10.2 Open String Field Expansion, Parity and Ghost Number
219
But, note that this equation contains much less information than the original (10.12):
as |
↓ is truncated from (10.40), a part of the equations of motion is lost. The
missing equation can be found by setting |
= 0 in (10.24) and must be imposed
on top of the action:
Q B | = 0.
(10.42)
It is called out-of-Siegel gauge constraint and is equivalent to the Gauss constraint
in electromagnetism: the equations of motion for pure gauge states contain also the
physical fields, and thus, when one fixes a gauge, these relations are lost and must
be imposed on the side of the action. This procedure mimics what happens in the
old covariant theory, where the Virasoro constraints are imposed after choosing the
flat gauge (if contains no ghost on top of | ↓↓, and then
Q B = 0 implies L n = 0,
see Sect. 8.3.3). Moreover, the states that do not satisfy the condition b 0 = 0 do not
propagate: this restricts the external states to be considered in amplitudes.
Remark 10.1 Another way to derive (10.40) is to insert {b 0 , c 0 } = 1 in the action:
S =
1
2
|Q B {c 0 , b 0 } | =
1
2
|Q B b 0 c 0 |
=
1
2
|{b 0 , Q B }c 0 | −
1
2
|b 0 Q B c 0 |
=
1
2
|c 0 L 0 | .
The drawback of this computation is that it does not show directly how the
constraints (10.42) arise.
Remark 10.2 (Generalized Gauge Fixing) It is possible to generalize the Siegel
gauge, in the same way that the Feynman gauge generalizes the Lorentz gauge.
This has been studied in [1, 2].
In this section, we have motivated different properties and adopted some
normalizations. The simplest way to check that they are consistent is to derive the
action in terms of the spacetime fields and check that it has the expected properties
from standard QFT. This will be the topic of Sect. 10.4.
10.2 Open String Field Expansion, Parity and Ghost Number
A basis for the off-shell Hilbert space H is denoted by {φ r }, where the ghost
numbers and parity of the states are written as
n r := N gh (φ r ),
|φ r | = n r mod 2.
(10.43)
219
But, note that this equation contains much less information than the original (10.12):
as |
↓ is truncated from (10.40), a part of the equations of motion is lost. The
missing equation can be found by setting |
= 0 in (10.24) and must be imposed
on top of the action:
Q B | = 0.
(10.42)
It is called out-of-Siegel gauge constraint and is equivalent to the Gauss constraint
in electromagnetism: the equations of motion for pure gauge states contain also the
physical fields, and thus, when one fixes a gauge, these relations are lost and must
be imposed on the side of the action. This procedure mimics what happens in the
old covariant theory, where the Virasoro constraints are imposed after choosing the
flat gauge (if contains no ghost on top of | ↓↓, and then
Q B = 0 implies L n = 0,
see Sect. 8.3.3). Moreover, the states that do not satisfy the condition b 0 = 0 do not
propagate: this restricts the external states to be considered in amplitudes.
Remark 10.1 Another way to derive (10.40) is to insert {b 0 , c 0 } = 1 in the action:
S =
1
2
|Q B {c 0 , b 0 } | =
1
2
|Q B b 0 c 0 |
=
1
2
|{b 0 , Q B }c 0 | −
1
2
|b 0 Q B c 0 |
=
1
2
|c 0 L 0 | .
The drawback of this computation is that it does not show directly how the
constraints (10.42) arise.
Remark 10.2 (Generalized Gauge Fixing) It is possible to generalize the Siegel
gauge, in the same way that the Feynman gauge generalizes the Lorentz gauge.
This has been studied in [1, 2].
In this section, we have motivated different properties and adopted some
normalizations. The simplest way to check that they are consistent is to derive the
action in terms of the spacetime fields and check that it has the expected properties
from standard QFT. This will be the topic of Sect. 10.4.
10.2 Open String Field Expansion, Parity and Ghost Number
A basis for the off-shell Hilbert space H is denoted by {φ r }, where the ghost
numbers and parity of the states are written as
n r := N gh (φ r ),
|φ r | = n r mod 2.
(10.43)