10.1 Classical Action for the Open String
213
It is straightforward to write the action in terms of the ket:
S =
1
2
φ|K |φ .
(10.9)
There is one hidden assumption in the previous lines: the definition of a scalar
product. A natural inner product is provided in the usual quantum mechanics by
associating a bra to a ket. Similarly, integration provides another definition of the
inner product when working with functions. We will find that the definition of
the inner product requires more care in closed SFT. To summarize, and to write
the kinetic term of the action, one needs the linearized equation of motion and an
appropriate inner product on the space of states.
10.1.2 Open String Action
The worldsheet equation that yields precisely all the string physical states |ψ is the
BRST condition:
Q B |ψ = 0.
(10.10)
Considering the open string field to be a linear combination of all possible onestring states |ψ
∈ H,
(10.11)
the equation of motion is
Q B | = 0.
(10.12)
Moving away from the physical state condition, the string field is off-shell and is
expanded on a general basis {φ r } of H. This presents a first difficulty because the
worldsheet approach—and the description of amplitudes—looks ill-defined for offshell states: extending the usual formalism will be the topic of Chap. 11. However,
this is not necessary for the free theory, and we can directly proceed.
Next, we need to find an inner product ·, ·· on the Hilbert space H. A natural
candidate is the BPZ inner product since it is not degenerate
A, B := =A|B ,
(10.13)
where A| = |A t is the BPZ conjugate (6.98) of |A, using I − . This leads to the
action:
S =
1
2
Q B =
1
2
|Q B | .
(10.14)
213
It is straightforward to write the action in terms of the ket:
S =
1
2
φ|K |φ .
(10.9)
There is one hidden assumption in the previous lines: the definition of a scalar
product. A natural inner product is provided in the usual quantum mechanics by
associating a bra to a ket. Similarly, integration provides another definition of the
inner product when working with functions. We will find that the definition of
the inner product requires more care in closed SFT. To summarize, and to write
the kinetic term of the action, one needs the linearized equation of motion and an
appropriate inner product on the space of states.
10.1.2 Open String Action
The worldsheet equation that yields precisely all the string physical states |ψ is the
BRST condition:
Q B |ψ = 0.
(10.10)
Considering the open string field to be a linear combination of all possible onestring states |ψ
∈ H,
(10.11)
the equation of motion is
Q B | = 0.
(10.12)
Moving away from the physical state condition, the string field is off-shell and is
expanded on a general basis {φ r } of H. This presents a first difficulty because the
worldsheet approach—and the description of amplitudes—looks ill-defined for offshell states: extending the usual formalism will be the topic of Chap. 11. However,
this is not necessary for the free theory, and we can directly proceed.
Next, we need to find an inner product ·, ·· on the Hilbert space H. A natural
candidate is the BPZ inner product since it is not degenerate
A, B := =A|B ,
(10.13)
where A| = |A t is the BPZ conjugate (6.98) of |A, using I − . This leads to the
action:
S =
1
2
Q B =
1
2
|Q B | .
(10.14)