String Theory and M-Theory

48 The bosonic string
2.5 Light-cone gauge quantization
As discussed earlier, the bosonic string has residual diffeomorphism symmetries,
even after choosing the gauge hαβ = ηαβ, which consist of all the conformal
transformations. Therefore, there is still the possibility of making an
additional gauge choice. By making a particular noncovariant gauge choice,
it is possible to describe a Fock space that is manifestly free of negative-norm
states and to solve explicitly all the Virasoro conditions instead of imposing
them as constraints.
Let us introduce light-cone coordinates for space-time 6
X ± = 1 √ 2 (X 0 ± X D−1 ). (2.123)
Then the D space-time coordinates X µ consist of the null coordinates X ±
and the D −2 transverse coordinates Xi . In this notation, the inner product
of two arbitrary vectors takes the form
v · w = vµw µ = −v + w − − v − w + +
v i w i . (2.124)
Indices are raised and lowered by the rules
v − = −v+, v + = −v−, and v i = vi. (2.125)
Since two coordinates are treated differently from the others, Lorentz invariance
is no longer manifest when light-cone coordinates are used.
What simplification can be achieved by using the residual gauge symmetry?
In terms of σ ± the residual symmetry corresponds to the reparametrizations
in Eq. (2.86) of each of the null world-sheet coordinates
These transformations correspond to
i
σ ± → ξ ± (σ ± ). (2.126)
τ = 1 + + − −
ξ (σ ) + ξ (σ ) , (2.127)
2
σ = 1 + + − −
ξ (σ ) − ξ (σ ) . (2.128)
2
This means that τ can be an arbitrary solution to the free massless wave
equation
∂2
τ = 0. (2.129)
∂2
−
∂σ2 ∂τ 2
6 It is convenient to include the √ 2 factor in the definition of space-time light-cone coordinates
while omitting it in the definition of world-sheet light-cone coordinates.