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(Pi - Pi)2 - ie P? ~ {Pi ~ P2)2 ~ »e M2 ~ (Pi + P2)2 - if-

In (14.106), each of the external four-momenta is on-mass-shell (as we just discussed), and hence Eq. (14.106) is identical to Eq. (9.90), illustrating that the path integral formalism and the operator formalism are equivalent. We have recovered the Feynman rules for <f>3 "tree" diagrams.

Next, we will see how the Feynman rules for "loops" emerge from the path integral. Again, we confine our discussion to the symmetric </>3 theory. In the next section we calculate the self-energy of the scalar particle, which was previously obtained using the operator formalism in Sec. 11.7. A study of this simple case also leads naturally to a discussion of disconnected diagrams and vacuum bubbles. The discussion will illustrate the similarities and differences between the method of path integrals and the operator formalism.

14.5 LOOP DIAGRAMS IN </>3 THEORY

In this section we calculate the propagator for a neutral, self-conjugate particle to order A2. The result for the free propagator was already given in Eq. (14.95), and now we will obtain the first loop contribution to the self-energy. As before, the propagator is but now the generating function includes interactions in contrast to the free generating function Zq used in Eq. (14.93).

The first non-zero contribution which depends on the coupling strength A occurs in second order and can be computed from Eq. (14.103), except that in this case it will be necessary to normalize the generator by dividing by Z[0] (it was sufficient to divide by ZQ[0] for the scattering problem we treated above but is no longer sufficient here). The term which is first order in X does not contribute to the propagator, because when it is inserted into Eq. (14.107), it contains an odd number of J derivatives, leaving at least one factor of J after differentiation and hence insuring that it is zero when ,7 = 0.

Now we carry out the six "internal" J derivatives in (14.103). Our analysis is similar to the steps following Eq. (14.103), except that only terms with two factors of J brought down from the exponential in Z0 or no factors of J will survive the final action of the two "external" J derivatives in (14.107). Since there are six internal derivatives to be evaluated, the terms with two factors of J require that four of the internal derivatives be paired [as defined in the discussion following Eq. (14.103)], and those with no factors of J require that all of the internal derivatives be paired.

Consider the terms with no factors of J first. These require three pairings, and since all of the derivatives are identical, there are only two distinct ways in which these pairings can be made. We may pair momenta from one interaction with momenta from the other, for example fci ♦-> k[ k2 <-► k'2 k3 <-> k'3 , or we may pair momenta within a single interaction, for example ki <-► k2 fc'i k'2 k3 «-► k3 .

There are 3! = 6 ways to make the first pairing and 3 x 3 = 9 ways to make the second, giving the following result:

1 64(ki + k2)6i(k[ + k'2)64(k3 + k'3) \ a r,,2 _ 1.2 _ „^1 r,,2 _ i./2 _ „-,i r„2 u2 r z,<uJJ

Z[0] w Ul J J 12(2tt)4 [h2 -k2- ie] [fi2 - k2 - ie] [p2 - k2 - ie iX\4mz\n fd^d% 1

Z{0] w 01 J J 8(27r)4 fi2 [n2 -k2- ie] [M2 - k'2 - ie]

the combined effect of the free propagator plus the second order contributions

Fig. 14.7 The elementary tadpole diagram

The new diagram, Fig. 14.6B, is another example of a "tadpole" diagram. The simplest example of such a diagram, and the one which originally suggested the name "tadpole," is the diagram shown in Fig. 14.7, which is proportional to

In general, a diagram is referred to as a tadpole diagram if it contains a factor of A(0), which corresponds to a loop which couples to an external particle in only one place. Such a loop arises from a factor of where both <p's come from the same interaction term in the Hamiltonian. In a theory in which the Hamiltonian is normal ordered, as in the operator formalism developed in Chapter 11, such terms cannot appear, but in the path integral formalism they arise naturally ; this is one difference between the two formalisms. In some sense, the path integral formalism does not permit normal ordering of the fields, which seems intuitive once one realizes that fields behave like c-numbers in the path integral formalism, and as such their order must be the same as multiplication by c-numbers.

We see that the operator formalism of Chapter 11 and the path integral formalism give different results for the self-energy of a neutral particle; the difference is the tadpole term shown in Fig. 14.6B. However, this lack of uniqueness has no physical consequences, because the tadpole 14.6B is a constant and will therefore be absorbed into the renormalization constants which are ultimately fixed by the physical charge and mass of the particles. The finite parts which remain after the renormalization is completed are identical.

Now, we return to the first term in iA(2i(p) given in Eq. (14.119) and previously ignored. This term corresponds to a disconnected diagram for the propagator; specifically, it gives the diagram shown in Fig. 14.8. This explains why it contributes only at zero momenta.

As we noted previously in Sec. 11.2, diagrams which are disconnected are a product of two (or more) independent lower order processes. In some cases these processes are unphysical and therefore do not contribute to the S-matrix. The

Fig. 14.8 The disconnected tadpole contribution to the propagator.

Fig. 14.8 The disconnected tadpole contribution to the propagator.

diagrams in Figs. 11.1 and 14.8 are examples of such processes. The disconnected tadpole, Fig. 14.8, when considered in isolation, corresponds to the simultaneous absorption and emission of a scalar particle by the vacuum, each of which is an unphysical process. In cases where each of the separate parts of a disconnected diagram are physical, they are more properly regarded as the simultaneous occurrence of more that one process, and not a proper contribution to a description of a single physical process. For all of these reasons, disconnected diagrams should be removed from the theory, and they can be removed systematically by working with the generating function W[J] instead of Z\J], where

We leave it as an exercise (Prob. 14.3) to show that the disconnected term in A'2'(p) is canceled when A^(p) is calculated from W, but that the connected term is unchanged.

14.6 FERMIONS

The extension of path integral techniques to fermions poses a special problem. An essential aspect to the description of Dirac fields is their anticommuting nature, and in the path integral formalism fields are c-numbers! What we need is a formalism or mathematics of anticommuting c-numbers. As it turns out, such a mathematics was developed by Grassmann in the latter half of the 19th century; the algebra of such anticommuting numbers is referred to as a Grassmann algebra.

The Grassmann numbers are constructed from real (or complex) numbers and generators. The generators are denoted Cit where i — 1 to N and N may be infinite. The generators satisfy

Note that Cf = 0. In applications, we are interested in functions of products of rj and 77, where

and V and tp = V^7o are Dirac c-number spinors and conjugate spinors (respectively) and Ci and Cj are independent Grassmann generators. If there is only one r) and one fj, then the most general function of fjrj is simply fiw) = ^2an (wn)n = a0 + aim

Differentiation will be defined to correspond to the removal of one power of a Grassmann variable, but the derivative anticommutes with other Grassmann variables, so we have

The rules for the integration of Grassmann numbers will be given shortly.

We now show how the anticommuting property of Grassmann numbers makes it possible to extend the path integral formalism to fermions. The first step is to find coherent states of the fermion annihilation operator, which we will denote by Bi (where i is the frequency of energy of the particle). This requires we find states with the following property:

At first it seems that it must be impossible to find a solution to this equation, because we know that fermion states can have at most one particle in any quantum state, and the coherent states (14.62) required a sum over states with an arbitrarily large number of particles in each quantum state. However, if the eigenvalue bi is assumed to commute with the state |b),

&i|&> = Ib)bi , then the anticommutation relations satisfied by the annihilation operators Bi imply that the eigenvalues bi must be complex Grassmann numbers and that the eigenvalues and annihilation operators also anticommute,

then

where the property 6f = 0 was used in the last step. These coherent states have a norm which is very similar to their Bose counterparts,

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