TOPOLOGY OF GAUGE FIELDS AND RELATED PROBLEMS
109
have to consider all possible terms in the lagrangian, having dimensions
four or less with coefficients having their natural magnitude. If so, we do
have a strong T-violation problem. An attempt to resolve this problem
led to an interesting suggestion [4]. It can be shown that if massless
fermions with broken chiral symmetry are present in the system, then
due to instanton effects Goldstone’s massless particles obtain some
mass (because of nonexact conservation of
and simultaneously the
0-term gets absorbed after redefinition of Goldstone’s field. This
consideration predicted a light isoscalar boson (which would have been
massless without instantons). Unfortuantely this particle, called the
axion, has not been found, and the strong 0-problem remains open.
Let us mention briefly some other interesting effects of instantons
together with the 0-term. It appears that their presence induces a small
electric charge
for a magnetic monopole, and in this case the
operator of electric charge differs from a gauge group generator by a
small constant.
Another interesting thing is a probably rich phase structure of the
theory as a function of 9. In the case of /i-fields, periodic 0-dependence
seems to have important consequences, explaining the quantized Hall
effect in metals.
Let us stress that 0-dependence of physical quantities is not to be
taken for granted from what we have said about it. It is quite possible
that due to dynamical effects, the contributions with q ^ 0 may become
absent in the infinite volume limit. Example of such a phenomenon is
observed in the case of plasma where only states with total charge zero
contribute in the thermodynamic limit (because of the enormous
Coulomb energy of other states). However, this option seems unlikely
in Non-Abelian gauge theory.
Another word of caution concerns the apparent periodicity in 0. It
might well happen that because of dissociation of instantons, states
with noninteger topological charge will give finite contibutions to the
partition function. These configurations certainly have infinite action,
but this can be compensated by the entropy. As a result, the
formal decomposition (6.30) will be untrue and we shall now have
a periodic 0-dependence.
All these questions are purely dynamical and their solution requires
some new concepts. In the next chapters we shall discuss what has been
done in this direction.
109
have to consider all possible terms in the lagrangian, having dimensions
four or less with coefficients having their natural magnitude. If so, we do
have a strong T-violation problem. An attempt to resolve this problem
led to an interesting suggestion [4]. It can be shown that if massless
fermions with broken chiral symmetry are present in the system, then
due to instanton effects Goldstone’s massless particles obtain some
mass (because of nonexact conservation of
and simultaneously the
0-term gets absorbed after redefinition of Goldstone’s field. This
consideration predicted a light isoscalar boson (which would have been
massless without instantons). Unfortuantely this particle, called the
axion, has not been found, and the strong 0-problem remains open.
Let us mention briefly some other interesting effects of instantons
together with the 0-term. It appears that their presence induces a small
electric charge
for a magnetic monopole, and in this case the
operator of electric charge differs from a gauge group generator by a
small constant.
Another interesting thing is a probably rich phase structure of the
theory as a function of 9. In the case of /i-fields, periodic 0-dependence
seems to have important consequences, explaining the quantized Hall
effect in metals.
Let us stress that 0-dependence of physical quantities is not to be
taken for granted from what we have said about it. It is quite possible
that due to dynamical effects, the contributions with q ^ 0 may become
absent in the infinite volume limit. Example of such a phenomenon is
observed in the case of plasma where only states with total charge zero
contribute in the thermodynamic limit (because of the enormous
Coulomb energy of other states). However, this option seems unlikely
in Non-Abelian gauge theory.
Another word of caution concerns the apparent periodicity in 0. It
might well happen that because of dissociation of instantons, states
with noninteger topological charge will give finite contibutions to the
partition function. These configurations certainly have infinite action,
but this can be compensated by the entropy. As a result, the
formal decomposition (6.30) will be untrue and we shall now have
a periodic 0-dependence.
All these questions are purely dynamical and their solution requires
some new concepts. In the next chapters we shall discuss what has been
done in this direction.