tion of left- and right-handed particles separately. Whether this leads to
a finite effect after removing the cut-off is another matter. It depends on
the compensation between the small values of amplitudes involving
p ^ Pf particles and the large number of particles in the Dirac sea.
Our computation which lead to (6.71) revealed that the compensation was exact in the ^ = 4 case. Although this computation was
performed for a specific regularization of the theory, its final and finite
result (6.71) can be shown to be regularization-independent. In more
than one dimension, fermions need spin in order to be relativistic. In
this case co = < tK. Spinlesss fermions cannot be relativistic, from spin
statistics theory.
We recognize in the right hand side of (6.71) the density of topological charge. As we have shown in Section 6.2 this can be written as the
divergence of some current JT^(x). One might have thought, therefore,
that the anomaly equation (6.71) does not break the conservation of
axial charge, but rather redefines this current: J -► — 47r^JT^.
This is not the case. If we consider a field with topological charge q
and integrate equation (6.71) over the 4-volume, we obtain:
TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
103
A25 =
d^x = 2q
(6.74)
Here:
Jo5 d^x =
and we denote by AQ^ the total change of Q5 for infinite time under the
influence of A^. Equation (6.74) shows that it is equal to twice the
topological charge of A^.
This amazing result means that there exists “compulsory” production of fermions and antifermions in topological nontrivial fields and
that the numbers of left- and right-handed particles
and
necessarily change.
We derived all this using the Euclidean formalism, but there is no
difficulty at all in understanding this effect in Minkowski space. To do
this, let us recall the Minkowskian interpretation of instantons. As we
have seen in Chapter 4, the instanton represents a tunnelling transition
between two states divided by a barrier (recall the double-well example). In the real time formulation we have very many different trajectories connecting these two states. However, since the classical action does
not have a corresponding extremum (since the transition is forbidden
a finite effect after removing the cut-off is another matter. It depends on
the compensation between the small values of amplitudes involving
p ^ Pf particles and the large number of particles in the Dirac sea.
Our computation which lead to (6.71) revealed that the compensation was exact in the ^ = 4 case. Although this computation was
performed for a specific regularization of the theory, its final and finite
result (6.71) can be shown to be regularization-independent. In more
than one dimension, fermions need spin in order to be relativistic. In
this case co = < tK. Spinlesss fermions cannot be relativistic, from spin
statistics theory.
We recognize in the right hand side of (6.71) the density of topological charge. As we have shown in Section 6.2 this can be written as the
divergence of some current JT^(x). One might have thought, therefore,
that the anomaly equation (6.71) does not break the conservation of
axial charge, but rather redefines this current: J -► — 47r^JT^.
This is not the case. If we consider a field with topological charge q
and integrate equation (6.71) over the 4-volume, we obtain:
TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
103
A25 =
d^x = 2q
(6.74)
Here:
Jo5 d^x =
and we denote by AQ^ the total change of Q5 for infinite time under the
influence of A^. Equation (6.74) shows that it is equal to twice the
topological charge of A^.
This amazing result means that there exists “compulsory” production of fermions and antifermions in topological nontrivial fields and
that the numbers of left- and right-handed particles
and
necessarily change.
We derived all this using the Euclidean formalism, but there is no
difficulty at all in understanding this effect in Minkowski space. To do
this, let us recall the Minkowskian interpretation of instantons. As we
have seen in Chapter 4, the instanton represents a tunnelling transition
between two states divided by a barrier (recall the double-well example). In the real time formulation we have very many different trajectories connecting these two states. However, since the classical action does
not have a corresponding extremum (since the transition is forbidden