is therefore truly remarkable that the quartic vertices drop out of the very expressions
for the cubic vertices already present in the theory, when these are taken to occur on the
same time-slice. This is not only true for the Gauge quartics, but also for the fermionic
quartics of equation (2-10). As was briefly touched on before, this occurs because of the
fluctuations of the q-s just when two Aq insertions are made on the same time-slice.
Recall that the left-hand side of Eq. 2-20 was independent upon where on the time-slice
the insertion was made (independent upon 1). Two insertion at k and I give
Dq (qk k-1) ( q 1q_) e- S (q I+ ii I (2 21)
where notation from Eq. 2-20 has been borrowed. The second term is the "quantum-fluctuation"
term and is an addition to the simple concatenation of two cubics. Notice that M above
is the total width of the strips which have the two insertions. Consider a situation as in
Figure 2-3, with i., labelling the widths of the various strips. The double insertions at
il and i2 produce a quantum fluctuation of 1/(M1 + M4). Taking into account the _.M4
pre-factor of the two cubic vertices and the 1/(M1 + M4) of the intermediate propagator
gives the combination [_M4/(M1 + M4)2. Adding the contribution for where the arrow
of Lightcone time on the intermediate particle goes the opposite way, we have the total
expression:
MM4 + MOM. 1 (M1- M4)( .) (222)
(Mi + M4)2 2 (Mi + M4)2
which is precisely the momentum dependence in expression (31) in Thorn [16] for one of
the quartic interactions. This very much simplified example serves to show how the
quartic seems to just miraculously fall out from the algebra. The above argument goes
In the paper [16] the author derives two Lightcone quartic vertices for the pure
Yang-Mills theory, the above one which contains the "Coulomb" exchange and
contribution from the commutator squared, and another one slightly more complex. The
vertices defined there depend on the polarization of the incoming particles.