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6-2-4 Euclidean Green Functions
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The Green functions of a (scalar) theory are functions of the invariant scalar products of their external momenta, which are real Lorentz four-vectors. As we have seen at the end of the preceding chapter, and as we shall see in the next section, these functions enjoy analyticity properties and may be continued to unphysical regions. Here, we dwell on the continuation to the euclidean region where field theory presents interesting analogies with statistical mechanical problems. We shall not formulate this euclidean theory ab initio, but rather derive its perturbative rules. Again, for simplicity, we deal with a scalar theory. Consider a proper function r(n)(p!, ... , Pn) and assume that its arguments satisfy the condition (euclidean region) (6-92)
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for any real (ud. The manifold satisfying this condition is a linear space-like subspace (a hyperspace) of momentum space. If we consider r(n) as a function of the invariants Pi' Pj (i, j = 1, ... , n - 1) let us compute the dimension of the total manifold in the absence of this constraint, and the one of the submanifold (6-92). We must, of course, remember that in a four-dimensional space, any set of more than four vectors is linearly dependent. Hence for n :2: 6, the P are not independent. Consequently, the invariants Pi' Pj belong to a manifold of dimension 4n - 10 for n :2: 4 (1 and 3 for n = 2, 3 respectively). If condition (6-92) is taken into account this restricts them to a submanifold of dimension 3n - 6 for n :2: 4 (1 and 3 for n = 2, 3). For n = 4, both have dimension 6 (for instance, s, t, u, pI, ... ,Pi, with L Pf = s + t + u), whereas for n :2: 5, the euclidean submanifold (6-92) has a dimension less than that of the whole space. In the euclidean domain (6-92), all the P are orthogonal to some time-like vector n. After a Lorentz transformation, we may choose n = (1,0,0,0) and thus p = 0 for all i. This enables us to associate to each Pi a vector Pi of a euclidean R4 space: in that frame p = p = 0 and Pi = Pi. Then
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If we now pick a diagram contributing to fin) and express its contribution according to Eq. (6-89), we may define Pv , the sum of the Pi entering the vertex v, and to each "cut" C of the diagram we associate [compare (6-87) and (6-88)J
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QUANTUM FIELD THEORY
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Figure 6-28 The Wick rotation in parametric space (a) or in momentum space (b). The initial contour of integration is indicated by a single arrow, the final one by a double arrow. On (b), the crosses <8> stand for the position of the poles k = [(k l + Lp + isJ 1/2.
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Therefore, on the manifold (6-92), we may rewrite (6-90) as
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We recall that the have implicitly a negative imaginary part. This, together with the positivity properties of the bracket of the exponent (for 0), allows us to rotate by -n/2 (i.e., clockwise) the integration contour in the complex A plane (Fig. 6-28). This is, of course, equivalent to a simultaneous ( - n/2) rotation of all the variables a in Eq. (6-89). It must be stressed that the is prescription has played a crucial role in telling us which rotation was allowed and that the Wick rotation, as it is usually called, is illegitimate if some happen to be negative. In momentum space [compare with (6-83)J, the Wick rotation amounts to a n/2 rotation (counterclockwise) of all k in the frame where all p = O. This is in agreement with the position of the poles of the propagator, located at
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