10.3 Path Integral Quantization
225
together with the gauge invariance
δ |B = Q B | 1 .
(10.78)
Note the difference with (10.70): while B =
B
↓ was the top component of the B
field, here, it is defined to be the down component, such that |B ↓ = |
B
↓ . However,
for the moment, we keep B to satisfy b 0 |B = 0.
Remark 10.4 (Decoupling of the Ghosts) Since the theory is free, the Faddeev–
Popov action (10.74) could be ignored and absorbed in the normalization because
it does not couple to the field. On the other hand, when interactions are included,
the gauge transformation is modified and the ghosts couple to the matter fields. But
this is true only for the C transformation (10.67a), not for (10.67b). Then it means
that ghosts introduced for gauge fixing (10.67b) will never couple to the matter and
other ghosts.
The invariance (10.67a) is a gauge invariance for C and must be treated in
the same way as (10.29). Then, following the Faddeev–Popov procedure, one is
lead to introduce new ghosts for the ghosts. But, the same structure appears again.
This leads to a residual gauge invariance, which has the same form. This process
continues recursively, and one finds an infinite tower of ghosts.
10.3.2 Tower of Ghosts
In order to simplify the notations, all the fields are denoted by n , where n gives
the ghost number:
• 1 := cl is the original physical field;
• 0 := C and, more generally, n with n < 1 are ghosts;
• 2 := B and, more generally, n with n > 1 are anti-ghosts.
The recipe is that each pair of ghost fields (( n+2 , , −n ) is associated to a gauge
parameter −n−1 with n ≥ 0. It is then natural to gather all the fields in a single
field
| =
n
| n
(10.79)
satisfying the gauge fixing constraint:
b 0 | = 0 ⇒ b 0 | n = 0.
(10.80)
For n ≤ 1, these constraints are gauge fixing conditions for the invariance δ | n =
Q B n . For n > 1, they arise by considering only the top component of the B field.
225
together with the gauge invariance
δ |B = Q B | 1 .
(10.78)
Note the difference with (10.70): while B =
B
↓ was the top component of the B
field, here, it is defined to be the down component, such that |B ↓ = |
B
↓ . However,
for the moment, we keep B to satisfy b 0 |B = 0.
Remark 10.4 (Decoupling of the Ghosts) Since the theory is free, the Faddeev–
Popov action (10.74) could be ignored and absorbed in the normalization because
it does not couple to the field. On the other hand, when interactions are included,
the gauge transformation is modified and the ghosts couple to the matter fields. But
this is true only for the C transformation (10.67a), not for (10.67b). Then it means
that ghosts introduced for gauge fixing (10.67b) will never couple to the matter and
other ghosts.
The invariance (10.67a) is a gauge invariance for C and must be treated in
the same way as (10.29). Then, following the Faddeev–Popov procedure, one is
lead to introduce new ghosts for the ghosts. But, the same structure appears again.
This leads to a residual gauge invariance, which has the same form. This process
continues recursively, and one finds an infinite tower of ghosts.
10.3.2 Tower of Ghosts
In order to simplify the notations, all the fields are denoted by n , where n gives
the ghost number:
• 1 := cl is the original physical field;
• 0 := C and, more generally, n with n < 1 are ghosts;
• 2 := B and, more generally, n with n > 1 are anti-ghosts.
The recipe is that each pair of ghost fields (( n+2 , , −n ) is associated to a gauge
parameter −n−1 with n ≥ 0. It is then natural to gather all the fields in a single
field
| =
n
| n
(10.79)
satisfying the gauge fixing constraint:
b 0 | = 0 ⇒ b 0 | n = 0.
(10.80)
For n ≤ 1, these constraints are gauge fixing conditions for the invariance δ | n =
Q B n . For n > 1, they arise by considering only the top component of the B field.