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The presence of photons in the initial state has raised the probability of emission. This is the basic result of stimulated emission. From Eqs. (4-48) it is also easy to check that the average number of radiated photons is stilI ii j
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The derivation of the S matrix in Eq. (4-13) relied on the assumption that the source was not an independent dynamical variable, i.e., that its time evolution was given once and for all. This method has left an arbitrary, source-dependent phase in S. It is therefore appropriate to set up now the general formalism and to compare its results in this specific case. The idea is to construct the operator that realizes the time-dependent canonical transformation relating the interacting field AI' to the incoming field Afn:
AI'(t, x) = U- 1(t) Afn(t, x) U(t)
(4-49)
As already discussed and expressed in Eq. (4-8), Afn is some weak limit of AI' as t -+ - 00, and, consequently, in that limit U reduces to the identity lim U(t) = I
t-+ - 00
(4-50)
In the canonical quantization, we aeal with the operators A(t) and n(t) that satisfy
at A(t, x)
i[ H(t), A(t, x)]
(4-51)
and an analogous equation for n(t). Here H(t) = H[ A(t), n(t), j(t)] denotes the hamiltonian. Similarly, the in-field Ain satisfies
where Hbn is the time-independent free hamiltonian [compare Eq. (3-120)] expressed in terms of the in-creation and annihilation operators. Let us derive the time evolution equation for the operator U(t). Unitarity requires that
U(t)]U- 1 (t)
+ U(t):r
U- 1 (t)=0
Moreover, it follows from Eq. (4-49) that
U(t)H(A(t), n(t),j(t))U- 1 (t) = H(Ain(t), nin(t),j(t))
at Ain(t, x) = at [U(t)A(t, x)U- 1 (t)] dU(t) = ---;it U- l(t)Ain(t, x)
+ iU(t) [H(A(t), n(t), j(t)),
A(t, x)] U -l(t)
INTERACTION WITH AN EXTERNAL FIELD
d + Ain(t, X) U(t) dt
= [ ------;[t U =
dU(t)
. + IH(Ain(t), 7rin(t),j(t)), Ain(t, X) ]
i[Hbn , Ain(t, X)]
with an analogous equation for 7rin. We conclude that the operator
commutes with every "in" operator and is thus a c number. This number will not contribute to normalized matrix elements of U (this point will be justified later), and we discard it hereafter. Then U satisfies the evolution equation
= [H(Ain(t), 7rin(t),j(t)) - Hbn] U
== Hint(Ain(t), 7rin(t),j(t))U == H/(t)U
(4-52)
In our case Hbn = H(Ain(t), 7rin(t), j) Ij= 0, and the interaction hamiltonian H/(t) vanishes withj. The evolution dictated by the operator U(t) will thus be referred to as the interaction representation. Notice that H/(t) depends on time both through the time dependence of j(t) and of Ain(t) and 7rin(t). Equation (4-52) may be solved by iteration of the corresponding integral equation:
U(t) = 1- i
roo dtl H/(t )U(td
(4-53)
where the boundary condition (4-50) has been taken into account, with the result that
(4-54) In these expressions, the ordering of the operators is important: it is related to the ordering of their time arguments. We are led to the general definitions of the time-ordered product of n operators (T product, for short):
T[ Al (t 1)'" An(tn)] =
L e(tpp t p2 , ... , tp)epApt(tpJ'" p
Adtp)
(4-55)
where the sum runs over all permutations P, the
efunction enforces the condition
QUANTUM FIELD THEORY
and ep denotes the signature of the fermion operators permutation involved in this product. (The latter point is of no use now, but will serve us soon.) Here, the T product is symmetric and we may write
(4-56) The last expression is symbolic. Its meaning is given by the previous line. However, T products of exponentials enjoy an important property, which may be proved easily. For t1 ::;;; t2 ::;;; t3,
T exp
t3 tl dt O(t) = i
T exp
it3 dt O(t) T exp i t2 dt O(t) t2 tl
If HI(t) ---+ 0 as t ---+ - 00, the operator U(t) of Eq. (4-56) does satisfy the requirement (4-50). This is what happens in our case if the current is adiabatically switched off in the remote past. The S matrix is defined as the limit
S = lim U(t) = T exp [- i
t-++oo
dt' HI(t')]
(4-57)
In the present case, HI =
Jj(x) Ain(X) d3 x, and we get
S = T exp [ - i
d x jl'(x) Afn(x) ]
(4-58)
In this expression, the T symbol contains the only reference to a definite time coordinate. This looks noncovariant, but owing to local commutativity the density I(X) == r(x)AI'(x) commutes with I(Y) if (x - y < O. Therefore a change of frame does not modify the expression of the T product. In every theory without derivative coupling we have
HI(t) =
d 3x I(X) = -
d 3x 2I(X)
(4-59)
where it is understood that XO stands for t and that all operators are in-operators. Finally, the general "covariant" expression for the S matrix reads
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