M 1486

We will reduce the integral for the general case of a complex field, where

Jdix{dlJi<p*(x)dtl(l){x) - (/i2 -ie)cf>*{x)cj){x)}

If the field is real (which is the case in the symmetric <j>3 theory), </>*(p) = 0(-p). Also, if the field is complex,

If the field is real, then J*(p) = J(-p) and only one of these terms is present, or alternatively, the two terms in (14.88) must be multiplied by Hence, for the symmetric </>3 theory, with real field <f>, the phase 0 of the exponential in Zq[J] is

6 = Jd*x {£0 + it?4>2 + J{x)<t>{x)} = \Jd4p {<*>*(p) [p2 ~p2+ «] 4>{p) + r(p)4>(p) + </>*(p) J(P)} This can be diagonalized by introducing

Putting this in the exponential gives

To obtain results in momentum space, we will re-express Cint using the following reduction:

,4j4Pld4P2 ^P3e_t(pi+p2+p3).x d4p\ dip2 d4p3

d4pi dip2 dip3 (27T)2

■6' {pi +P2 + P3)<t>(Pl)<t>{P2)<fr{P3)

Now, as a second example of the use of path integrals, we calculate the M-matrix for the elastic scattering of two particles. We will only obtain the result to second order in the couplings, but many of the steps, including the method for extracting the .M-matrix from the path integral, are quite general. Recall that the second order result for this case was already obtained in Sec. 9.7, Eq. (9.90).

From the discussion in Sec. 14.2, we know that the S-matrix is proportional to the path integral, but in field theory a single Lagrangian describes many different interactions, so we must develop a way to project out the particular channel in which we are interested. Recall that the initial and final states are constructed from the vacuum by the action of creation operators and that these creation operators r xn be obtained from the field operators by projecting out the coefficient of the positive frequency part of the field. The form of this projection depends on the properties of the field. For scalar fields we use the orthogonality of the KleinGordon wave functions, and continuum normalization, to obtain a\p)

VMp)W dt

We will also need to express this projection operator in terms of momentum space variables, which can obtained by substituting the momentum space form of $ into the above expression, giving af(p) = -i J

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