This is a beautifully simple, covariant formula for the configuration space propagator of a spin zero particle. It shows that the propagator depends only on Xi — X2, and hence justifies Eq. (9.27). It displays the propagator as the four dimensional Fourier transform of a simple function of the square of the virtual four-momentum of the particle. The integration is over all four components of k2, and k2 ^ ¡i2. We say that the four-momentum of the propagating particle is "off-mass-shell," or simply "off-shell." Inserting this result into Eq. (9.30) gives the following second order result for the .M-matrix which describes 1 + 2 —► 1+2 scattering in this simple theory:

-A1A2

This calculation of the .M-matrix was more lengthy that the calculation of the decay amplitude in Sec. 9.2 but was still reasonably easy. The calculation of more complicated processes can be very tedious if done in this way (as we shall ' ¿e!), and in the late 1940's Feynman discovered a very beautiful diagrammatic way to organize perturbative calculations so the results can be obtained much more easily. In the next section we begin our discussion of the famous Feynman rules.

One of our principal objectives as we continue to study interactions will be to extract the Feynman rules for the calculation of the .M-matrix. For the rest of this chapter and during the next two, we will calculate results directly from field theory the "hard" way, but as we do so, we will pause at the end and introduce new Feynman rules illustrated by the particular calculation just completed. In this way we will demonstrate explicitly how most of the Feynman rules arise in lowest order calculations and also develop an understanding of how results can be obtained directly from the field operators using perturbation theory. A summary of the Feynman rules we will introduce is included in Appendix B, and the reader is encouraged to refer to this frequently.

'Two of the original papers, [Fe 49], are reprinted in a nice introductory volume by Feynman <1961).

Rule 2: a factor of

H2 — k2 — it for each internal line of four-momentum k corresponding to the virtual propagation of a scalar particle.

• Four-momentum is to be conserved at each vertex, like current flowing through a circuit.

Note that the decay calculated in the last section also followed these rules (refer to Fig. 9.3).

The Feynman rules fall into two general categories. There are rules which are very general and apply to all theories and rules which are specific to a particular theory. Rules 0 and 2 above are examples of the first type, and Rule 1 is of the second. Each theory has its own characteristic interactions, and for each type of interaction there is a Rule 1 with a unique structure. Many of these are summarized in Appendix B. Rule 0 may seem frivolous; if the factor of i is omitted from every amplitude, it will be of no consequence for any prediction since all observables depend on the absolute square of an amplitude. However, omitting this factor is inconvenient from a theoretical point of view, since many amplitudes known to be real (or imaginary) will not have the expected property unless it is included. Finally, the propagator is a key building block in the construction of Feynman amplitudes, and the propagator given above (Rule 2) is the correct propagator to use in any Feynman amplitude which includes an internal scalar particle, and in this sense it is a general result. We will discuss some of the properties of the propagator now.

We will first rederive the explicit form for the propagator given in Eq. (9.37) using a different technique. This new derivation will give us experience with the treatment of matrix elements of field operators and insight into the physics which goes into its definition.

We begin by showing that the propagator iA(i) = (0|T ($(x)<0(0)) |0) [recall Eq. (9.31)] satisfies the following inhomogeneous KG equation:

To prove this, note that the field operators satisfy the KG equation (they are constructed from its plane wave solutions), so the only non-zero contributions must come from the time derivatives of the ^-functions associated with the time-ordered product. Recalling that

9.5 CALCULATION OF THE CROSS SECTION

Compare this to

Hence, for any colinear frame,

which demonstrates the covariance of EiE2v, and hence da.

Now evaluate da in the laboratory (LAB) frame, where m2 is at rest. The initial expression is da

The ¿-function is now far from trivial. Recalling that 6 is the scattering angle in the LAB system and using

where, by three-momentum conservation, p'2 = (*-*') = (k' sin 6)2 + (k — k' cos 6) = k'2 + k2 — 2kk' cos 8 ,

gives

) fixed

= (Ei + m2 — E[)k'+ E[(k' - kcosO) = (Ei +m2)k' - E[kcos0 . (9.60)

It turns out to be convenient to express this in terms of the square of the four-momentum transfer, which can be written in many different ways and will be useful in later applications:

q2 = (k' - k)2 = 2mj - 2EXE[ + 2kk' cos 6 = (p' - p)2 = 2ml - 2m2E2 = 2m2(m2 - E2) — 2m2(E'1 — Ei) . (9.61)

9.6 EFFECTIVE NONRELATIVISTIC POTENTIAL 257

Note that the recoil factor may now be significant. We will return to this formula later.

We can use Eqs. (9.29) and (9.54) to relate M to the nonrelativistic potential (in momentum space). Recall that the nonrelativistic ¿"-matrix [from Schiff (1968), for example] for non-forward scattering is

where T is the reduced nonrelativistic scattering amplitude (which plays a role similar to M) and the second equation expresses the result in the first Born approximation. The momentum transfer is q = (k — k'), and V(q) is the potential in momentum space

V(q) = J d3rV{r)e^ r . In terms of these quantities, the differential cross section is

where m is the reduced mass, rai2

77117712 m] + 7712

To extract the effective nonrelativistic potential from the relativistic theory, assume that V should be chosen to give the correct result for M and da/dil in the first Born approximation. Equating the relativistic formula (9.54) to the nonrelativistic formula (9.70) gives, in the nonrelativistic limit,

7711 + 7712

Ei + E2 J 4miTO2 nonrel 4mim2

where the sign, which cannot be determined from the cross sections (which involve the square of both quantities), is fixed by a comparison of the nonrelativistic scattering amplitude T, given in Eq. (9.68), and the relativistic scattering amplitude M, defined in Eq. (9.29). In both cases the factor relating these quantities to the S-matrix is a factor of —i (because of Rule 0), so they have the same sign.

Hence the effective coordinate space potential between the charged particles mi and TO2, which arises from the exchange of the neutral particle in the <j>3 theory investigated above, is obtained by Fourier transforming the result (9.38),

Imi77"l2 J (27T

4mim2 J (27t)3 ¡i2 + q2 where E\—E[ = 0 in the CM system. Recalling the Yukawa integral, Eq. (4.67),

we obtain

This gives finally

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