1.2 String Theory
7
− −−−−−−− →
Fig. 1.5 Trajectory x
μ
c (τ, σ ) of an open string in spacetime (worldsheet). It begins and ends at
the lines parametrized by x i (σ ) and x f (σ ). The worldsheet is topologically a rectangle and is
parametrized by (τ, σ ) ∈ [τ i , τ f ] × [0, ,]
where x μ is the centre-of-mass position of the string and p μ its momentum. 4
Canonical quantization leads to the usual commutator:
[x
μ , p
ν
] = iη
μν .
(1.2)
With respect to a point-particle for which only the first two terms are present, there
are an infinite number of oscillators α
μ
n and ¯
α
μ
n which satisfy canonical commutation
relations for creation n < 0 and annihilation operators n > 0
[α
μ
m , α
ν
n ] = m η
μν δ m+n,0 .
(1.3)
The non-zero modes are the Fourier modes of the excitations of the embedded string.
The case of the open string is simply obtained by setting ¯
α n = α n and p → 2p. The
Hamiltonian for the closed and open strings read respectively
H closed = −
m 2
2
+ N + ¯
N − 2 ,
(1.4a)
H open = −m
2
+ N − 1,
(1.4b)
where m 2 = −p μ p μ is the mass of the state (in Planck units), N and ¯
N (level
operators) count the numbers N n and ¯
N n of oscillators α n and ¯
α n weighted by their
4 In the introduction, we set α = 1.