INSTANTONS IN ABELIAN SYSTEMS
63
ordinary quantum mechanics with the action:
di
1
^ = 2
(4.56)
This action can describe a free particle on a line or a free particle on a
circle. The spectrum is continuous in the first case and is discrete in the
second. The difference arise because in the first case we integrate over
continuous x{t) while in the second we should allow 27r-jumps.
We have to decide on physical grounds what version of QED is
realized in Nature. The main reason why we believe in the periodic (or
compact) version of QED is based on the empirical fact of charge
quantization. We know that the ratio of any two electric charges is a
rational number. Let us show that this is a necessary consequence of
compact QED, while in the noncompact version it would be an
unexplained mystery.
The qualitative argument is that, as we saw in Chapter 3, the electric
flux (which is an analogue of angular momentum) is quantized. Since
charged particles are sources of electric flux, which according to Gauss’
theorem must be equal to their charges, we conclude that possible
charges are quantized as well.
To be more explicit, let us consider two charged fields,
with a unit
charge and Xx
charge e. The Lagrangian for these fields is of the
form:
= Z (iAi+6
+ xl+8
+ c.c.
X . 6
The form of (4.57) is dictated by gauge invariance:
(4.57)
(4.58)
From (4.57) we see that if ' is well defined in our phase space only if e is
integer, which is needed for periodicity in >1, g. Notice also that the
period of the free action (4.54) defines a natural unit of charge.
In noncompact QED the flux is continuous and there are no reasons
for charge quantization. Another important thing concerning compactness is that the two above-mentioned options are present only in the
abelian case. For any Non-Abelian group the fact of compactness or
noncompactness can be seen from its Lie algebra. For instance if the
gauge group is SU{2) we cannot formulate the noncompact version at
all. If we consider the noncompact group St/(1,1), we get gauge quanta
with negative norm. Therefore, when we consider QED as arising as a
63
ordinary quantum mechanics with the action:
di
1
^ = 2
(4.56)
This action can describe a free particle on a line or a free particle on a
circle. The spectrum is continuous in the first case and is discrete in the
second. The difference arise because in the first case we integrate over
continuous x{t) while in the second we should allow 27r-jumps.
We have to decide on physical grounds what version of QED is
realized in Nature. The main reason why we believe in the periodic (or
compact) version of QED is based on the empirical fact of charge
quantization. We know that the ratio of any two electric charges is a
rational number. Let us show that this is a necessary consequence of
compact QED, while in the noncompact version it would be an
unexplained mystery.
The qualitative argument is that, as we saw in Chapter 3, the electric
flux (which is an analogue of angular momentum) is quantized. Since
charged particles are sources of electric flux, which according to Gauss’
theorem must be equal to their charges, we conclude that possible
charges are quantized as well.
To be more explicit, let us consider two charged fields,
with a unit
charge and Xx
charge e. The Lagrangian for these fields is of the
form:
= Z (iAi+6
+ xl+8
+ c.c.
X . 6
The form of (4.57) is dictated by gauge invariance:
(4.57)
(4.58)
From (4.57) we see that if ' is well defined in our phase space only if e is
integer, which is needed for periodicity in >1, g. Notice also that the
period of the free action (4.54) defines a natural unit of charge.
In noncompact QED the flux is continuous and there are no reasons
for charge quantization. Another important thing concerning compactness is that the two above-mentioned options are present only in the
abelian case. For any Non-Abelian group the fact of compactness or
noncompactness can be seen from its Lie algebra. For instance if the
gauge group is SU{2) we cannot formulate the noncompact version at
all. If we consider the noncompact group St/(1,1), we get gauge quanta
with negative norm. Therefore, when we consider QED as arising as a