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a transformation with unit jacobian, and write the reduced Bethe-Salpeter amplitudes as XP(Xl, X2) = e- iP . x Xp(x) (10-38) xAxt, X2) = e iP . x Xp(x) According to (10-33) X and X are not to be confused with wave functions but stand rather for generalized form factors. The normalization conditions are therefore not straightforward, since they involve the relative time variable Xo. This innocent-looking question has been the subject of a long elaboration. The role of normalization is, of course, to provide the correct relation between the X function and the four-point Green function. Furthermore, it is essential in selecting the proper solutions to Eq. (10-35). We return to the inhomogeneous Eq. (10-29). Introducing the overall momentum of the pair (1,2) through
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S(12)(X 10 X 2, 1, Y2 [p)=fd4aeip.aS(12)(X 1 +a, x +a"y y) 'y 2 ,10 2
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the bound state and its CPT transform contribute a pole in the variable p 2 :
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S(12)( . [P) _ iXP(Xlo X2)jp(ylo Y2) Xt, X2,Y1, Y2 p 2 _ M2 + ie
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with R regular in the vicinity of p 2 = M2. The factorization property of the pole residue is crucial for the interpretation in terms of bound state and has to be suitably generalized in the case of degeneracies. We iterate the equation (D + V)S(12) = -1 in the form S(12)(D + V)S(12) = - S(12l, insert (10-39), and use the fact that (D + V)X = i(D + V) = 0 to compare the residues of both sides at p2 = M2. The result is symbolically
i(D + V)X . . I1m 2 2 = I P - M
p2 .... M2
or, equivalently, the covariant expression
i[a!/l (D + V)
where the left-hand side involves an integral over relative variables. In general the normalization condition depends on the "potential" Vas opposed to the nonrelativistic case. It is useful to have these equations also written in momentum space. Let p denote the conjugate variable to x. According to (10-37) we have
+ P2
where P1 = 1]lP + P and P2 = 1]2P - P refer to momenta carried by the fields <P1 and <P2 (Fig. 10-5). We have here no natural definition of the relative momentum as given by the separation of variables in a nonrelativistic motion, which results from the specific choice 1]1.2 = m1,2/(m1 + m2). In the relativistic case this choice may, however, be a good candidate for the purpose of comparison. Using the same symbols for the Fourier transformed quantities and taking translational invariance into account, we have
xp(p) =
e ip . XXp(x) d4 x
[(1]l P
+ p - miJ [(1]2 P -
p)2 -
m~J Xp(p) + (~~~ V(p, p'; P)xp(p') = 0
p)2 -
jp(P) [(1]1 P
+ p - miJ [(1]2 P -
m~J + (~~~ jp(p') V(p', p; P) =
(10-42) 0
The exchange of a scalar particle of mass p and coupling constants gl and g2 to particles 1 and 2, that is, such that the corresponding interaction lagrangian
pz = 'T/zP- P Pz
Figure 10-5 Homogeneous equation for the bound-state amplitude X.
Yin! =
(g1 <pi <P1
+ g2 <pi <P2)<P, leads in the Born approximation to
, Vl (p, p ; P)
ig l g2 p - p')2 - J1 2
+ It; .
independent of P. Explicitly, the normalization condition reads
f(~~4 (~~~
jp(p') a!/l {[l]lP
+ p - miJ [(1]2 P -
p)2 -
m~J(2n)4b4(p -
p') (10-43)
+ V(p', p; P)}xp(p) = 2iP/l
With Eqs. (10-35) and (10-40), or equivalently (10-42) and (10-43), we have now the basic ingredients to study some definite models.
10-2-2 The Wick Rotation
Bound-state equations are derived in Minkowski space. In his early study, Wick was led to an analytic continuation to euclidean variables which was at the origin of the Wick rotation. It is easy to justify this procedure perturbatively, i.e., precisely, ignoring the possibility of new singularities such as those investigated here. Some care is required in order to extend it to the present situation. The essential physical point is to insure that stability criteria are met. Of course, the desirability of using this trick is to bring the equation in a form convenient for a simpler analysis. Let us proceed in a straightforward manner by writing, for X = 0,
xp(x) = B(xO) f(x, P) jp(x)
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