Pictures Paths Particles Processes

November 7, 2013 149
diagram 2 : u 1 Γ 4 u 2 v 3 Γ 5 v 6 u 5 Γ 6 u 4
diagram 3 : u 1 Γ 7 u 4 v 3 Γ 8 v 6 u 5 Γ 9 u 2 ,
Clearly, we have left out an enormous amount of detail here, and the Γ’s can be
anything. Note that we have written the three diagrams in such a way that the
conjugate spinors u 1 , v 3 and u 5 are in the same order in each diagram : this is
always possible. Now, we see that to go from diagram 1 to diagram 2, the positions
of u 4 and v 6 must be interhanged, whereas one can go from diagram 1 to
diagram 3 by, say, interchanging first u 2 and u 4 , and then u 2 and v 6 . Therefore,
diagram 1 and 3 have no relative minus sign, and diagram 2 has a minus sign
with respect to 1 and 3. In actual practice, the determination of the relative
signs can be made even easier ; simply decide on some preferred ordering of
all your u’s, v’s, u’s and v’s , and compare the ordering in your given diagram
with your preferred one. Note that, since spinor sandwiches always contain two
spinors, spinor sandwiches may be interchanged at will without destroying this
simple rule.
Before finishing this section we want to make an important observation.
The loop and interchange minus signs as we have discussed them depend on the
structure of the diagrams, and not on the type of the Dirac particles ; even if a
neutrino and a top quark were interchanged, the minus sign would crop up 38 .
The minus signs depend only on the fact that they are Dirac particles, that is,
spin-1/2 fermions. No notion of ‘identical particles’ is relevant here.
5.4.4 The Pauli principle
Let us consider a possible experiment in which we attempt to produce two Dirac
particles of the same type (two electrons, say), with exactly the same momentum
and spin. Any such process is, in principle, described by Feynman diagrams.
We can say immediately that the number of diagrams must be even, since for
every diagram there must be a corresponding one in which the two electons
are interchanged. Now, if the momenta and the spins of the two electrons
are precisely the same, they will be described by identical conjugate spinors,
and in fact the two diagrams of the pair will have exactly the same value —
apart from the relative minus sign ! The total amplitude is therefore identically
zero. We conclude that it is not possible two produce two Dirac particles in
exactly the same state. By considering incoming electrons, we can also conclude
that it is not possible to observe two Dirac particles if they are in exactly the
same state, since the observation process is also describable (presumaby !) by
Feynman diagrams. This is the Pauli exclusion principle 39 .
38 Of course, the interactions in the theory may be such that no such interchange is possible :
but this is beside the point.
39 Note that I do not comment on the possibility that electrons in identical states might simply
exist : they would not be observable by any process describable by Feynman diagrams.
Their only influence could arise through some non-diagrammatic process, involving possibly
gravity since that appears not to be amenable to diagrammatics. Of course, classical quan-