracting Hamiltonian the same observable evolves according to

Note that the free observable O0(i) is not equal to the interacting observable 0(t) because the free time translation operator J/0 is not equal to the full time translation operator i/totai- This is because when t > to, the time at which the interaction is turned on, Ui / 1. However, because of our definition (3.9), there is a simple connection between these two quantities:

0(t) = U^Oo(to)Ut otai = U^U0lOQ(t0)UQU, — Uf1Oo(t)Ui .

Hence the connection between the free observable and the interacting observable is

and the operator U¡ converts free observables into interacting observables, and vice versa.

We can find the operator U¡ from the relations

3.1 TIME EVOLUTION AND THE S-MATRIX which are the infinitesimal equivalents of Eqs. (3.10) and (3.11). Hence

where the last term was simplified using Eq. (1.34). However, H is a function of the fields <f> (a particular subset of the physical observables O),

where the square brackets [ ] will be used whenever we wish to express H as a function of field quantities and round brackets ( ) are used to express H as a function of t. Since H can be expanded in powers of <f>,

[Ho, Co«] = ("^f7/"1 + H0 + H¡ [0o«]) , O0(t) {^fUf1 +iHi[<t>o}) ,O0(t) =0 .

This is a remarkable equation. Because the Hi in this equation is a function of the free fields <j>o, the equation allows us to determine the interaction time translation operator entirely in terms of the free fields. Since Co is any operator in a complete set, and since any operator which commutes with all operators in a complete set must be a multiple of the identity (Schur's Lemma), this means that, just as in Eq. (1.35),

where Eq is a complex number which can depend on time. Hence we obtain an equation for Uj, dUj dt

62 INTERACTION OF RADIATION WITH MATTER This can be written as an integral equation

62 INTERACTION OF RADIATION WITH MATTER This can be written as an integral equation

which builds in the initial condition Ui(t0, t0) = 1. This is a very beautiful result. It gives [// in terms of Hj only, and Hi is a function of the free fields 4>o{t), which are known.

Now we use perturbation theory to solve Eq. (3.20). If Hi is small, we may solve the equation by iteration:

{//(Mo) = 1 -t f dt' H,{t') + {-i)2 f dU I'dt2HI{tl)H,{t2) + --- .

Note that Hi(t\) does not necessarily commute with Hi(t2)\ the order of terms in the double integral is important, and the later time stands to the left. If we define the time-ordered product r(H/(ti)Jf/(t2)) = Hiit^fhi^diU-^+Hd^Hiit^eih-U) , (3.22)

where 8(x) = 1 if x > 0 and is zero if x is negative, then the double integral may be "symmetrized,"

fdt j f

Jta Jt0

1 dt2 H /(il) H /(t2)= [ dt2 (2 dh H ¡{t2)H,ih) Jta Jta t rt

Ui(t,t0) = l-ijt dt' Hi(t') + {-^- J jl dti dt2 T (H/(îi)H/(î2)) + • • • •

Since there are n! time orderings for a time-ordered product of n terms, the expansion looks like an exponential and may be formally written

However, because each of the terms in the power series expansion (3.24) of the exponential is time ordered, the terms cannot actually be summed up into a closed form, and (3.25) should be regarded only as a shorthand for the original infinite sum (3.24).

The ¿'-matrix will be defined by choosing N(oo) = (0|t/;(oo, -oo)|0), so that

where a and /? are non-interacting states of H0 and |0) is the ground state. In nonrelativistic atomic theory, this state is a direct product of the ground state wave function of the atom and the photon vacuum (Fock state with no photons).

There are important reasons why we choose to normalize S by dividing by (0|i//(oo, — oo)|0). First, we show that this number must have unit modulus:

where c is a c-number. To prove this use the facts that U is unitary (which is a consequence of the conservation of probability) and that the ground state is stable (otherwise it would not be the ground state). Stability of the ground state implies that

where we assume that there is only one vacuum state. Hence, from UjUi = 1, it follows that

1 = <0|0> = (OlCZ/tZ/lO) = J2(0\U]\P)(P\Uj\0)

This proves the result.

Normalizing the S-matrix elements by this phase factor ensures that they are independent of any overall c-number phases. For example, if a c-number is added to Hi, then the time translation operator is changed to

and this phase becomes infinite as t -10 —i• oo. But this multiplicative factor cancels in Spa, since it occurs both in the numerator and in the denominator. Thus this cancellation is very useful, since it works even for c's which are infinite. An infinite c-number, which might occur order-by-order in U and which would otherwise disturb our concentration, is seen to be irrelevant since it exponentiates and cancels from S, and we may therefore ignore c-number infinities when calculating S. In the context of the time evolution of states, this provides the justification for dropping the electron and nucleus Coulomb self-energies, as discussed in Sec. 2.3. In addition, we now can justify dropping the additional E0(t) which arose in the derivation of Eq. (3.20) and use H] instead of Hj.

3.2 DECAY RATES AND CROSS SECTIONS

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