8
1 Introduction
mode index n:
N =
n∈N
nN n ,
N n =
1
n
α −n · α n ,
¯
N =
n∈N
n ¯
N n ,
¯
N n =
1
n
¯
α −n · ¯
α n .
(1.5)
With these elements, the Hilbert space of the string theory can be constructed.
Invariance under reparametrization leads to the on-shell condition, which says that
the Hamiltonian vanishes:
H |ψ = 0
(1.6)
for any physical state |ψ. Another constraint for the closed string is the levelmatching condition
(N − ¯
N) |ψ = 0 .
(1.7)
It can be understood as fixing an origin on the string.
The ground state |k with momentum k is defined to be the eigenstate of the
momentum operator which does not contain any oscillator excitation:
p
μ
|k = k
μ
|k ,
∀n > 0 : α
μ
n |k = 0 .
(1.8)
A general state can be built by applying successively creation operators
|ψ =
n>0
D−1
μ=0
(α
μ
−n )
N n,μ |k ,
(1.9)
where N n,μ ∈ N counts excitation level of the oscillator α
μ
−n . In the rest of this
section, we describe the first two levels of states.
The ground state is a tachyon (faster-than-light particle) because the Hamiltonian
constraint shows that it has a negative mass (in the units where α = 1):
closed : m
2
= −4 ,
open : m
2
= −1 .
(1.10)
The first excited state of the open string is found by applying α −1 on the vacuum
|k:
α
μ
−1 |k .
(1.11)