String Theory and M-Theory

36 The bosonic string
2.4 Canonical quantization
The world-sheet theory can now be quantized by replacing Poisson brackets
by commutators
This gives
Defining
[. . . ] P.B. → i [. . . ] . (2.53)
[α µ m, α ν n] = [α µ m, α ν n] = mη µν δm+n,0, [α µ m, α ν n] = 0. (2.54)
a µ m = 1
√ m α µ m and a µ†
m = 1
√ m α µ
−m for m > 0, (2.55)
the algebra satisfied by the modes is essentially the algebra of raising and
lowering operators for quantum-mechanical harmonic oscillators
[a µ m, a ν†
n ] = [a µ m, a ν†
n ] = η µν δm,n for m, n > 0. (2.56)
There is just one unusual feature: the commutators of time components
have a negative sign, that is,
a 0 m, a 0†
m = −1. (2.57)
This results in negative norm states, which will be discussed in a moment.
The spectrum is constructed by applying raising operators on the ground
state, which is denoted |0〉. By definition, the ground state is annihilated
by the lowering operators:
a µ m|0〉 = 0 for m > 0. (2.58)
One can also specify the momentum k µ carried by a state |φ〉,
|φ〉 = a µ1†
m1 aµ2† m2
which is the eigenvalue of the momentum operator p µ ,
· · · aµn† mn |0; k〉, (2.59)
p µ |φ〉 = k µ |φ〉. (2.60)
It should be emphasized that this is first quantization, and all of these states
(including the ground state) are one-particle states. Second quantization
requires string field theory, which is discussed briefly at the end of Chapter 3.
The states with an even number of time-component operators have positive
norm, while those that are constructed with an odd number of time-