220
10 Free BRST String Field Theory
The corresponding basis of dual (or conjugate) states {φ c
r } is defined by (6.145):
φ
c
r |φ s = δ rs .
(10.44)
The basis states can be decomposed according to the ghost zero-modes
|φ r = |φ ↓,r + |φ ↑,r ,
b 0 |φ ↓,r = c 0 |φ ↑,r = 0.
(10.45)
Finally, each state ψ ↑ ∈ c 0 H can be associated to a state
ψ:
|ψ ↑ = c 0 |
ψ ↓ ,
b 0 |
ψ ↓ = 0,
N gh (ψ ↑ ) = N gh
ψ ↓
+ 1.
(10.46)
More details can be found in Sect. 11.2.
Any field can be expanded as
| =
r
ψ r |φ r ,
(10.47)
where the ψ r are spacetime fields (remembering that r denotes collectively the
continuous and discrete quantum numbers). 1
Obviously, the coefficients do not carry a ghost number since they are not
worldsheet operators. However, they can be Grassmann even or odd such that each
term of the sum has the same parity, so that the field has a definite parity:
∀r :
|| = |ψ r | |φ r |.
(10.48)
If the field is Grassmann odd (resp. even), then the coefficients ψ r and the basis
states must have opposite (resp. identical) parities, such that || = 1.
Since the parity results from worldsheet ghosts and there would be Grassmann
odd states even in a purely bosonic theory, it suggests that the parity of the
coefficients ψ r is also related to a spacetime ghost number G defined as
G(ψ r ) = 1 − n r .
(10.49)
The normalization is chosen such that the component of a classical string field
(N gh = 1) is classical spacetime fields with G = 0 (no ghost). We will see later
that this definition makes sense.
1 The notation is slightly ambiguous: from (10.45), it looks like both components of φ r have the
same coefficient ψ r . But, in fact, one sums over all linearly independent states: in terms of the
components of φ r , different basis can be considered; for example, {φ ↓,r , φ ↑,r }, or {φ ↓,r ± φ ↑,r }. A
more precise expression can be found in (10.54) and (10.56).
10 Free BRST String Field Theory
The corresponding basis of dual (or conjugate) states {φ c
r } is defined by (6.145):
φ
c
r |φ s = δ rs .
(10.44)
The basis states can be decomposed according to the ghost zero-modes
|φ r = |φ ↓,r + |φ ↑,r ,
b 0 |φ ↓,r = c 0 |φ ↑,r = 0.
(10.45)
Finally, each state ψ ↑ ∈ c 0 H can be associated to a state
ψ:
|ψ ↑ = c 0 |
ψ ↓ ,
b 0 |
ψ ↓ = 0,
N gh (ψ ↑ ) = N gh
ψ ↓
+ 1.
(10.46)
More details can be found in Sect. 11.2.
Any field can be expanded as
| =
r
ψ r |φ r ,
(10.47)
where the ψ r are spacetime fields (remembering that r denotes collectively the
continuous and discrete quantum numbers). 1
Obviously, the coefficients do not carry a ghost number since they are not
worldsheet operators. However, they can be Grassmann even or odd such that each
term of the sum has the same parity, so that the field has a definite parity:
∀r :
|| = |ψ r | |φ r |.
(10.48)
If the field is Grassmann odd (resp. even), then the coefficients ψ r and the basis
states must have opposite (resp. identical) parities, such that || = 1.
Since the parity results from worldsheet ghosts and there would be Grassmann
odd states even in a purely bosonic theory, it suggests that the parity of the
coefficients ψ r is also related to a spacetime ghost number G defined as
G(ψ r ) = 1 − n r .
(10.49)
The normalization is chosen such that the component of a classical string field
(N gh = 1) is classical spacetime fields with G = 0 (no ghost). We will see later
that this definition makes sense.
1 The notation is slightly ambiguous: from (10.45), it looks like both components of φ r have the
same coefficient ψ r . But, in fact, one sums over all linearly independent states: in terms of the
components of φ r , different basis can be considered; for example, {φ ↓,r , φ ↑,r }, or {φ ↓,r ± φ ↑,r }. A
more precise expression can be found in (10.54) and (10.56).