244
11 Introduction to Off-Shell String Theory
A dual state φ c
r is also expanded:
φ
c
r = φ
c
↓↓,r + φ
c
↓↑,r + φ
c
↑↓,r + φ
c
↑↑,r ,
(11.45)
and the components satisfy
φ
c
↓↓,r | c 0 = =φ
c
↓↓,r | ¯
c 0 = 0,
φ
c
↓↑,r | c 0 = =φ
c
↓↑,r | ¯
b 0 = 0,
φ
c
↑↓,r | b 0 = =φ
c
↑↓,r | ¯
c 0 = 0,
φ
c
↑↑,r | b 0 = =φ
c
↑↑,r | ¯
b 0 = 0.
(11.46)
The indexing of the basis is chosen such that
φ
c
↓↑,r | ==φ
c
↓↓,r | ¯
b 0
φ
c
↑↓,r | ==φ
c
↓↓,r | b 0
φ
c
↑↑,r | ==φ
c
↓↓,r | ¯
b 0 b 0 ,
(11.47)
such that
φ
c
x,r |φ y,s = δ xy δ rs ,
(11.48)
where x, y =↓↓, ↑↓, ↓↑, ↑↑.
Consider the Hilbert space H − ; then, a basis state must satisfy
b
−
0 |φ r = 0 ⇒ b 0 |φ r = ¯
b 0 |φ r .
(11.49)
The expansion (11.42) gives the relation
φ ↑↓,r + φ ↑↑,r = φ ↓↓,r + φ ↓↑,r
(11.50)
such that
φ r = 2(φ ↓↓,r + φ ↓↑,r ).
(11.51)
For convenience, the factor of 2 can be omitted (which amounts to rescaling the
basis states).
11.3 Off-Shell Amplitudes
In this section, we provide a guideline of what we need to look for in order to write
an off-shell amplitude. The geometrical tools will be described in the next chapter
and the construction of off-shell amplitudes in the following one.