where, in the last expression, it is understood that the projection operator V(t,p) involves an integration over the virtual energy po on which the operator $>(p)

depends and somehow projects in onto the physical energy of the particle, which is w(p) (the details of how this works out will be given shortly). Note also that the projection is carried out at a particular time t.

Using the position space projection operator (14.99), the S-matrix, in the notation of Fig. 9.7, is where the particles have momenta pi + p2 -* p\ + p2, and Ptotai is the product of the four individual projection operators which project the S-matrix from the vacuum expectation value in (14.101). The time ordering symbol is added for free because all of the fields internal to Uj are already time ordered and are at times between ±oo, and the projections of the initial and final fields commute, so their time order does not matter. Now, as we discussed in Sec. 14.3, this vacuum expectation value of a time-ordered product can be calculated from the normalized generating function Z[J). Hence, casting this time-ordered product into momentum space and using the momentum form of the projection operators (14.100), we obtain the following expression for the S'-matrix which describes elastic scattering:

The projection operator 'Ptotai is now the product of four momentum space projection operators with the form given in (14.100) and is not the same as the operator given in Eq. (14.101). Also, in the process of going from the initial expression (14.101) to our final expression (14.102), all operators are replaced by their c-number equivalents, as has been discussed in detail in the previous sections.

Equation (14.102) is a specific example of the general formula for the scattering matrix in the path integral formalism. In the general case, there is one derivative with respect to J for each particle in the initial state and one with respect to J* for each particle in the final state and projection operators for each external particle. We will discuss the form of the projection operator for fermions later.

We specialize to non-forward scattering, where p[ / pi and p2 / p2. This means that, when computing the above derivatives, we must involve the interaction terms, because the free generating function only contains products of the form J*(p)J(p), in which both initial and final momenta are necessarily equal. The

=VH°°,P2)'P](o°<P'i)'P(-°o,Pi)'P(-<x><P2)

=Ptotai (0\T(^(x'2)^(x'l)U,^(x1)^(x2))\0) , (14.101)

second order generating function, which we denote by Z<2), is 7(2)1 n-f -\2 A2 f d4kld4k2d4k3d4k[d4k'2d4k'3

where Z0 is the free generating function (14.92), which contains the product J*(p)J(p) = J(-p)J(p). The second order elastic scattering amplitude therefore results from 6 "internal" differentiations from Eq. (14.103) and 4 "external" differentiations from Eq. (14.102), for a total of 6 + 4 = 10 differentiations of J on Z0, after which all J's are set equal to zero. Because Z0 ~ exp(J2), each differentiation by J brings down a factor of J, which must be eliminated eventually if the final result is to be non-zero when J —> 0. Therefore, 5 of the 10 derivatives must act directly on Z0, bringing down 5 powers of J, which are then eliminated by the 5 remaining derivatives. Hence, all the derivatives must be "paired" so that the factor of J brought down by one is eliminated by the other. When two derivatives are "paired," their momenta must sum to zero because they act on a single J(—p)J(p) term. Therefore the external derivatives cannot be paired with each other because the scattering is in the non-forward direction, requiring p\ / p[, etc. Hence each external derivative must pair with one internal derivative, leaving two internal derivatives to pair with each other. However, only internal derivatives from different interaction terms can pair; any terms which might arise from a pairing of derivatives within the same interaction are zero. To see this, note that \ the derivatives with respect to J(ki) and J{k2) act on the same J(-p)J(p) term, for example, they will force k\ + k2 = 0 and the delta function will then force k3 = 0. Since one of the external four-momenta must pair with k3, it will be therefore also be zero, which is impossible. Now, since all the derivatives are identical (i.e., the J's are identical even if their arguments are not), there are many ways to obtain the final answer. As we have just shown, the only restriction on how the derivatives are evaluated is that one derivative from each interaction must pair, and there are therefore 3x3 = 9 identical possibilities. Therefore the action of the 6 internal derivatives gives

(-if J(-kQ J(-fca) J(-k[) J{—k'2) 64{k'3 + k3)Z0[J} X [k\ -n2+ it)[kl - M2 + ie}[k[2 — /i2 + ie][k'22 — p.2 + ie\[k2 — p? + ie] A2 f d4k1d4k2d4k'1d4k'2 64(k1 + k2 + k[ + k'2)

ie where, by convention, the 9 identical terms have been expressed as a factor of 9 times the term with fc3 and k'3 paired and the factor Z0[J] has been set to unity in the last step, anticipating the fact that none of the 4 external derivatives will act on it, and it will become unity when J 0 in the final step. Note also that Z(2) [0] = 0, showing that this factor does not contribute to the overall normalization factor Z[0], justifying the use of Z0[0] (instead of ZQ[J)/Z[0]) in the above equation. Next, differentiating this four times, as required by Eq. (14.102), gives 4!=24 terms, which can be organized into three different terms, each multiplied by 8. Because the two final momenta are fixed by differentiations with respect to J(—p\) and J (-p'2), and the initial momenta by differentiations with respect to J(p\) and J(p2), the argument of the delta function becomes p\ +P2 —p[ —p'2 regardless of how the derivatives act, and the different terms are distinguished only by different values of the quantity A = $(ki + k2 - k\ - k'2)2. Only three different values of A are possible, arising as follows:

Hence, after the four external derivatives are computed, and the remaining J's are set to zero, we have

dJ'iAWtfjsjipdsjfa) J=0 . A2

Pi, |
P'2 |
—► |
k\, k'2 |

Pi, |
P'2 |
ki, k2 | |

-P2, |
P'2 |
- |
k[, k'2 |

~P2, |
P'2 |
- |
k\, k2 |

Pi, |
~P2 |
- |
1y v kI> k2 |

Pi, |
-p2 |
-> |
h, k2 |

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