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d4x [if(x)l/J(x)
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(4-73)
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xexp [ - ffd4Xd4Yif(X)<0ITl/J(X)ifr(y)10)Yf(y)]
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The only difference with the corresponding formula for Bose fields (4-64) lies
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in the absence of the factor i in front of the last exponent. This is due to the fermion charge. Had we started with a charged, non hermitian Bose field A, we would have written 2'I(X) = [A*(x)j(x) + j*(x)A(x)] and obtained two equal contributions in the exponent <01 H d4 x d4 y T[2'I(X)2'/(y)] 10), and therefore no factor i either. The explicit relations between time-ordered and normal products follow when expanding (4-73) and taking into account the anticommutation of the 11, fj. For example, to second order in the fields, we get
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+ ij/(X)I1(X)] [fj(y)I{I(y) + ij/(Y)I1(Y)] + ij/(X)I1(X)] [fj(y)I{I(y) + ij/(Y)I1(Y)]:
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d4 x d4 y : [fj(x)l/I(x)
d x d y fj(X) <01 T[ I{I(x)ij/(y)]
0 1) I1(Y)
and after identification
TI/I(x)l/I(y) = : I{I(x)I/I(Y): Tij/(x)ij/(y) = : ij/(x)ij/(y): TI{I(x)ij/(y) = : I{I(x)ij/(y): Tij/(x)I{I(y) =
0) + <01 TI{I(x)il/(y) 1 0) : ij/(x)I{I(y): + <01 Tij/(x)I{I(y) 1
(4-74)
We note that the two last expressions are equivalent, since
T[ I{I e(x)ij/q(y)] = - T[ij/q(y)I{I e(x)]
and 11 are spinor indices). For the sake of compactness, it is convenient to include in the argument of I{I the space-time coordinate, the Dirac index, and a discrete index which distinguishes I{Ia.(x) from ij/a.(x). With this convention, the general expression of Wick's theorem for Fermi fields reads
One sums over all distinct contractions, and (ip = 1 is the signature of the permutation which transforms {1, ... , n} into {1, ... -;i2;, ... ;k;p,"" n, k pl , .,j( P 2J Let us illustrate this rule on the T product of four fields:
T{I{I(1)1/I(2)1/I(3)1/I(4)} = : 1/I(1)1/I(2)1/I(3)I{I(4): - : I{I(1)I{I(3):
+ : I{I(1)I{I(2): <01 TI{I(3)I{I(4) 10)
0 0) <01 TI{I(2)I{I(4) 1 + : I{I(1)I{I(4) : <01 TI{I(2)I{I(3) 1) + : I{I(2)I{I(3): <01 TI{I(1)I{I(4)IO) - : I{I(2)I{I(4): <01 TI{I(1)I{I(3) 10) 0 0 + : I{I (3) I{I(4) : <01 TI{I(1)I{I(2) 1) + <01 TI{I(1)t/i(2) 1) <01 TI{I(3)I{I(4) 10) - <01 TI{I(1)I{I(3) 10) <01 TI{I(2)I{I(4) 10) + <01 TI{I(1)I{I(4) 10) <01 TI{I(2)I{I(3) 10)
INTERACTION WITH AN EXTERNAL FIELD
Equation (4-75) may be specialized to the vacuum expectation value:
<01 T!/J(1) !/J(2n -1)10) = 0 <01 T!/J(1) !/J(2n) 10) = L
distinct terms
<01 T!/J(Pzn- d!/J(Pzn) 10)
2"f Bp <01 T!/J(Pd!/J(Pz) 10) ..
(4-76)
permutations P
<01 T!/J(PZn-l)!/J(PZn) 10)
The expression on the right-hand side of (4-76) is a pfaffian, i.e., the square root of the determinant of the antisymmetric 2n x 2n matrix with generic element
<01 T!/J(i)!/JU) 10).
4-2-3 General Case
In the general case, where both fermion and boson fields are involved, the expression of the Wick theorem may be easily derived. Distinct fermion fields anticommute, distinct Bose fields commute, and a boson field is assumed to commute with a fermion field (the latter point is just a matter of convention). We then decide to use as above a compact notation: !/J denotes either a boson or a fermion field, with a discrete index to distinguish its nature. It must then be clear to the reader that the formula (4-75) is still valid, provided we understand that the sign (Jp is the signature of the permutation of the Fermi fields only. This result follows directly from the generating function (4-73), which has the same form whatever the fields are. The Wick theorem discussed in this section has a close relationship to the computation of the integral of a polynomial of several variables, with a gaussian weight. This analogy is not fortuitous, as we shall see later in the discussion of functional integrals.
4-3 QUANTIZED DIRAC FIELD INTERACTING WITH A CLASSICAL POTENTIAL 4-3-1 General Formalism
After this algebraic interlude we shall now discuss another case of a quadratic lagrangian, namely, the electron-positron field in the presence of a classical electromagnetic field. Electron-positron pairs will possibly be created in this external field. This is the physical counterpart of the problem discussed in Sec. 4-1, whereas the creation or annihilation of single electrons by the anticommuting source of the last section was its mathematical counterpart. Besides interesting phenomena such as pair creation -which has clearly no
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