The famous Yukawa potential therefore arises from the exchange of a particle in rield theory and is a natural consequence of the simplest interaction.

There are three general features of this result which are of considerable significance and should be noted for future reference:

• V{r) is attractive. This is a general feature of scalar particle exchange.

• The strength of V(r) depends on the coupling constants AjA2, which in <p3 theory have the dimensions of energy (mass).

• The range of V(r) depends on the mass of the exchanged particle, fi.

The extraction of effective interactions from field theory is a major industry, and we will soon see how the famous one-pion exchange (OPE) potential is derived. The study of effective interactions is continued in Chapter 12.

In this section we discuss some of the additional complications which arise when identical particles are present in the initial or final state. The cj>3 theory introduced in the first section will be used to illustrate the discussion, but all of the results will be quite general.

For identical particles we must be careful how the states are normalized. For a two-particle state, we have

where M is a normalization constant to be fixed shortly, and we will adopt the convention that the order of the variables in the "bra" is the same as the order of the operators, which matters only for fermions [recall Eq. (7.28)]. For "kets" we will use the same rule, so that

In either case,

+ for bosons

- for fermions.

The normalization constant M is fixed by the completeness requirement

PlP2

which implies

PlP2

Put with identical particles

Substituting (9.82) into the completeness relation (9.81) and doing the sums over Pi and p2 give

2A/"4 [5fcifc',5fc2fci ± ¿fcifc^fcafc'J = AT2 [^k'/kjfcJ ± ¿idfc^fcjfc'J , which leads to the requirement

Now the factor l/\/2 is inconvenient and can be omitted if we use the convention, for identical particles only, that pi > p2, or in the center of mass system where both momenta always have precisely the same magnitude, will be the momentum which has a polar angle 6\ < ir/2. In this case the second Kronecker delta ô^k' Sk^ = 0' and the above derivation would give N = 1. This is the convention we will adopt. This convention can be extended to more that one identical particle, in which case the momenta are restricted by pi > P2 >■■■ pn, or 81 < 02 • • with this restriction the n-particle normalization also equals unity.

This convention requires care when integrating over final states in the calculation of cross sections or decay rates. If we choose to integrate over the full solid angle 47r then, for a two-body final state, we are counting both p\ > P2 and P2 >pi, so we must divide by 2. [Note that this factor would arise naturally if we had used AT = l/\/2 for final states (which is appropriate if we place no restriction on the final momenta), giving a factor of N2 = | in cross sections and decay rates.] For an n-body final state, we can again integrate over all final momenta if we divide by a factor of n!. These factors are called statistical factors. In summary, our convention for the treatment of identical particles involves two new rules:

• for identical particles take hi = 1 and order the momenta

• when integrating over final states, ignore the ordering and divide by the statistical factor n\.

With these rules in mind, we will now calculate the elastic scattering of two particles of type-1 (i.e., 1 + 1—>1 + 1) to lowest order in 4>3 theory. Our beginning .s very similar to Eq. (9.23) for the ¿»-matrix,

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