108
GAUGE FIELDS AND STRINGS
for the Yang-Mills theory, and the analogous Lagrangian for the /i-field
(6.89)
1
, ie
where 9 in both cases is a new coupling constant. Due to the presence of
instantons, physical transition amplitudes will depend on 6. For
instance, the vacuum to vacuum amplitude will be
Z= X
(6.90)
where by
we denoted the functional integral over the fields with a
fixed value of q.
The extra terms in the above expressions are purely imaginary in
Euclidean space for the following reasons. We have to have a real action
in Minkowski space. As we change i -► — ii we have to change A„ -► A„
and Aq-^iAq, because Aq transforms as d/di. Hence electric and
magnetic fields change as:
The two terms in (6.88) change as:
Tr Fl, = E^
-h E^)
TrF^/F^,= -2H E-^-2iH E
(6.91)
(6.92)
So, in the Euclidean action (6.88) we obtain i in the second term when
we transform the real Minkowskian action.
Topological “0-terms” in the action create a problem when we apply
the theory to describe strong interactions. If they really do contribute to
the physical amplitudes, then the whole theory loses invariance under
time reversal (since
E is F-odd). This implies that for some reasons
the coupling 0 must be zero or extremely small.
Certainly the existence of this problem somewhat depends on one’s
personal philosophy. One can take the view that in the cut-off* region,
i.e. at the Planck length, we have a gauge Lagrangian which preserves
T-invariance. Then 0 = 0 from the very beginning and no problem of
strong T-noninvariance arises. It is interesting, however, to consider
another point of view, according to which there are no special symmetries at the Planck length and they persist in the low energy region only
for dynamical reasons, namely because only renormalizable interactions contribute significantly for large scales. Accepting this view, we
GAUGE FIELDS AND STRINGS
for the Yang-Mills theory, and the analogous Lagrangian for the /i-field
(6.89)
1
, ie
where 9 in both cases is a new coupling constant. Due to the presence of
instantons, physical transition amplitudes will depend on 6. For
instance, the vacuum to vacuum amplitude will be
Z= X
(6.90)
where by
we denoted the functional integral over the fields with a
fixed value of q.
The extra terms in the above expressions are purely imaginary in
Euclidean space for the following reasons. We have to have a real action
in Minkowski space. As we change i -► — ii we have to change A„ -► A„
and Aq-^iAq, because Aq transforms as d/di. Hence electric and
magnetic fields change as:
The two terms in (6.88) change as:
Tr Fl, = E^
-h E^)
TrF^/F^,= -2H E-^-2iH E
(6.91)
(6.92)
So, in the Euclidean action (6.88) we obtain i in the second term when
we transform the real Minkowskian action.
Topological “0-terms” in the action create a problem when we apply
the theory to describe strong interactions. If they really do contribute to
the physical amplitudes, then the whole theory loses invariance under
time reversal (since
E is F-odd). This implies that for some reasons
the coupling 0 must be zero or extremely small.
Certainly the existence of this problem somewhat depends on one’s
personal philosophy. One can take the view that in the cut-off* region,
i.e. at the Planck length, we have a gauge Lagrangian which preserves
T-invariance. Then 0 = 0 from the very beginning and no problem of
strong T-noninvariance arises. It is interesting, however, to consider
another point of view, according to which there are no special symmetries at the Planck length and they persist in the low energy region only
for dynamical reasons, namely because only renormalizable interactions contribute significantly for large scales. Accepting this view, we