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Chapter 5
Dirac particles
5.1 Pimp my propagator
5.1.1 Extension of the propagator and external lines
So far we have been studying particles that can carry only a limited amount of
information : such a particle is completely specified once we have determined
its identity and its momentum. In this chapter we shall start increasing the
number of properties that particles can carry, by examining how the Feynman
propagator can be modified. Since the pole structure of the propagator is closely
connected with the causality of the theory, and must be used to derive Newton’s
first law in the approximation of propagation over macroscopic distances, we will
not mess around with the denominator of the propagator. The generalizations
we shall propose therefore concern themselves with the numerator, and are of
the form
1
T (p)
i¯h
p 2 − m 2 → i¯h
+ iɛ p 2 − m 2 + iɛ , (5.1)
where T (p) is some object that informs us that the particle propagating is not
as simple as we have seen so far, but has additional properties. What those
properties are depends, of course, on the choice of T (p).
Now, one very important observation is in order here. The particle propagator
never occurs in isolation, but always between two vertices, where the
particle is ‘produced’ and where it is ‘absorbed’ 1 . This implies that, as long as
we have not committed ourselves to particular vertices, a change in the propagator
may be compensated to some extent by a change in the vertices. For
instance, suppose that T (p) is a simple number: then the predictions of the
theory will remain unchanged if we opt to multiply the vertices by T (p) −1/2 .
1 It may be realized that this statement holds true also in the case of external lines, if it is
kept in mind that these are defined in the square of the matrix element.
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