Pictures Paths Particles Processes
Pictures Paths Particles Processes
Pictures Paths Particles Processes
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140 November 7, 2013<br />
now forced by the negativity of the energy to write<br />
−/p + m = − ∑ s<br />
v(p, s) v(p, s) . (5.79)<br />
The sign flip in the projection operator is of course precisely that which turns<br />
a particle description (with negative energy, moving backwards in time along<br />
the orientation of the propagator) into the antiparticle description, with positive<br />
energy. The truncation argument then tells us that v(p, s) must be the<br />
factor associated with the production, and v(p, s) must be associated with the<br />
annihilation, of the antiparticle. There remains the question of where to put<br />
the left-over Fermi minus sign. Consistently, we may decide to keep it with the<br />
v, in which case we arrive at the following Dirac Feynman rules :<br />
k<br />
↔ i¯h<br />
/k + m<br />
k · k − m 2 + iɛ<br />
internal lines<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
p,s ↔ √¯h u(p, s)<br />
outgoing particle<br />
p,s<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
↔ √¯h u(p, s)<br />
incoming particle<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
p,s<br />
↔ √¯h v(p, s)<br />
outgoing antiparticle<br />
p,s<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
00000 11111<br />
↔ − √¯h v(p, s)<br />
incoming antiparticle<br />
Feynman rules, version 5.2 (5.80)<br />
The awkward-looking minus sign is usually subjected to the argument that any<br />
matrix element containing an incoming antiparticle will have the factor −v in<br />
each of its diagrams, and since we are interested in absolute values squared<br />
anyway, there would appear to be little harm in deleting this overall minus sign<br />
from the Feynman rules : and this is what is commonly done. A little reflexion,<br />
though, will remind us that the sign of the amplitude’s real part is fixed by<br />
unitarity, and now we have changed it ! Clearly, the minus sign will be back to<br />
haunt us later on.<br />
5.3.5 The spin of Dirac particles<br />
We shall now determine the spin of Dirac particles. Although the fact that they<br />
have two orthonormal spin states strongly suggests that they have spin-1/2,
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