Lectures on String Theory

– 41 –
where on the r.h.s. p µ is not an operator and |0〉 denotes the zero-momentum ground
state. Plane-waves are not square-integrable functions but they form a basis of a
generalized Hilbert space and they are assumed to be normalized as
〈p|p ′ 〉 = δ(p − p ′ ) .
4.1.1 Virasoro algebra
Let us now investigate the algebra of the operators L m in the quantum case. We
first assume that the normal ordering constant a = 0. We start by computing
∞∑
∞∑
)
[α m, µ L n ] = 1 [α µ 2 m, : αpα ν n−p,ν :] =
(mδ 1 2
m+p α µ n−p + mα pδ µ m+n−p = mα µ n+m .
Then we have
p=−∞
[L m , L n ] = 1 2
p=−∞
∞∑
[: α pα µ m−p,µ :, L n ] .
p=−∞
We write down the normal ordering explicitly (to simplify the notation we write the
Lorentz summation index on the same level)
[L m , L n ] = 1 2
= 1 2
+ 1 2
0∑
[α pα µ m−p, µ L n ] + 1 2
p=−∞
0∑
p=−∞
∞∑
p=1
pα µ p+nα µ m−p
} {{ }
p=q−n
∞∑
[α m−pα µ p, µ L n ] =
p=1
+(m − p)α µ pα µ m−p+n
(m − p)α µ m−p+nα µ p + pα µ m−pα µ n+p
} {{ }
p=q−n
In the underbraced terms we make a change of summation index p = q − n and get
[L m , L n ] = 1 ( 0∑
(m − p)α µ
2
pα µ m+n−p +
+
p=−∞
∞∑
(m − p)α m+n−pα µ p µ +
p=1
n∑
(q − n)α q µ α µ m+n−q
q=−∞
+∞∑
q=n+1
(q − n)α µ m+n−qα µ q
Without loss of generality we assume that n > 0. then we have
[L m , L n ] = 1 ( 0∑
(m − n)α µ
2
pα µ m+n−p +
+
p=−∞
∞∑
p=n+1
(m − n)α µ m+n−pα µ p +
n∑
q=1
q=1
(q − n) α µ q α µ m+n−q
} {{ }
.
not ordered!
)
.
n∑
)
(m − q)α m+n−qα µ q
µ .