JHEP11(2017)172
at each factorization channel. We find that the result perfectly agrees with the four-point
amplitude we computed in section 4 directly from string theory. Details of the four-point
dilaton amplitude are summarized in appendix C.
2 Quantization of closed string
The core of this paper is to challenge the conventional route to the quantized string theory.
So, we shall begin our considerations from the basics of string theory. In this section, we
redo the first-quantization of closed bosonic string, paying special attention to the choice
of worldsheet vacua for left-moving and right-moving sectors as well as center-of-mass
zero modes.
2.1 Conventional route: quantization over conventional vacuum
Our starting point is the Polyakov formulation of closed bosonic string theory [3], whose
worldsheet action is given by
S = −
1
4πα
0
Z
Σ
dτdσ
√
−hh
ab
∂
a
X
µ
∂
b
X
ν
η
µν
. (2.1)
Here, Σ is the Lorentzian worldsheet parametrized by (τ, σ). The worldsheet action is
a functional of the metric h
ab
and the scalars X
µ
. This action is invariant under the
worldsheet diffeomorphism. This local symmetry is fixed by imposing the conformal gauge
condition,
√
−hh
ab
= η
ab
= diag(−1, 1). The equations of motion for the worldsheet metric
is then reduced to the constraints
T
ab
= ∂
a
X
µ
∂
b
X
ν
−
1
2
η
ab
η
cd
∂
c
X
µ
∂
d
X
ν
= 0 . (2.2)
Later, we will separately treat the Faddeev-Popov ghosts and the BRST quantization. The
canonical momenta Π
µ
conjugate to X
µ
are given by
Π
µ
(τ, σ) =
1
2πα
0
∂
τ
X
µ
(τ, σ) . (2.3)
In the conformal gauge, the remaining fields on the worldsheet are the string coordi-
nates X
µ
(σ, τ). Their equations of motion in the conformal gauge read
X
µ
(τ, σ) =
∂
2
τ
− ∂
2
σ
X
µ
(τ, σ) = 0 (µ = 0, 1, ··· , D − 1). (2.4)
We impose the periodic boundary condition in the σ direction,
X
µ
(τ, σ) = X
µ
(τ, σ + 2π), Π
µ
(τ, σ) = Π
µ
(τ, σ + 2π) , (2.5)
and find the most general closed string solution as a sum of arbitrary left-moving and
right-moving profiles
X
µ
(τ, σ) = X
µ
L
(τ + σ) + X
µ
R
(τ − σ). (2.6)
– 7 –
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