Kiefer C. Quantum gravity

GENERAL INTRODUCTION 281
The quantities α µ n are the Fourier components (oscillator coordinates) and obey
α µ −n =(α µ n) † due to the reality of the X µ . The solution (9.3) describes a standing
wave.
In view of the path-integral formulation, it is often convenient to continue the
worldsheet formally into the Euclidean regime, that is, to introduce worldsheet
coordinates σ 1 ≡ σ and σ 2 ≡ iτ. One can then use the complex coordinate
with respect to which the solution (9.3) reads
√
α
X µ (z, ¯z) =x µ − iα ′ p µ ′ ∑
ln(z¯z)+i
2
z =e σ2 −iσ 1 , (9.4)
n≠0
α µ n
n
(
z −n +¯z −n) . (9.5)
For a closed string the boundary condition X µ (σ) =X µ (σ +2π) is sufficient.
The solution of (9.2) can then be written as
X µ (σ, τ)=X µ R (σ− )+X µ L (σ+ ) , (9.6)
where we have introduced the lightcone coordinates σ + ≡ τ + σ and σ − ≡
τ − σ. The index R (L) corresponds to modes which would appear ‘rightmoving’
(‘leftmoving’) in a two-dimensional space–time diagram. Explicitly one has
√
X µ R (σ− )= xµ
2 + α′
α
2 pµ σ − ′
+i
2
∑
n≠0
α µ n
n e−inσ− (9.7)
and
√
X µ L (σ+ )= xµ
2 + α′
α
2 pµ σ + ′ ∑ ˜α µ n
+i
2 n e−inσ+ ,
n≠0
(9.8)
where ˜α µ n denotes the Fourier components of the leftmoving modes. One can also
give a formulation with respect to z and ¯z, but this will be omitted here. It is
convenient to define
√
α
˜α µ 0 = αµ 0 = ′
2 pµ
for the closed string, and
α µ 0 = √ 2α ′ p µ
for the open string.
In Section 3.2, we introduced the string Hamiltonian; see (3.59). Inserting
into this expression the classical solution for X µ ,oneobtains
for the open string, and
H = 1 2
∞∑
n=−∞
α −n α n (9.9)