TAix1Ax2

<71<J293<74 " \/l6i(?3<f94 EqiEqi L6

where use was made of the fact that the two terms with x\ <-> x2 are identical and can be accounted for by multiplying by 2. Using the same pairings we worked out before gives us

2 1 diXXdiX2

(0|T [Ai(xx)A:'(x2)) |0> x [u(f_)1ivlP+j\ [s(*+bM*-)] • (10.72)

Fig. 10.4 Feynman diagram for the production process showing the flow of charge.

Rule 3: for fermions, assemble the incoming fermion spinors, vertex operators, and outgoing fermion spinors in order along each fermion line to make a well-formed matrix element. In particular:

• multiply from the left by u(p_,s~) for each outgoing fermion with momentum p_ and spin

• multiply from the right by u(A_,s_) for each incoming fermion with momentum k- and spin s_.

• multiply from the right by v(p+, s+) for each outgoing an-tifermion with momentum p+ and spin s+.

• multiply from the left by v(k+,s+) for each incoming antifermion with momentum k+ and spin s+.

Note the peculiar fact that v is associated with incoming ant ¿particles, yet must be on the left to make a Lorentz invariant matrix element. Similarly, v is associated with outgoing antiparticles but must be on the right. The labeling of antiparticles given in Fig. 10.4 helps suggest this ordering. On the electron side of the diagram, the direction of the momentum of the incoming positron is reversed, suggesting that the incoming positive charge is to be regarded as equivalent to an outgoing negative charge. The negative electron charge flows into the vertex along the electron line with momentum and "out" of the vertex along the positron line with momentum -k+. This flow of negative charge is the same as the ordering of the Dirac indices:

flow of negative charge.

Similarly, the order in which the matrix element is constructed follows the flow of negative muon charge. In this case the incoming negative charge is carried by the positive muon flowing backward and into the vertex with momentum —p+ and by the negative muon flowing out of the vertex with momentum p_ :

order of Dirac indices w(p_)(+*'e7/Xp+) •<=>■ <= (10.77)

flow of negative charge.

10.3 ANTIPARTICLES: e+e~ fi+ft~ 301

Warning: The momentum in the v or v spinors is the actual physical momentum of the antiparticle.

(iii) Sign ambiguity: If we choose the p+ to be the particle and the p~ to be the antiparticle, the current would be of the opposite sign, and we would have for the muon u{p+){-ie^)v{p_) .

Taking the transpose of this expression and using Eq. (5.38), from which the relations

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