Lectures on String Theory

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4.2 Quantization in the physical gauge
Quantization of strings in the physical (light-cone) gauge is perhaps the most straightforward
way to obtain a restriction on the space-time dimension as well as to understand
the spectrum of physical excitations.
Using the Poisson brackets and the basic quantization rules we can easily get the
table of the basic commutator relations of the light-cone string theory.
[ , ] p + p − p j x j x − αm j αm
−
p + 0 0 0 0 i 0 0
p − 0 0 0 − i pi
p + − i p−
p + − 2πT mα j p + m − 2πT mα − p + m
p i 0 0 0 − iδ ij 0 0 0
x i 0 i pi
iδ ij 0 0 i δij δ m
p + 4πT
x − − i i p−
0 0 0 0 i α− p + m
p +
αn i 2πT
0 nα i p + n 0 − i δij δ n
0 nδ ij √
δ 4πT
4πT n+m p +
αn − 2πT
0 nα − p + n 0 − i αi n
p +
− i α− n
p +
i αi m
p +
nα i n+m
− √ 4πT
p + mα i n+m [α − n , α − m]
Tab. 2. Canonical structure of the light-cone modes. The variable
p − is essentially the Hamiltonian: p − = 2πT H . The commutators
p +
involving ᾱ variables are the same.
We would like to point out the following commutator
√
4πT
[αn, i αm] − =
p + nαi n+m .
One of the most important commutators of the light-cone theory is [αm, − αn − ]. It can
be computed precisely in the same way as [L m , L n ] of the previous section. We find
the same result as before except we have now only d − 2 transversal fields which
contribute to the central charge term with the factor d − 2 instead of d
[α − m, α − n ] =
√
4πT
p +
The normal ordering ambiguity
(m − n)α− m+n + 4πT
(p + ) 2 d − 2
12 m(m2 − 1)δ m+n . (4.14)
α − n → α − n −
√
4πT
p + aδ n,0
leads to the change as
√
4πT
[αm, − αn − ] =
p (m − + n)α− m+n + 4πT ( d − 2
(p + ) 2 12 m3 + 2am − d − 2 )
12 m δ m+n .