M j iea0

in precise agreement with Eq. (11.21b).

488 PATH INTEGRALS Fermion Loops

We close this discussion by considering fermion loops in a theory with a scalar meson coupled to a fermion. This is similar to the <f>3 case previously discussed, except that now there are two different kinds of particles. The interaction term is

Replacing ^ilxj) by derivatives, using Eq. (14.156), gives

The sign in the exponent depends on the order in which the t) and fj derivatives are written; for the order given in the above expression the sign is the same as for the generating function Z\J] of <f>3 theory.

To illustrate the difference between closed fermion and boson loops and obtain the factor of —1 for closed fermion loops found previously in Sec. 11.5, we calculate the self-energy of a scalar particle which arises from a closed fermion loop, similar to that drawn in Fig. 14.6A (the drawing for the two cases is identical, but the meaning of the lines and vertices is different). The calculation is sufficiently different from that given in Sec. 14.5 that a new calculation is necessary. In order to track the difference between a charged scalar loop and a fermion loop, we will keep track of sign changes which arise from the interchange of the Jrassmann numbers by multiplying by a factor of f = -1 for each interchange. Then, after the calculation is complete, we can recover the result for a charged scalar loop by changing £ —► 1 (and making other changes which we will discuss below).

Drawing on our experience with the previous calculation, the second order contribution to the scalar particle self-energy comes from the term

where the momenta of the tp fields are denoted by pi and p2, the momenta of the ip fields are denoted by p[ and p'2, and momenta of both scalar fields are outgoing, so that their source terms are j(k). Now, since the source terms occur only in the combinations fj(p)r,{p) and J(k)J(-k), the only non-zero, connected terms come from the pairing of p2 with p'v pi with p'2, and kx and k2 with the external momenta k. Hence the terms we need are z(2)

J dVidViffci d*p'2 d4k2

x Si(p'2 - pi)64(p2 — p\) {iSa0(pi)iS0a(p2)}Zo[J,f},r]} - ■ (■««)

where the extra factor of £ = -1 arose in the second step when we needed to pass the S/6ria derivative through f?7 to act on r)6. [Two minus signs (£2) also arose in the first step, but these do not change the overall sign.] The self-energy E introduced in the last step is i

¿4(P2 -Pl-k) {iSaß(pi)iSßa(P2)} tr {(m+ fi)(m+ i> + ft)}

Note that the final form of Z given in Eq. (14.161) has the same structure as Eq. (14.116), permitting us to identify iE(fc) as the loop contribution to the self-energy of the scalar particle, which we are seeking. Equation (14.162) exhibits features of fermion loops which were previously encountered in Sec. 11.5:

• a trace must be taken over the product of Dirac propagators.

• there is an additional factor of -1 which arises from the closed fermion loop.

The result for a loop involving charged scalar particles is easily obtained from (14.162). The calculation is the same except f = 1, and scalar propagators must be substituted for Dirac propagators. We obtain the result

1W -*7& -»»-fcH^-fr+ »)■-!«) ' <14163)

Note that there is no symmetry factor of | if the scalar particles are charged.

This concludes our introductory discussion of the use of path integrals to quantize field theories. Note that the formalism allows us to use c-number fields (but Dirac fields must be described by anticommuting Grassmann numbers) and that the results are equivalent to the operator formalism we presented in Chapters 9-11. In the next chapter we use the techniques developed here to quantize QED and QCD, and we will discuss the Standard Model.

PROBLEMS

14.1 Obtain the path integral given in Eq. (14.81) from the path integral Eq. (14.77) by transforming from the coordinates {qi,Pi} to the coordinates {<Aq,7tq} using the transformations given in Eq. (14.78). This can be accomplished by working through the following steps:

(a) From the relations prove the completeness and orthogonality relations given in Eq. (14.80).

(b) Prove that the volume integration is invariant by showing that, for each time "slice,"

where M2 = 1 and can be ignored.

(c) The first term in the exponential in Eq. (14.77) may be symmetrized by integrating half of the expression by parts,

14.2 Using the Feynman rules worked out in Chapter 11 or given in Appendix B, obtain the correct integrals corresponding to the two diagrams in Fig. 14.5. (Don't overlook the symmetry factors). Can the same results be obtained from Eq. (14.113)? (It is an interesting exercise to use the Feynman rules to construct these diagrams, but do not forget that all such diagrams can be neglected because they cancel when the generating function is renormalized.)

(b) Prove that the volume integration is invariant by showing that, for each time "slice,"

Show that

Show that

14.3 Using the generating function W[J), defined in Eq. (14.125), calculate the propagator for a neutral scalar meson to second order in the coupling constant A, and show that it contains no disconnected pieces. Compare your answer with the result given in Eq. (14.122).

14.4 From the eigenvalue condition (14.130) and the anticommutation relations {Bi,Bj} = Sij, prove that the eigenvalues of the coherent states, must be Grassmann numbers and that the Grassmann numbers also anticommute with the annihilation operators.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 15

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