192
8 BRST Quantization
Light-Cone Parametrization
The two scalar fields X 0 and X 1 are combined in a light-cone (if 0 = −1) or
complex (if 0 = 1) fashion
X
±
L =
1
√
2
X
0
L ±
i
√
0
X
1
L
.
(8.54)
The modes of X ± are found by following (7.50): 2
α
±
n =
1
√
2
α
0
n ±
i
√
0
α
1
n
,
n = 0,
(8.55a)
x
±
L =
1
√
2
x
0
L ±
i
√
0
x
1
L
,
p
±
L =
1
√
2
p
0
L ±
i
√ 0
p
1
L
,
(8.55b)
The non-zero commutation relations are
α
+
m , α
−
n
= 0 m δ m+n,0 ,
x
±
L , p
∓
L
= i 0 .
(8.56)
This implies that negative-frequency (creation) modes α
±
−n are canonically conjugate to positive-frequency (annihilation) modes α ∓
n . Note the similarity with the
first-order system (7.134).
For later purposes, it is useful to note the following relations:
2 p
+
L p
−
L =
p
0
L
2 + 0
p
1
L
2 = 0 p
2
,L ,
(8.57a)
x
+ p
−
+ x
− p
+
= x
0 p
0
+ 0 x
1 p
1 ,
(8.57b)
n
α
+
n α
−
m−n =
1
2
n
α
0
n α
0
m−n + 0 α
1
n α
1
m−n
.
(8.57c)
In view of the commutators (8.56), the appropriate definitions of the light-cone
number N ±
n and level operators N ± are
N
±
n =
0
n
α
±
−n α
∓
n ,
N
±
=
n>0
n N
±
n .
(8.58)
The insertion of 0 follows (7.63). Then, one finds the following relation:
N
+
+ N
−
= N
0
+ N
1 .
(8.59)
Using these definitions, the variables appearing in L 0 (8.33)
L 0 =
L
⊥
0 − m
2
,L
2
− 1
+
L
0
(8.60)
2 For 0 = 1, this convention matches the ones from [5] for X 0 = X and X 1 = φ. For = −1, this
convention matches [12].