String Theory and M-Theory

2.3 String sigma-model action: the classical theory 35
The terms linear in σ cancel from the sum X µ
R + Xµ
L , so that closed-string
boundary conditions are indeed satisfied. Note that the derivatives of the
expansions take the form
where
∂−X µ
R
∂+X µ
L
= ls
= ls
+∞
m=−∞
+∞
m=−∞
α µ me −2im(τ−σ)
(2.44)
α µ me −2im(τ+σ) , (2.45)
α µ 1
0 = αµ 0 =
2 lsp µ . (2.46)
These expressions are useful later. In order to quantize the theory, let us
first introduce the canonical momentum conjugate to X µ . It is given by
P µ (σ, τ) = δS
δ ˙ Xµ
= T ˙ X µ . (2.47)
With this definition of the canonical momentum, the classical Poisson brackets
are
P µ (σ, τ), P ν (σ ′
, τ)
P.B. =
X µ (σ, τ), X ν (σ ′
, τ) = 0,
P.B.
(2.48)
P µ (σ, τ), X ν (σ ′
, τ)
P.B. = ηµνδ(σ − σ ′ ). (2.49)
In terms of X ˙ µ
˙X µ ν ′
(σ, τ), X (σ , τ)
P.B. = T −1 η µν δ(σ − σ ′ ). (2.50)
Inserting the mode expansion for X µ and ˙ X µ into these equations gives the
Poisson brackets satisfied by the modes3
α µ m, α ν
n
P.B. =
α µ m, α ν
n
P.B. = imηµνδm+n,0 and
(2.51)
µ
α m, α ν
n = 0. (2.52)
P.B.
3 The derivation of the commutation relations for the modes uses the Fourier expansion of the
Dirac delta function
δ(σ − σ ′ ) = 1
π
+∞ X
n=−∞
e 2in(σ−σ′ ) .