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0 L <01 T[ A (Xk/>,) A (XkI'J] 10)'" <01 T[ A(Xk,',P-!) A(Xk,',,)] 1) + ...
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(4-65)
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In these formulas, the caret above a term means that it is to be omitted from the product, and the sum runs over all permutations that lead to different expressions. In words, Eq. (4-65) expresses the T product as the sum of all possible normal products where some pairs of fields have been omitted and replaced by their contraction, i.e., the vacuum expectation value of their T product. It is a good exercise to prove directly the identities (4-65). This is most easily done by induction. We also let readers convince themselves that they may be generalized to the T product of distinct fields TAl (Xl)' .. An(xn). They may be further extended to an expression of the form
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T: [A(Xl)" . A(Xk)]:' .. : [A(x/)' .. A(xn)] :
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with the restriction that only contractions between distinct normal ordered products occur. Identities similar to (4-65) may be written in order to express the ordinary product Al (x 1) ... An(xn) in terms of normal products. The only modification is that now contractions represent the vacuum expectation value of the ordinary product
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A 1(Xl)'" An(xn) =
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L: A 1(X1)'"
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~ Akl(XkJ" Ak2P (Xk 2 J" A(xn):
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0 {<Ol AdxkJAk2 (Xk,) 1)." <01 Ak2P_l(Xk2P_) Ak2P (Xk 2)
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+ permutations}
QUANTUM FIELD THEORY
Finally, analogous identities connect T products and ordinary products. In that case, the contraction stands for a retarded Green function; for instance,
T A(x)A(y)
= A(x)A(y) + e(yO -
XO) [A(y), A(x)]
From Eq. (4-65), it follows that
<01 T[A(xd A(X2p-d] 10) = 0
0) <01 TA(Xl) A(X2p) 1 =
L <01 TA(xpJA(xP2) 10 ) ... p
x <01 T A(XP2P_JA(xP2) 10 )
(4-66)
The symmetric form of the right-hand side has been called "haffnian" by Caianiello. Using (4-66), Eq. (4-65) may be rewritten in the weaker form
TA(Xl) A(x n )=
L p=o
[nI2]
kl< <k 2p
L: A(Xl) A(Xk)" A(Xk2J A(x
n ):
(4-67)
4-2-2 Fermi Fields
When Fermi fields are involved the only modifications to be made in Eq. (4-65) are the signs. As in the Bose case, we shall derive an identity relating the generating function of the T products to the one of normal products. The derivation for boson fields was based on the identity (4-15). In order to deal only with commutators, when fermion fields are present, we introduce anticommuting sources ij and 1'/ for IjJ and lif. These quantities anticommute among themselves as well as with IjJ and lif. We insist on their purely mathematical role (they are elements ofa Grassmann-i.e., anticommuting-algebra). They enable us to write the identity (4-68) [ij(x)ljJ(x), lif(y)I'/(Y)] = ij(x){ ljJ(x), lif(y)} I'/(Y) All other commutators of ijljJ, lifl'/ vanish. We now introduce the fictitious interaction lagrangian (4-69) and the corresponding S matrix
S = Texp i
d4 x [ij(x)ljJ(x)
+ lif(x)l'/(x)]
(4-70)
Since the commutator [2'J(x), 2'J(Y)] commutes with 2'J(z), we may use again our favorite identity (4-15) and get
T exp i
x exp { -
f iff
d4 x [ij(x)ljJ(x)
+ lif(x)l'/(x)] =
{i fd x(ij(x)ljJ(x) + lif(x)l'/(x))}
d4 x d4 y e(xO - yO) [ij(x)ljJ(x)
+ lif(x)l'/(x), ij(y)ljJ(y) + lif(y)I'/(Y)] }
(4-71)
INTERACTION WITH AN EXTERNAL FIELD
In the same way, we may decompose ifl/J + l/JYf into a sum of annihilation, positive frequen~y part (superscript +), and creation part (superscript -), and write exp i f d4x [if(x)l/J(x)
+ ifr(x)Yf(X)]
= exp i f d4x [if(x)l/J(-)(x)
+ ifr(-)(x)Yf(X)]
x exp i f d4x [if(x)l/J(+)(x) x exp
+ ifr(+)(x)Yf(X)]
f f d4x d4y [if(x)l/J(-)(x)
+ ifr(-)(x)Yf(X), if(y)l/J(+)(y) + ifr(+)(y)Yf(Y)] }
(4-72)
As above, we replace the c number:
d4x d4y {e(xO - yO) [if(x)l/J(x)
+ ifr(x)Yf(x), if(y)l/J(y) + ifr(y)Yf(Y)]
+ ifr(-)(x)if(x), if(y)l/J(+)(y) + ifr(+)(y)Yf(Y)]}
- [if(x)l/J(-)(x)
by its vacuum expectation value, that is,
<01 f f d4x d4y{e(xO - yO) [2'1(X), 2'1(Y)]
2'1(Y) 2'1 (x)} 10 )
<01 f
4 4 0 f d x d y T[2'1(X)2'1(y)] 1 )
Owing to the anticommutation of the Yf, this is nothing but
0) d4x d4y [if<x(x) <01 e(xO - yO)l/Jix)ifr/b) - e(yO - XO)ifrp(y)l/J<x(x) 1 Yfp(y)
+ Yf<x(x) <01 e(xO yO)ifr<x(x)l/Jp(Y) - e(yO - XO)l/Jp(y)ifr<x(x) 10)ifp(y)]
Notice that the other terms in l/Jl/J or l/Jl/J have disappeared in the integration. We recognize the definition of the T product of Fermi fields [see Eqs. (3-173) or (4-55)]:
4 4 0 d x d y [if<x(x) <01 Tl/J<x(x)ifrp(y) 1) Yfp(y)
+ Yfix) <01 Tifr<x(x)l/Jp(y)IO)ifp(Y)]
Both terms give the same result, after an interchange of the variables x and y under the integration sign. Finally, we obtain
T exp i f d4x [if(x)l/J(x)